My Library - Books



My mathematical library, and related matters thereof, as of 8 January 2019, of an continuous update, rather than yearly, as previously, primarily of books and articles, but also of letters, pamphlets, reviews, patents, theses, puzzles per se, jigsaw puzzles, exhibition catalogues, conferences, videos, notes, reports, newspaper articles, reviews, interviews, obituaries and genealogy matters.


This is a personal collection of references with notes and annotations for my own mathematical researches especially as regards tessellations and Escher-like aspects, to which it is inclined, and that may come in useful for other researchers. Dates in bracket are of date of obtaining the publication. Occasionally a book is referenced that is not in my possession but is desirous of a mention for a variety of reasons, although this is made clear in the text. On occasions a book is referenced that is way too advanced for me to be of any use, generally from a bibliography. A typical example would be from Tilings and Patterns by Branko Grünbaum and G. C. Shephard. This is simply to have ‘seen and noted’ so I can rest easy that there is nothing that may otherwise have been of interest.


This listing was begun in 2006, and continues to the present day. Note that the text can be considered a perpetual work in progress, due to its very nature, of additional books and articles, and more, coming to light. The length and depth of each entry depends broadly on the significance of the book/article, albeit I am not always consistent in this desire. Not all entries have comments, due to time constraints. Occasional typos and omissions have simply to be accepted as the inevitable outcome of a work of such length and depth. Some of the entries have additional biographical detail on the author, generally taken from Wikipedia. This is for the sake of general interest, when the author's background is not generally known. Further, although I strive to be consistent, this is not always possible; I do indeed have other matters to attend! 


A few clarifications to what may otherwise appear as obscure text:

1. Many references to the Cairo tiling can be seen, referring to my special interest in this tiling.

2. Some entries begin with ‘from a library sale’. Typically, with other would-be purchasers present, it is best to keep the book and examine in detail later in the sale, or later at home. Typically, the books are of a price that is inconsequential, often pennies, or no more than a pound. Such relatively small sums of money is thus inconsequential, and thereby offers up books of possible interest (in whatever subject) that perhaps I would not normally pay full or even half price for. The sum involved being inconsequential, and can thus be written of if it proves of no use.

3. Some entries begin with ‘chance’ or ‘speculative purchase’. This refers to a book seen in a temporal sense, at a car boot sale or other such like sale, only available on that particular one-off occasion. Save for an obvious interest, typically for a book of possible interest, there is simply not the time to stand examining every page of a lengthy book for the topic of interest, and of which given that the price is usually inconsequential, a pound or under, and so rather than losing the opportunity, it is considered prudent to obtain the book, the sum involved being inconsequential, and can thus be written of if it proves of no use.

4. Some references to jigsaw puzzles may seem otherwise obscure in a mathematical context, but what is a jigsaw puzzle if not a tessellation? Here however, these are with historical matters or mathematics of some kind underlying them.

5. Some entries have background detail as to the author, often taken from Wikipedia. This is typically where the author is perhaps less well known, and I desire to know a little of the background.

On occasions, entries are discussed simply, for the sake of brevity, with a series of short stock phrases. The key:

‘Interested bystander’. Within my core interest in tessellation, even in my specialised field, there are aspects that although I am interested in, I do not undertake anything in the way of active study, generally of a popular level, such as with geometric dissections and polyominoes. Rather, I simply ‘stand back’ and observe and admire as an ‘interested bystander’ would, leaving the study to those of much greater ability than my low-level efforts.


‘Of peripheral interest’. Some references here are included despite on the face of it having nothing to do with tessellation. Some of these include such as pavements, where these are or can be said to be, only token references, or indeed strictly none at all. However, at all times, there does indeed remain some underlying tessellation interest, even if not obvious.


‘Seen and noted’. In the course of my studies, many books and articles contain references to additional works, some of a more obvious interest than others. However, not all of these are strictly relevant, some being at too advanced a level for my limited mathematical understanding. However, only by examining these can this matter typically be determined for certainty. This being so, such books and articles are simply stated as ‘seen and noted’, with the implication that these are of an advanced nature, of no use to me. Rather than leaving the matter open-ended, I can now ‘rest easy’ in the knowledge that the reference is superfluous to my needs.

‘Of pavement interest’. Typically, but not necessarily, of lesser mathematical interest. Broadly, this can be described as an off-shoot of tessellation, in which pavements, and by extension, roads, are studied. Such matters are thus likely of a lesser degree of interest to the typical reader here. These range from nothing on all on tessellation (such as pavement materials, history, techniques etc.) to considerable overlaps. However, within a broad tessellation interest, they nonetheless remain of a broad relation, hence their inclusion here. Others may disagree. But is is my listing...

‘Of vision interest’. Typically, of lesser mathematical interest. Broadly, this can be described as a side interest of mine (not that I am active in the field), in which vision, and related matters, are studied. Such matters are thus likely of a lesser degree of interest to the typical reader here. As such tessellation, even when it occurs here, which is rarely, is insignificant. However, one aspect relating in a more obvious way is figure and ground. Within a broad tessellation interest, this field nonetheless remains of a broad relation, hence its inclusion here. Others may disagree. But is is my listing...




Abas, Syed Jan and Amer Shaker Salman (with forewords by Ahmed Mousafa (Arabic calligrapher) and Sir Michael Atiyah). Symmetries of Islamic Geometrical Patterns. World Scientific 1995. (12 December 2009)

Small format hardback, 396 pp. From p. 140 onward, of diagrams only. A scholarly, although still popular account of Islamic tilings. I like this book more than most others on the theme, which generally lack rigor. Snippets of interest include: Khatem Suleimani (eight-pointed star and cross, Soloman’s Seal) pp. 14-15.  This is also known as ‘Breath of the Compassionate’, in Chorbaci. However, this appears to be ‘unofficial’ titling; upon searching using Abass’s term, there is next to nothing, and what there is merely refers to Abbas. A further oddity is the description caption of ‘Soloman’s Seal;’ this generally refers to a pentagram of hexagram, rather than this eight-pointed star. Fig Leaf (Maple Leaf) pp.102. has a good bibliography, with a few references not given elsewhere. No Cairo tiling.

Wikipedia gives: The Seal of Solomon (or Ring of Solomon; Arabic: خاتم سليمان‎ Khātam Sulaymān)


Abbott, David (general editor). The Biographical Dictionary of Scientists. Mathematicians. Blond International 1985 (22 June 2003)

Popular account of a scientific series (includes Astronomers, Engineers and Inventors), with here mathematics. Has a glossary and index. Many entries on geometry. That said, most entries are outside my area of direct interest and understanding, but it all still makes for pleasant read.


Abbott, P. Geometry. (Teach Yourself Books) The English Universities Press Ltd. 1962 First printed 1948 (19 July 1992) and Hodder and Stoughton. 1981 (26 July 1992)

Typical generic geometry text book of the day, one of many that I have; simply, one would have sufficed. The reason for obtaining the book was to be able to look up any geometric construction as and if required, but I do not believe that I have used this in any way.


Abbott. Algebra (20 January 1987)


Abbott, P. and C. E. Kerridge. National Certificate Mathematics. Volumes 1 and 2 Technical College Series. The English Universities Press Ltd 1961 (19 July 1992, 26 July 1992, 21 June 1992)

Textbook, somewhat advanced. Of no practical use


Aczel, Amir D. Fermat’s Last Theorem. Unlocking the Secret of an Ancient Mathematical Problem. Penguin Books 1997 (16 July 2007)

Popular account of the historical quest; many digressions and good yarns.


Adams, D. M. Inorganic Solids: Introduction to Concepts in Solid-state Structural Chemistry.

Wiley-Blackwell, 1974. (First saw, or at least recorded, 24 September 1987, at college library)

A minor study, in which the crystal studies are shared with other books of a like nature.


Adler, Irving. Mathematics. The Story of Numbers, Symbols and Space. Golden Press New York, 1958. (3 September 1995)

Juvenile, 56 pp. 811 year-old range. No tessellation. Oddly, not structured at all, being without introduction and contents. Broadly, ‘pictorial mathematics’, of a wide range. Note that Adler was also active in the field of adult mathematics, and was a prolific author. From Wikipedia:

A book Adler wrote for adults in 1958, The New Mathematics, was important in the ‘New Math’ curriculum reform movement, and led to his frequent appearances at educational meetings throughout Norht America.


————. Groups in the New Mathematics. Dobson Books Ltd. First published in Great Britain 1968 (21 February 1998).

Of no interest! From a library sale, on the off-chance of possible interest. Said to be of an intermediate level, not technical, but not too simple. No tessellation.


Agostini. Franco. Visual Games. Guild Publishing by arrangement with Macdonald & Co 1988. (5 February 1994)

Minor Escher text pp. 8081, Waterfall, Sky and Water I pictures. Bizarrely, the Sky and Water I print is asymmetrically cropped!


————. Mathematical and Logical Games. Macdonald (sic) & Co. 1983 (27 July 1992).

Escher’s Ascending and Descending, p. 34, Mobius Strip II, p. 74, no text, just captions.


Ahrens, W. Mathematische Unterhaltungen und Spiele [Mathematical Recreations and Games], Leipzig 1901 (Downloaded from Internet archive 21 April 2015)

From a reference in Bradley (and MacMahon). General maths recreations, in the style of Rouse Ball. Like Rouse Ball, very little on tiling, barely worth mentioning. 424 pages.


Ainsley, Robert. Bluff your way in Maths. Ravette Limited 1988. (9 June 2002)

Small format paperback, 62 pp. A Bluffer’s Guide per se can be seen in many other subjects.

Various topics and biographies of  mathematics covered, briefly, with a dry wit. As might be imagined, a popular account, albeit diagram free. No tessellation/Escher. Of interest p. 39, on the Coriolis effect, on the fallacy(?) of water spiraling in different hemispheres. Ainsley asserts that it is due to the bath shape. To investigate. Of coffee-time reading only.


Albarn, Keith and Jenny Miall Smith. Diagram. The Instrument of Thought. Thames and Hudson, 1977 (26 July 2015)

Mystic nonsense throughout, in the ‘best’ tradition of Keith Critchlow. Unreadable save for skimming each page. A favourite ruse here is to quote well-known scientist/mathematicians to give the book a perceived credibility. Minor aspects of tessellation, within ‘perception’, pp. 40, 43 and Islamic design, pp. 6667.

Appleton, Le Roy H. Indian Art of the Americas. Charles Scribner Sons, New York, 1950 (Internet Book Archive, one-hour loan) (7 June 2021)

From a reference in Tilings and Patterns. A brief mention of the book and an illustration on p. 329 in the context of folk art, with two others. Numerous colour plates on patterns in the broad sense, in three sections, but nothing of any real significance per se as regards tilings.

Alexanderson, Christopher, Sara Ishikawa and Murray Silverstein. A Pattern Language. Towns, Building, Construction. Centre of Environmental Structure, Berkeley California. Volume 2 of a 3 volume set (PDF) Oxford University Press, 1977(12 September 2018)

From a reference of the architecture site Henn, in relation to a mooted parquet deformation reference. Upon research, I found the book as a PDF. A weighty tome, of 1,171 pages! Much to my annoyance, after viewing the whole book, I was not able to find the reference! If there is a parquet deformation reference, it can only be most minor. However, whilst searching, I found much of the book most interesting indeed in a general sense, and ideally I would very much like to read this (ideally as a book, rather than a pdf). However, the sheer length puts me off (with time involved), as good as the material appears to be! Further, I then looked more on the background of Alexanderson, of whom previously I was unaware of. I then became he was still alive, had a website (with in situ tilings), and with more books to his name, of which I notice his interest in tilings per se. His name then began to appear in tiling papers. Likely now, fully primed, I will find references to him in existing papers.


Amiraslan, I.  Azerbaijani Tessellations, 2006 WANTED

Anderson, Fiona. Tweed (Textiles that Changed the World). Bloomsbury Academic, First edition 2016 Google Books (March 2019)

Textile tessellation interest. Specifically concerning the Shepherd's Check and Sir Walter Scott’s supposed popular wearing as trousers, pp. 3134. In short, Anderson largely debunks this oft-made claim


Anderson, Paul and Deborah Curry. Imagined Worlds. Stories of Scientific Discovery. Ariel Books British Broadcasting Corporation, 1985. (28 August 1996, but seen much earlier, in April 1989)

Various essays on scientific discovery by eminent scientists, including Roger Penrose. of general interest overall, with a tiling aspect of Chapter 9 (by Deborah Curry), Beyond Space-Time, pp. 161180, on Penrose, with a small tiling interest; Penrose chickens p. 177, and Escher's print Waterfall, p. 179, along with a popular discussion of Penrose tiles.


Andrew, H. E. Laye, editor. Reader’s Digest Manual of Handicrafts. Reader's Digest Association Limited, 1980. First saw, or at least recorded, 2 December 1987, Scartho library.

Although not a book on mathematics, including on this listing as it has a tiling, of overlapping circles, of a minor nature, p. 97, of which I briefly studied (a single sheet). As such, inconsequential, in both mathematics and study thereof. 432 pages.


Angel, Henry. Plane and Solid Geometry. William Collins, Sons & Co., Limited 1885 (21 June 1992)

Typical geometry book of the day. Begins simply, from first principles, and then discusses more technical matters. The only tiling is on p. 26, a problem in copying a given tiling (square and octagon). I seem to have collected many instances of this ‘type’ in the early 1990s; any one really suffices for my needs.


Anon. Bricks and Concrete. By (author) the editors of Time-Life Books. 1984. First saw 21 December 1987, Scartho library

Although not a book on mathematics, including on this listing as it has brick tilings, p. 76, of which I briefly studied (a single sheet), with proto Escher-like additions,  on 21, 23 December 1987 and 5 January. As such, inconsequential.


Anon. Design and Designing.  Publisher not given

From a reference on a 17 January 1989 sheet. However, upon looking for the exact title, there is no book with this name that correlates to the date of study. Perhaps this was rather a chapter title? Whatever, the study was inconsequential, of just a tracing of a jigsaw puzzle in outline.


Anonymous. Magic Snake Shapes. Corgi 1981 (14 July 1991)

Small format paperback, of 96 pp, brought out at the height of this Rubik Cube spin-off craze. A brief, single page introduction, followed by pictorial instances of representational shapes that the Magic Snake can be formed. No instructions as such, save for two instances.


Anonymous. Technical Drawing. Letts (22 March 1987)

From a reference of early maths studies, of 1986-1987.


Anonymous. Letts O Level Geometry (23 March 1987)

From a reference of my early maths studies, of 19861987.


Anonymous. The New Mathematics

From a reference of my early maths studies, of 19861987 (17 December 1986).


Anonymous. Longman Maths 1 (13, 18 February 1986) and 2 (12 February 1987)

From a reference of my early maths studies, of 19861987.


Anonymous. Maths to 16 (1 January 1987) by Bennet?


Anonymous. Practical Plane and Solid Geometry


————. The Sociable or One Thousand and One More Amusements. 1858 New York, Dick & Fitzgerald (Downloaded from Internet 10 June 2014). 375 pages

From Stegmann’s site. Best describes a series of ‘parlour games’, such as acting and magic tricks, popular of the day. Mathematically light with two small chapter on mathematical games: Fireside Games for Winter Amusements pp. 274–284, Puzzles and Curious Paradoxes 286300. Answers to Puzzles and Paradoxes 301318. These contain loose geometric dissections, but nothing of particular note.


————. Tricks Played on Hand and Eye’ The UNESCO Courier, Vol., 19, no, 5 (1966), p. 14. 

(Note the year commonly given, 1964 (Locher, Schattschneider), is incorrect, it is 1966, as given by all authors where this is quoted; all copying from one another, likely from ? Locher is correct)

Somewhat of a disappointment, no text of note, with only two of Escher's pictures used, Belvedere and Waterfall.


————. Mathemagic. Childcraft Volume 13. World Book Childcraft International, Inc. 1979 (21 February 2004)

Juvenile. Occasional polyhedra, no tessellation.


————. Oddities. In Words, Pictures and Figures. Reader’s Digest Association Limited 1975. (July 1996 and 20 August 2003? The year is semi-legible). Two copies

Small-format ‘booklet’ 48 pages. Escher prints and minor essay pp. 2528: Belvedere, Waterfall, and Ascending and Descending

Also see a later companion booklet, of 1988.


————. Nuffield Mathematics Teaching Project. 1971. (22 August 2004)

A series of ‘work card’ packs: Area (contains the Cairo pentagon, without reference to Cairo), Similarity 1, Similarity 2, Number Patterns, Topology, Number Patterns.


————. The Universal Encyclopedia of Mathematics. With a foreword by James A. Newman. Pan Books Ltd, 1976. First published in Great Britain 1964 by George Allen & Unwin Ltd. Translated and adapted from the original German Meyers Rechenduden, published 1960 by Bibliographisches Institut in Mannheim (1 April 1993)

Substantial small format paperback, of 715 pp! Academic in tone, although said to be, in the publishers note (p.6), as ‘… intended for the man in the street…’! However, this is way beyond the stated audience. Formula biased. No tessellation. Nothing that can be said to be recreational. Kepler’s star polyhedra pp. 270271. Of only possible reference use. I have no plans to re-read.


————. The World of Shape & Number. Marshall Cavendish Learning System. First published 1970 (6 February 1994).

Advanced juvenile. Popular account on shape and number, with much of interest. No tessellation though!


————. Artfile Patterns. Phaidon Press Limited 1990 (14 May 2005).

Patterns only, no text. Occasional tessellations


————. The Alhambra and the Generalife. (11 July 2004)

Looks like tour guide book (I also have another, different book of the same title) no date given, perhaps page torn out…


————. Mathematics in Primary Schools. Schools Council Curriculum Bulletin No. 1. HMSO

Tilings p. 55, one diagram of octagons of interest.


————. Visual Elements 3. Marks and Patterns Clip Art. Columbus Books c 1989. (2 April 1994)

Strictly a pattern book, rather than mathematics. Book 3 of 10 in a series of a ‘visual elements’ premise. As such, of very little interest; tiling is of no substance, it being subsumed among general wall paper type patterns.


————. Visual Illusions. Reader’s Digest 1988. (20 August 2003? The year is semi-legible)

Small footprint booklet, 48 pages. Escher pp. 2031, Day and Night. Broadly a retelling of existing illusions. Also see a later companion booklet, of 1975.


Appleton, Le Roy H. Indian Art of the Americas. Charles Scribner Sons, New York, 1950. Internet Book Archive, one-hour loan (7 June 2021)

From a reference in Tilings and Patterns. A brief mention of the book and an illustration on p. 329 in the context of folk art, with two others. Numerous colour plates on patterns in the broad sense, in three sections, but nothing of any real significance per se as regards tilings.

Apsley, Brenda (Devised by). Coloring Patterns: Fun Patterns. World International Publishing Limited. 1993 (1 April 1993).

Juvenile. Also see an accompanying book, of the same nature. A child’s colouring book, almost of a five-year-old level! Looking at both books again, I am at a loss as to why I obtained these, and furthermore at full price! The diagrams are as intended for their audience, of no challenge. That said there is the occasional diagram (tessellation) of interest - see p. 29 here, and p. 20 of Picture Patterns below. I can only think that I thought I had not seen these tilings, and so may as well have these books at a relatively low price.


————. (Devised by). Coloring Patterns: Picture Patterns. World International Publishing Limited. 1993 (1 April 1993).

See above.


Armstrong, Tim. Make Moving Patterns. How to Make Optical Illusions of Your Own. Tarquin Publications. 1982 (16 February 1991 (used) and 18 February 2007 (intact)

Square format paperback, 56 pp. Gives instructions for composing geometrical illusions from acetate overlays with a series of grids, designed to be cut out and experimented with. Text is light. Not advanced in any way. No tessellation. To what extent these ideas are original with Armstrong is not made clear. Also see his other book, Colour Perception.

Briefly studied in 1991, although not to an extensive extent. Of mild interest previously, but perhaps less so now (2018), and for a long while.


————. Colour Perception. A Practical Approach to Colour Theory. Tarquin Publications 1991 (30 April 1994)

Not strictly mathematical, but has the occasional crossover.


Arnold, Arnold. Winners…and Other Losers in War and Peace. Paladin Grafton Books. 1989 (12 March 1999)

As such, this is on game theory, and is of a heavy read, overwhelming of text, 431 pp., with mathematical aspects in the appendix. Likely obtained, for a bargain price, on the off chance of later usefulness. However, it has not proved so! I doubt very much if I even began to read this, never mind the whole book. Even more so today, I simply do not have the time to read. I do not recall any references to Arnold or this book. However, there is indeed a true board game link; he designed the Parker Brothers logo.

From Wikipedia:

Arnold Ferdinand Arnold (February 6, 1921 January 20, 2012) was an author, game designer and cyberneticist, known more for the fame of his relatives and wives in later life. His first and only legal wife, Eve Arnold, was known for photography. His second partner, who he never married, was writer Gail E. Haley. Arnold's two brothers-in-law were Theodor Gaster and Peter Drucker.

… Arnold followed his eldest sister to the United States where he gained work as a writer and cartoonist. He was drafted into the U.S. military in 1941… Arnold was also a successful and well known advertising and commercial designer, and created the famous Parker Brothers swirl logo, first used in 1964. He created and designed many innovative educational and teaching games for leading game designers through the 1960s. He also designed classical record covers for EPIC Records during the 1950s. … During the 1980s and 90's, Arnold published several books, but never again had a financially successful career. He moved back to Petersfield in 1998, where his health rapidly declined. He died in 2012, from complications of sepsis and pneumonia.


Arnold, George and Frank Cahill. The Magician’s Own Book or The Whole Art of Conjuring. 1862. New York, Dick & Fitzgerald, 18 Anne Street, London (Downloaded from Internet 18 June 2014)

As recommended on Rob Stegmann’s site, although indeed on magic, has much recreational mathematics; especially see sections on geometric aspects: ‘Curious Tricks in Geometry’ pp. 256-266, ‘Curious and Amusing Puzzles’, 266290.


Ashcroft, Mike. Mathematics GCSE Passbook .1988. (15 October 1995)

Tessellations p. 130, barely worth the mention. Textbook.


Ashurst, F. Gareth. Founders of Modern Mathematics. Frederick Muller Ltd; First Edition 1982. 128 pages. Small format. First seen at Nunsthorpe library, although the author and date was not recorded, but must have been of 1987. The author was subsequently found by internet research.

Seemingly briefly studied, possibly of just a single sheet, likely, as the discussion would have been of a more advanced nature, with as I recall biographies of advanced mathematicians. However, some pages were more accessible than others, with space filling curves, of which I copied the diagrams and text verbatim. However, there is nothing original here on my part.

The book has long been deleted from the library stock, and of which although it available of an accessible price, £7.79, I am not in urgent need of it and I am not actively pursuing this, no matter how ideal it might otherwise be.

Audsley, W & G. Outlines of Ornament in the Leading Styles. London, Samson Low, Marston, Searle, and Rivington, 1881. Internet Book Archive (14 July 2021)

Subtitled ‘Selected from Executed Ancient and Modern Works. A Book of Reference for the Architect, Sculptor, Decorative Artist, and Practical Painter’.

From a reference in Tilings and Patterns. See p. 14 (introduction). The book is one of many of their generic listings on pattern and design. It does not as such appear to have been discussed in the main text.

As such, the book is typical of the day i.e. Owen Jones. Nothing in the way of the diagrams appears new, or has tilings that are of particular interest (Cairo, houndstooth etc). Only skim viewed. Ideally I would read and study, but I have no time.

Augarde, Tony. The Oxford Guide to Word Games. Oxford University Press 1984. (26 May 1996)

Not strictly mathematical, but related in a sense, with word play. 26 chapters.

d'Avennes, Prisse. L'art arabe d'après les monuments du Kaire depuis le VIIe siècle jusqu'à la fin du XVIIIe. J. Savoy & Cie editeurs (Paris) 1877, 296pp. Downloaded from Bnf Gallica (20 July 2020) Translated: Arab art from the monuments of Cairo from the 7th century to the end of the 18th century

From a reference in Tilings and Patterns and mentioned in the minor listings on tilings in every civilizations and cultures, p. 463, but apparently not in the main text of the book again. Somewhat of a letdown in terms of tiling matters. Mostly text rather than illustrations. And mostly on ornament rather than tilings. What tilings there are have essentially been seen in other like books of the day.



Bachet, de Mesiriac. Problemes plaisans et delectables qui se font par les nombres. 1612. A. Labosne Paris 1884 (downloaded from internet 5 May 2015)
From reference in MacMahon. As this title suggests, this is wholly on numbers; no tiling whatsoever.

Bager, Bertel. Nature as Designer. A Botanical Art Study. Reinhold, New York, 1966. Translated by Albert Read. Foreword by Harry Martinson. Originally published in Sweden under the title Naturen som formgivare, 1961. 175pp. (Viewed on Internet Archive, one-hour loan, 28 July 2021) From a reference in Tilings and Patterns and mentioned in the listing ‘Patterns in nature and science’, p. 14, but is not discussed further in the text. Unashamedly a pictorial book, and without mathematical tilings as such. Lavishly illustrated. Would be an interesting (and worthwhile) read, but I don’t have the time!

Bain, George. Celtic Art. The Methods of Construction. Dover Publications, New York, 1973. First published by W. MacLellan, Glasgow, 1951 (Viewed on Internet Book archive, one-hour loan, 3 August 2021)

From a reference in Tilings and Patterns and mentioned in the listing ‘patterns and design’, p. 14, and briefly in the main text pp. 214 and 234 but is not discussed further. Skim viewed thumbnails only. Nothing in the way of tiling as such, albeit with obvious crossovers. As expected, of limited interest. I do not intend to pursue this further.

Bain, Iain. Celtic Knotwork. Constable London 1991. (3 June 1993)

Baker, Lyndon et al. The Art Machine Pattern Book. Leapfrogs Ltd. 1990.


Ball, Johnny. Think of a Number. British Broadcasting Corporation 1979. (16 February 1995)

Soap bubbles p. 59.


————. Johnny Ball’s Think Box. Puffin Books, 1982 (17 January 1998)

For children. An assembly from the BBC TV children's series, of  ‘Think of a Number’, and spin-offs, ‘Think Again’, ‘Think Backwards’ and ‘Think This Way’ with the material assembled as a book. As such, there does not appear to be anything original here, with Martin Gardner credited as the main inspiration. Of most interest, relatively speaking, is pp. 70-71, on tiling with quadrilaterals.


————. Wonders Beyond Numbers. A Brief History of all Things Mathematical. Bloomsbury Sigma, 2017, Grimsby Library (7 October 2017)

Popular account. Has many interesting titbits, some new to me. However, the length of the book (480 pages) mitigates against a considered reading, and so some pages were merely skimmed. Some pages of special interest include the Golden Ratio, pp. 50-51, where he gives, for me a new explanation. Kepler pp. 306314 plate in Harmonice Mundi  p. 313, Tessellation, pp. 457458, albeit a lightweight treatment. Escher, pp. 428429, again lightweight. Pavement and Alhambra tilings on colour plate. Ideally requires a more leisurely read once more.


Ball, Phillip. Designing the Molecular World. Chemistry at the Frontier. Princeton University Press 1994 (19 February 1998).

Chapter 4, pp. 111141 has much on quasicrystals and Penrose tiling. Escher’s page and minor text 128129.


Ball, W. W. Rouse and Coxeter, H. S. M. Mathematical Recreations and Essays. (thirteenth edition). Dover Publications, Inc.1987. (30 April 1994)

Surprisingly light on tessellation, pp. 105107 only.


Banchoff, Thomas F. Beyond The Third Dimension. Geometry, Computer Graphics, and Higher Dimensions. (Distributed) W. H. Freeman and Company 1990. (30 April 1994)

A little hard to describe, the book consist of advanced concepts in geometry at a largely popular level, profusely illustrated. Loosely stated it is of dimensions higher or lower than three. No tessellation.


Barber, Frederick, et al. ‘Tiling the Plane’. Faculty Advancement in Mathematics Module, Lexington, Mass., 1989 LOOK FOR. (Reference in Comap)


Barnard, D. St P. Figure it Out. Pan Books Ltd 1973 (20 September 1992).



Barr, Stephen. Experiments in Topology. John Murray, London. 1965 (9 July 1994)

Largely popular account, although I do not actively pursue the topic. Of most note is Chapter 3, ‘The Shortest Moebuis Strip’ pp. 3239 and Chapter 7, ‘Map Colouring’ pp. 8897. The Moebius idea is so simple, and yet I had never thought of this! Some of the material is taken from others.


Barratt, Krome. Logic & Design in Art, Science and Mathematics. The Herbert Press 1989. First edition 1980. (24 April 2016)

First saw, and studied, in 1993, at Grimsby Art School library. Decided to actively obtain subsequently (2016) upon a desire to review the study I had previously done. Upon receiving, my memory of the book had dimmed. I’m not quite sure what to make of it. I’m not too sure of Barrett’s, a designer, maths knowledge. It appears to be a compilation from other sources, with next to nothing of originality. The book drifts, in that one topic is introduced, before yet another, and another…. In short, it is too ambitious in scope; there is nothing is in depth or substance. The bibliography is at least extensive. Only minor tiling matters, of no consequence pp. 47, 53, 6667, 7071, 196197. It has other minor aspects of interest. As such, I have no plans to ‘study’ this once more.


Barrow, John D. Pi in the Sky. Counting, Thinking and Being. Penguin Books 1992 (22 July 2001).

Small format paperback, 317 pp, of limited interest; semi-popular, not easily described, of mostly ‘mathematical philosophical musings’. Barrow is perhaps better known as an astronomer. Text heavy, with only occasional diagrams. No tessellation, Escher. Numbers from different countries p. 44, Four-colour conjecture pp. 227234. Overall, it’s ‘interesting’, but the time involved now (2018) to re-read this would be disproportionate as to any benefits gained, of which I doubt.


————. The Infinite Book. Vintage 2005 (24 January 2015)

Has brief tiling matters, with of significance the Cairo tiling p. 16, although without attribution, and Penrose tiles. Also has minor reference to Escher, pp.130131, with his print Sphere Spirals, referring to loxodromes.


Beard, R. S. (Colonel) Patterns in Space. Creative Publications Inc. 1973.

On geometry aspects, of nine chapters: Polygons, Tessellated Polygons, Polyhedra Patterns, Golden Section, Fibonacci numbers and related Drawings, Conics and Curves, Spirals, Triangle relationships, Primitive Triangles, Miscellaneous. A lot of formulae given, although the premise is of a diagram led book. Despite a chapter on ‘Tesselated Polygons, 2342, not really concerning tessellations per se, but more of ‘patches’, and in general geometric constructions. The work on tessellations is taken from, or was inspired by. Beard’s article in Scripta Mathematicae, of the same title, which is reprinted in the book. Much of the formulae are too complicated for me, but nonetheless the diagrams are largely accessible. However, the book largely flatters to deceive.

Beaumont, Roberts. Color In Woven Design. Whittaker & Co, First Edition 1890, Second edition 1912 (Internet (2019?)

Textile tessellation interest. Of houndstooth and weave interest.

Beer, Arthur and Peter Beer (editors). Vistas in Astronomy. Four Hundred Years Proceedings of Conferences held in honour of Johannes Kepler. Vol.18. Pergamon Press. 1975. (c. 2001)

A major collection of articles (of 1034 pages!) arising from the conference. Perhaps somewhat surprisingly tessellations, and to an extent polyhedra, are not really discussed. Instead, this is really more of his astronomical work. Chapter 11 is described as ‘Kepler as Mathematician and Physicist’. Of most interest here is Coxeter’s essay ‘Kepler and Mathematics’ pp. 661670. Also see Chapter 14, pp. 861-876 ‘Kepler’s Crystallographic Ideas and his Tract ‘The Six-Cornered Snowflake’ by I. I. Shafranovskii, which touches on circle packing, and is illustrated.


Begelman, Mitchell and Martin Rees. Gravity's Fatal Attraction: Black Holes in the Universe. Cambridge University Press (Google Books, 16 June 2015)

Use of Escher's Circle Limit, Angels and Devils pp. 8081.


Bekkering, Betsy and Geert Bekkering. Stukje Voor Stukje: Geschiedenis van de Legpuzzel in Nederland. 1988 (In Dutch) Translated: Piece by Piece: A History of the Jigsaw Puzzle in the Netherlands (20 March 2016) Translation Printed 27 January 2017

Obtained in regards of interests in cluster puzzles, albeit with Bekkering telling me in a mail of 2014 that there is nothing there in this field. Has detail on background of Simplex pp.  5758, and also on p. 30. Although there is nothing here on cluster puzzles per se, nonetheless it is of interest for background details of Dutch puzzle history.


Bekkering, Geert. Spaß und Geduld: zur Geschichte des Puzzlespiels in Deutschland. (In German). Translated: Fun and patience: the history of the puzzle in Germany. Husum. 2004 (20 March 2016) Translation Printed 27 January 2017

Obtained in regards of interests in cluster puzzles, with a Bekkering connection. Again, as above, there is nothing of direct interest. However, there is indeed peripheral interest, of what I surmised may occur, hence the speculative purchase. P. 56 has Heye Profi puzzle, which uses an adaptation of Escher's running man tessellation (without due credit), and of which upon further investigation can be seem to have been applied to numerous other puzzles by the company. Pp. 56-57 give a history of the company. Pp. 66, 68, 90, 92 are interesting in that they show a ‘wavy square’ tessellation cut, of 1914.


Bell, E. T. Mathematics Queen and Servant of Science. G. Bell & Sons Ltd. 1966 (24 October 1996 or 1998)


Bell, Marc. Marc Bell Presents the Magical World of M. C. Escher. Boca Raton Museum of Art January 20–April 11, 2010 (15 December 2014)

Nominally a catalogue of a Escher exhibition at the Boca Raton Museum of Art museum, although of the nature of a book. Has many unpublished drawings taken from microfiche. With essays by Salvatore Iaquinta (‘The Reluctant Pop Culture Phenom’ (sic), ‘Escher Memories: How Italy Shaped the Future’ and ‘Compass Card’), Federico Guidiceandrea (‘Filling The Void’) and Willem F. Veldhuysen (‘The Magical Work of M.C. Escher’). That by Iaquinta on the ‘Compass Card’ print is interesting, although whether his observations/conjectures is indeed correct needs confirmation.


Bell, R. C. The Boardgame Book. Marshall Cavendish Books First printing 1979, second printing 1983. (26 June 2016)

Oversize. A sumptuous presentation, and widely quoted as the bible of board games. Although board games are not a matter of undue concern, I do indeed I have a passing interest, and about half of these are surprisingly new. Nothing particularly of a mathematical nature, although of course there is no reason to be so!


————. Discovering Old Board Games. Shire Publications Ltd 1980 (18 February 2007)

Small format paperback.


Bellos, Alex. Alex’s Adventures in Numberland. Dispatches from the Wonderful World of Mathematics. Bloomsbury Publishing Ltd, 2010. Titled in the US as Here’s Looking at Euclid. (27 July 2014).

A personal wander around mathematical aspects of interest to the author, of an overwhelmingly popular level. Occasional references to Escher, pp. 244 and 392 hyperbolic geometry, with Circle Limit IV. Phi, pp. 299-301 (and colour plates), with Gary Meisner interview. Martin Gardner pp. 250-253, plus lots of general interest. Sam Loyd pp. 237240, Henry E. Dudeney pp. 240242. Typical Bellos, of a delightful read.


————. Can You Solve My Problems. A case book of ingenious, perplexing and totally satisfying puzzles. Guardian Books, 2016 (3 February 2018)

Described as, in the introduction, ‘… a curated collection of 125 brainteasers from the last two millennia, linked with stories about their origins and influence.’


Bellos, Alex and Edmund Harriss. Snowflake Seashell Star. Canongate Books Ltd, 2015 (September 2015)

Mathematical coloring book, not paginated. Of note is my contribution to this, of the fish tessellation on back cover and two works inside; ‘Nested Fish’ again, and ‘Interflocking Birds’ (Bellos' witty description). Of note is Harriss' parquet deformation (with likely Bellos title) ‘De-four-mation’, of four non-periodic tilings positioned in a corner, which morph left to right and top and bottom. Beat that if you can!


Belur, Ashwin and Blair Whitaker. A Practical Solution to Rubik’s Magic. Corgi Books 1986 With a foreword by Erno Rubik. (Two copies, 27 September 1992 and 5 February 1994)

Small format paperback, of just 32 pp. Gives instructions as to Rubik’s Magic, a latter day addition to the cube theme, perhaps best described of a folding rectangular 2 x 4 array and ring premise. From memory, I do not believe I have ever tried ‘seriously’ to solve this.

Seeing as I was unfamiliar with Belur and Whitaker I had a look on the web. Apparently there was controversy with Rubik, despite him writing the foreword. From The New York Times, 20 October 1986:

Dr. Erno Rubik, the Hungarian inventor who bedeviled millions when his beguiling Mondrian-colored cube became a phenomenal best seller in the early 1980's, is himself bedeviled by two University of Pennsylvania computer-science graduate students who are cashing in on the spiraling popularity of his newest brain twister, which has been on the market for only two weeks.

The first of 500,000 copies of a 32-page book by the graduate students explaining how to solve the new puzzle, Rubik's Magic, are to be shipped Monday to bookstores around the country, several months before the inventor can finish his own book containing the solution. It is scheduled for publication early next year.

The students, Ashwin Belur and Blair Whitaker, hope their $2.95 book, ''Rubik's Magic: The Solution,'' will be carried piggyback on the fast-growing craze for the palm-sized puzzle, a series of squares and rainbow-colored rings that is mathematically even more difficult to solve than its predecessor, Rubik's Cube. More than 185 million copies of the cube have been sold around the world….


Bergamini, David and the Editors of TIME-LIFE Books. Mathematics. Time-Life 1969, 1970. First published 1963 (16 July 1995 Hardback, 21 March 1998 Paperback).

This is really ‘The Story’ of mathematics, rather than of an expository nature as the title implies. Much of interest, although detailing this is not the most straight forward task. No tiling. (False) references are made to the golden ratio appearing in paintings, pp. 9497, of which Mario Livio in The Golden Ratio p. 164 rebuts. For instance, it’s just ludicrous the figure of St Jerome.

This is a paperback, also see hardback, in possession.


Beyer, Jinny. Designing Tessellations: The Secrets of Interlocking Patterns. Contemporary Books 1999 (11 December 2007)

Jinny Beyer, a patchwork designer, and not, by nature, a mathematician, or at least a natural one, gives her thoughts on designing tessellations, and much more than the title otherwise suggests. Her background pervades the book, of a patchwork premise. Strictly, I do not know what to make of this. There is the potential for a good book here, but this is not it! In short, I think she addresses too many aspects beyond her understanding (albeit well-intentioned, with the non-mathematician patchwork worker in mind), of which she attempts to cover ‘all’, from history to basics to Escher and more. There are many aspects here that I have issues with. To begin, even the title! There are no ‘secrets’ as such in the sense of information being withheld. Another is the text is lacking in exactness in various places, too numerous to list all. I content myself with her definition of a tile, illustration 1.3, p. 4. Chapter 10, a digression to the Escher aspect, is a veritable disaster. She simply does not understand the issues. Anyone who can be proud of ‘houses’, pp. 206 and 222 reveals her lack of understanding of them. Likely a house, being a popular patchwork motif, was thus obviously chosen, but this does not excuse poor practise in design. The other tessellations, some by others, generally lack merit. However, a cat (by Beyer) ‘Tessellating Sue’ is at least respectable. The ‘pure’ tilings are better in terms of worth. Chapter 11, on Metamorphosis, is not really as such; the transitions are far too abrupt, being nothing more than abutments. Aside from the content per se, the book lacks an index, and so thus finding specific aspects is trying. It really is most frustrating trying to separate the wheat from the chaff here!

Interestingly, the cover and title pages features the Pólya tile C4, but without my bird motif.

Has many instances of Escher’s periodic drawings: Birds E128; E120/121 Birds and Fish; E24 Birds and Fish E25 Reptiles, all p. 3; Reptile E25, p. 127; E73 Flying Fish, p. 134; E128 Birds, p. 203, E90 Fish, p. 205, Fish and Boat E72, p. 219; E120/121 Birds and Fish, p. 220; Fish E119, p. 221; Bat/Bird/Bee/Butterfly E81, p. 224, E85, p. 225.

Prints: Reptiles, p. 228, Metamorphosis I, pp. 236–237.

Sketch: wall mosaic in the Alhambra, p. 202.

Cairo tiling, but not attributed as such, p. 144.


Bezuszka, Stanley, Margaret Kenney and Linda Silvey. Tessellations: The Geometry of Patterns. Creative Publications 1977 (15 October 1994)

School age level, with ‘activities’. ‘Skew’ Cairo tiling, on triangular grid, p. 38. No Escher-like tessellation discussion at all.


Bibby, John. Mathematics Resource Guide. No.4 (Year Unstated)


Bigalke, von Hans Günther and Heinrich Wippermann. Reguläre Parkettierungen. Mit Anwendungen in Kristallographie, Industrie, Baugewerbe, Designund Kunst Gebundene Ausgabe – 1994. LOOK FOR, WANTED


Bigalke, von Hans. Heinrich Heesch: Kristallgeometrie, Parkettierungen, Vierfarbenforschung (Vita Mathematica) (Gebundene Ausgabe) WANTED 

Birkhäuser Verlag, 1988 Translated: Heinrich Heesch: Crystal Geometry, Tiling, Four-Color Research, 320 pages.


Billings, Robert W. and Robert Williams. The Infinity of Geometric Design Exemplified. One Hundred Designs and their Foundations Resulting From One Diagram. London 1849 On line (not downloadable), seen at Hathi Trust (24 April 2015)

From a reference in Bradley. Quoted on p. 6. Of limited interest, if at all. The book is ostensibly about tracery designs, something of which is strictly outside of tiling matters. Tracery (rather than tiling) seems to be Billings’ main interest, he has one other book, at least, on the subject.


————. The Power of Form Applied to Geometric Tracery. London 1851. (24 April 2015)

From a reference in Bradley. Of limited interest, if at all. See comments above.


Bilney, Bruce. Plato’s Jewels. The Five Regular Solids. OZZigami Pty Ltd 1997 (19 February 2010)

Gift of Bruce Bilney. Self-published booklet of 32 pages. Promoting his own ‘Spectrochrome’ Platonic models. Occasional digressions from polyhedra, with stereo and tessellations.


Bingham, Jane. Illusion Art. Heinemann Library 2008 (17 November 2018, Cleethorpes Library)

Implied as for teenagers on the back cover. A look at various aspects of illusion art, of 56 pp. A somewhat lightweight treatment. M. C. Escher features prominently, of 16-17 (primarily), 20-21, 35, 40-42  (in passing) and cover (Waterfall). However, the research is particularly poor here, with Escher described as from Belgium! And the ‘find out more’ page gives J. L. Locher’s name as Locker, and misspells Doris Schattschneider without the n. From this, likely they will be other errors throughout too. The book is perhaps atypical of others, in the illusions shown many I have not seen before, and with a name I have not seen before, notably with John Kay, p. 39, of ‘The Lawyer and the Client’, although I am familiar with the illusion. The section ‘puzzling patterns’, pp. 40-43  discusses tessellation, with an illustration of Patrick Snels' work. Overall, even for a teenager, far too lightweight.


Birtwistle, Claude. Mathematical Puzzles and Perplexities: How to Make the Most of Them. Allen & Unwin, 1971. First saw c. 23 July 1987. Not in possession

A book that was briefly studied in 1987, but is of no consequence. Unavailable, save for a mad price, £180!

Black, Mary E. the sett and weaving of Tartans. Lily Mills Co, 1959 PDF (February 2019)

Textile tessellation interest. A small book, of 47 pp, readable. On tartans primarily and despite the title, has much on houndstooth-related matters, notably with the Shepherd's Check, pp. 9-11. Very nice.

From Wikipedia: Mary Ellouise Black (1895–1988), an occupational therapist, teacher, master weaver and writer, created almost single-handedly a renaissance in crafts in Nova Scotia in the 1940s and 1950s. Her best-known book, ‘’The Key to Weaving", was published in 1945 and has since run to 3 editions and numerous printings. Its clarity is without parallel, and, more than half a century later, it remains a handweaver's prime source of information.

Blackie, Alex B. Wood Pavement; Its Origin and Progress, London, Sherwood, Gilbert and Piper, 1843. Available Online. (c. 8 May 2019)

Of pavement interest. On wood block paving. Everything one would wish to know! (Skim Read). Reference to hexagonal blocks, pp. 25–26, 28, 35–36, 39, ‘41’, 48–49, 54, 56, 72. David Stead mentions.


Block, J. Richard and Harold E. Yuker. Can You Believe Your Eyes? BCA 1991 (14 September 1996)

Not mathematical per se, but as it includes maths related aspects, such as ambigrams, I thus include here. Very pleasing indeed. 20 chapters, replete with interest. To list favourites is invidious.


Bonanni, A. P. P. Ricreatione dell’ochio e della mente, nell’Osservatione delle Chiocciole, Roma, 1681 WANTED

Circle packing reference, as given by D’Arcy Thompson


Bourgoin, J. Arabic Geometrical Pattern & Design. Dover Publications, Inc.1973. (9 April 1993)

No Cairo.


Boles, Martha and Newman, Rochelle. Universal Patterns. Book 1. Pythagorean Press 1992 (19 November 1994).

The first of a * book series. It’s somewhat difficult to describe the premise of this book, due to a fragmentary nature of topics covered; likely aimed at a secondary school level. Prominent throughout are ‘compass constructions’, of a basic level, useful as an immediate resource. Occasional reference is made to pattern in the real world. Note that this a book on patterns in the general sense; that is, it is not focussed on tiling.


Boles, Martha and Newman, Rochelle. The Surface Plane. Book 2. Pythagorean Press 1992 (3 June 1993)

Similar in spirit to Book 1, with compass constructions. Of the two, this is more directly related to my interest, with chapter 4 on tiling, pp. 130169, and other tiling instances scattered throughout the book.


Bolt, A. E. and J. E. Hiscocks. Machines, Mechanisms and Mathematics. Mathematics for the Majority. Chatto & Windus, 1971 (22 August 2004)

One book of the seven-part ‘Mathematics for the Majority’, series, of which I have two. The book seems to have been compiled by a ‘project team’, with one primary author stated. The books are stated as ‘Chatto & Windus for the Schools Council’, which thus gives the intended audience. The topic of this book is out of my mainstream interest, but it still has isolated aspects of interest. Also see Mathematical Patterns by T. M. Murray-Rust for another in this series. Note that patterns here is used in the broad sense, and is not of tiling.


Bolt, Brian. The Amazing Mathematical Amusement Arcade. Cambridge University Press 1987 (9 June 2002)

130 ‘popular’ mathematical puzzles, with answers. Stated as from ‘a resource book written for teachers’, but the title is not given.  Possibly, this is the reference below. Also states ‘many of the puzzles have a very long history, other are original…’. However, upon an admittedly cursory look, I cannot see any that I am not familiar with. No plans to re-read, being one of many of the same compilation nature.


————. Mathematical Activities. A resource book for teachers. Cambridge University Press 1987 (18 July 2009).

154 activities, of a recreational nature, pitched at a middle school level, with answers. Especially see Activities 39 Tessellations, p. 28 and Activity 40, Tessellations and art, p. 29. Has Escher’s Swans and Horseman periodic drawings. Unfortunately, Swans is overlaid with an incorrect grid. Also answers pp. 147148 beginners, any quadrilateral will tessellate rule. Also see Activity 76 The Pentominoes, pp. 5677. Other aspects are of interest.


————. A Mathematical Pandoras Box. Cambridge University Press, 1993 (27 June 2016) PDF

Seems to be, as with his other works, a compilation from other sources.


Bossert, Patrick. You Can Do the Cube. Puffin Books 1981 (27 September 1992)

Small format paperback, of 112 pp., brought out at the height of the Rubik Cube craze. Simple step-by-step instructions. However, although I have likely tried this at the time, this was to no avail!  The situation has not progressed since then, not that I have tried for a long while. As much as I would like, time forbids a new round of study. I see that Bossert was just twelve-years-old at the time! What became of him is not clear.

Bourgoin, Jules. Grammaire Élémentaire De L'Ornement: Pour Servir À L'Histoire, À La Théorie Et À La Pratique Des Arts Et À L'Enseignement. Delagrave, Paris, 1880, 207 pp. (Internet Archive, 24 August 2021)

Translated: Ornamental Elementary Grammar: For Use In History, Theory And Practice Of The Arts And Teaching. In French, seemingly no English translation.

From a reference in Tilings and Patterns, p. 657, and mentioned in the listing of failure in the context of a systematic mathematical study p. 12 (not discussed further in the text). However, ‘failure’ here is in relative terms, as according to Grunbaum and Shephard stricter structures; as is, there is much here worthy of study.

Quite exactly what Bourgoin is striving for here is unclear, as I am let down by my French, as well as the ‘inconvenience’ of reading a PDF. However, it is clear that this is a very nice book indeed. A (simple) combinatorial approach to tiling is adopted, similar in style to the later work of Percy MacMahon in New Mathematical Pastimes, but whether all the tiles will tile is unclear. Indeed, there is not a tiling diagram as such in the book!

Boyer, Carl B. A History of Mathematics. (Second edition, revised by Uta C. Merzbach) John Wiley & Sons Inc. 1991 (25 April 1998)


Boys, C. V. Soap Bubbles Their colours and forces which mold them. Dover Publications Inc. (19** reprint of 1959 edition) (18 October 1995)


Bradley, Amos Day. The Geometry of Repeating Design and Geometry of Design for High Schools. Bureau of Publications Teachers College, Columbia University, New York City. 1933, and 1972 reprint. (17 January 2011)
As oft quoted by Doris Schattschneider. What is Bradley’s own work or not is not of the greatest clarity; I suspect he is borrowing from the other references.Has much of interest. P. 123 Cairo-like diagram, dual. Has a good bibliography.


Bradley, Chris. Cairo. Berlitz pocket guide. Berlitz Publishing/Apa 2008. (24 April 2016)

Possible Cairo tile sighting at Azbakkiyyah Gardens, p. 28.


Brandreth, Gyles. The Big Book of Optical Illusions. Carousel Books 1980 (7 September 1997). Juvenile. Standard fare.

Not a ‘big book’ at all; standard paperback size!


————. The Big Book of Puzzles and Games. Treasure Press. (First Published in Great Britain as four separate titles by Carousel Books) 1989. (Day not stated, July 1999)


————. The Complete Puzzler. An ingenous compilation of tricky riddles and cunning conundrums. Panther Books,1984. First published  by Robert Hale Limited, 1992. (Possession date not recorded)

A part original, part from other sources small format paperback compilation, as is made clear in the foreword, including puzzles of Lewis Carroll, Sam Loyd and Henry Dudeney, although the attributions not made clear in the puzzles themselves. Brandreth’s ‘originals’ seem few and far between. Has 20 various puzzles within the full remit of the term. Dissection puzzles pp. 3943, Geometric Puzzles pp. 5456, Pentominoes pp.8792. In style, the book is similar to many other compilations. In short, a fun work, not of a scholarly nature, having not seen it referenced.


Brest, Hillary et al. The Stella Octangula Activity Book. Key Curriculum Press 1991. (30 April 1994)

Various activities and investigations of the Stella Octangula, including blackline masters (nets)

Also see companion book The Platonic Solids Activity Book, Ann E. Fetter et al.


Brett, Michael, and Werner Forman. The Moors: Islam in the West. HarperCollins 1984.

(Seen c. September 1987, but not in possession).

Note that this book (title only) was recorded on a menu card,  stated from the central library, in conjunction with other Islamic tiling books of the day, 1987. There is no recorded studies as such. I cannot now recall this in any way.


Briggs, William. Second Stage Mathematics. The Organised Science Series. University Correspondence College Press. c. 1900? (20 June 1993)

Typical generic maths text book of the day; way beyond me, on Euclid, Algebra and Trigonometry. One of many that I have; simply, one would have sufficed. The reason for obtaining the book was to be able to look up any maths/geometric construction as and if required, but I do not believe that I have used this in any way.


Bringhurst, Robert. The Elements of Typographic style Second edition. Hartley & Marks, publishers. 1992 (3 November 2018)

Some ‘page size mathematics’ Chapter 8, pp. 143178.


Briscoe, Susan. The Ultimate Sashiko Sourcebook. Patterns, Projects and Inspirations. David and Charles 2005. Cleethorpes library. First saw a few years ago, but never got around to borrowing (16 February 2019)

Simply stated, a book on ‘Sashiko’, a term of Japanese hand stitching. It is replete with tiling images. Especially see p. 67, ‘Yatsude asanoha’ (eight-lobed hemp leaf) from which a Cairo tiling can be derived. P. 87 has a tiling that can be shaded as a outstretched human figure! Furthermore, for the first time (I believe) I have realised that what appears to be traditional Japanese patterns of an isometric appearance are not so, but rather are drawn on a rectangular grid. From the author’s website:

Sashiko (pronounced shash-ko) literally meaning 'little stab' or 'little pierce' is a traditional Japanese hand stitching technique that can be used to strengthen, repair, add warmth to or simply decorate fabric refers to the small running stitch that is worked to build up distinctive decorative patterns, of which there are hundreds. The book begins by exploring the origins of the technique to strengthen clothes and to make them warmer. Getting Started describes everything you need to begin stitching, including selecting suitable fabrics and threads, marking out patterns on the fabric, as well as the stitching technique itself. Ten project chapters show how easy it is to use sashiko patterns to make beautiful items for the home. The main focus of the book is the step-by-step detail in the pattern library, showing you exactly how to mark and stitch each individual pattern with ease. Finally a gallery of work by contemporary Japanese textile artists from Yuza Sashiko Guild provides extra inspiration.

Although inself, from the title alone, one would not expect this to be of any real interest, the book is replete of tiling patterns, and so worthy of study. Further, as such, it is invaluable (i.e. readable) in the Japanese-English context. 

Biography: Susan Briscoe is a textile artist, quilter, teacher and author of numerous books, including Japanese Quilt Blocks to Mix and Match, The Ultimate Sashiko Sourcebook, 21 Sensational Patchwork Bags and 21 Terrific Patchwork Bags. She is also a specialist dealer in Japanese textiles, Kimono and Kimono fabric.

Brockett, Anna. Draw Patterns. Adam & Charles Black 1981. (15 May 2005)

Juvenile, 12+. No Cairo.


Brown, James. Shiny Touch Farm. Bibliographic detail is next to non existent. Web research gives 2011 and publisher Walker (13 or 20 April 2013)

Minor tessellation reference of a dog. An infant’s book, found by pure chance upon a visit to Cleethorpes library, where in the sale section this was placed prominently, my attention drawn to a symmetrical drawing of cows on the front cover. Various other animals (pigs, sheep, duck, horse) are arranged ‘close fitting’ in a symmetrical arrangement. Curiosity aroused, upon looking inside, a tessellation of a dog, seen before, on the internet, but by whom I can’t recall.

No credit was given in the book. Symmetry is evident throughout the whole book, of just 12 pages, but the dog is the only tessellation per se.


Brown, Richard (ed.). 30-Second Maths. The 50 most mind-expanding theories in mathematics, each explained in half a minute. Ivy Press, 2012 (18 March 2017)

Popular account. With contributions by Richard Brown, Richard Elwes, Robert Fathauer, John Haigh, David Perry and Jamie Pommersheim., No tessellation. Disconcertingly Brown himself (presumably) makes a schoolboy mistake on matters of astronomy, referring to the ‘dark side of the moon’ (meaning the far side), p. 83. Has isolated instances of interest.


Brown, Richard G. Transformational Geometry. Dale Seymour Publications 1973. (24 October 1998)

Escher’s periodic drawings on cover, swans, and p. 36, Beetles and Flatfish p. 45, Swans, and p. 83 Fish. As such, there no tiling per se whatsoever! Discuses algebraic operations, which goes over my head, or at least as I so desire to study.


Brissenden, T. H. F. Mathematics Teaching. Theory in Practice. Harper & Row, Publishers, Ltd 1980 (19 February 1998).

The thinking behind teaching. No tessellation.


Britton, Jill and Walter Britton, Teaching Tessellating Art. Activities & Transparency Masters

Dale Seymour Publications 1992 (9 February 2010)

Aimed at a school-age level, 12+ years. Much use is made of Escher's work, both tessellations and prints, E 25, 35, 44, 63, 67, 75, 96, 97, 104, 105, 117, and Reptiles, Metamorphosis I. Use is made of students’ work, the quality of which varies. Broadly, it discuses procedures for creating Escher-like tessellations, and also with early computer programs, now somewhat dated.


Bronowski, Jacob. The Ascent of Man. British Broadcasting Corporation 1976 (24 October 1993)

Tilings occasionally discussed, Alhambra, Chapter 5, The Music of the Spheres pp. 155188.

Buchsbaum, Ralph. Animals Without Backbones. University of Chicago Press. Eleventh Impression 1947. First published 1938. (May 2019). Available on the Internet Archive:

Of peripheral Escher interest. Said (and confirmed by Sherry Buchsbaum, the daughter of the author in a reply to a blog posting, below), to be the book that Escher used for his Flatworm drawing references. Although obviously non Escher per se, it is included here in relation to him. From Sherry Buchsbaum:

Escher was definitely influenced by Elizabeth Buchsbaum's drawing of planaria. This can be seen in the chapter heading drawing for Chapter 10 and 12 and following drawings in Animals Without Backbones... Chapters 10 p. P.109, Chapter 12 p. 124. From Amazon: Animals Without Backbones has been considered a classic among biology textbooks since it was first published to great acclaim in 1938...

Brückner, Max. Vielecke und Vielfläche: Theorie und Gesschichte. (Translated: Polygons and Polyhedra) Leipzig: B. G. Teubner, 1900 (Downloaded from Internet archive 10 April 2015)

Frequently quoted in tiling concerns, such as by Schattschneider. On polyhedra. Highly technical, with much abstruse text, albeit liberally illustrated with line drawings, and latterly plates and polyhedral models. Of interest as regards tiling p. 109 dual tiling (Cairo) p. 158.


Buckwell, Geoff. Mastering Mathematics. Macmillan master series. Macmillan 1991. (11 September 2000)

Textbook, for beginners, of a broad range, with the equivalent of 2 + 2 to calculus! Minor tessellation pp. 94-95, with one diagram is of interest, in that this stumped me in my early days (in a different book), of a octagon and two squares, as a unit to be tiled. Or was it a octagon and one square?


———— . Work out GCSE Maths. Macmillan (September 1987?)

Note that I have various doubts as to this book, recorded on a shared sheet filed in Cundy and Rollett. Seemingly, part of a series, although the chronology does not correlate…


Bunch, Bryan. Reality's Mirror: Exploring the Mathematics of Symmetry. New York: Wiley, 1989. (13 September 2014)

From an Escher reference in Schattschneider’s Visions…. Somewhat disappointing in this regard, with a most lightweight treatment indeed of Escher, with two small discussions, as ‘Eschervescence’ Part 1 pp. 8185 (Fish and Frog Optimist/pessimist Birds and Fish) and Part 2 pp. 118121 (Pegasus, Birds) but without any new insights. There is one enigmatic matter concerning a tessellation of Escher's (Pegasus) in which Bunch states, p. 120 ‘… once flew along the cover of a book on crystals…’, but this is not sourced. I am unfamiliar with this. Upon looking for Bunch’s details online to ask him, of an initial look, as of  2017 there is nothing on him. he appears to be more of a science writer than a mathematician per se. The book itself is very much in the spirit of Gardner’s The Ambidextrous Universe, of which in the preface Bunch defers to.

Burden, I., J. Morrison, and John Twyford. Design & Designing? Longman 1989? (17 January 1989)

As such, there are various uncertainties here as to the book, borrowed from Grant Thorald library, due to it being poorly referenced of the day. Upon a internet search, I have only found one other book with this title, but as this is a subsequent publication to the date recorded here, 17 January 1989,  it cannot be this one. Further, I am not even sure of the title – a page number precedes the title, and so possibly this is a chapter reference instead. No publisher was given.

Whatever, the book can hardly be of any importance; the study, on a 10 January 1989 sheet headed by A. Racinet ornament  studies, consists solely of a well-known jigsaw tiling seemingly traced in which I remark upon the opposite side square feature.

Burn, Bob. Sorting by Symmetry. Patterns with a Centre. Association of Teachers of Mathematics 2005. (13 June 2009)

As sent by Bob Burn.

Borrego, John. Space Grid Structures. Skeletal Frameworks and Stressed-Skin Systems

Cambridge, M.I.T. Press, 1968, 200pp. (One-hour loan on Internet Archive, 12 August 2021)

From a reference in Tilings and Patterns and briefly discussed in passing on p. 111 (as regards a blatant error on the numeration of Archimedean tilings), but the book is then not discussed further in the text.

In the discussion Grünbaum & Shephard take Borrego to task for an error regarding Borrego’s assertion of an ‘extra’ Archimedean tiling and ‘documented’ by a drawing.

As such, as regards tiling, there is very little here, and what there is is of just basics, almost of first principles, not to mention the error outlined above, which does not exactly fill one with confidence in the content in the round. Pp. 132, 134–135, 194–197 have basic tilings. The book is typical ‘structural topology’.

A skim view proved more than sufficient; I am not planning on pursuing the book further.


————. The Design of Tessellations. Cambridge University Press 1987 (14 April 1993)

Non-attributed Cairo tiling, sheet 30, shown as line drawing, equilateral, no text. Drawing tessellations on a microcomputer, the BBC (B).


Burn, D. V and E. W. Tamblin. Arithmetic Itself. A Junior Teach Yourself Book. English Universities Press, 1965 (First saw, or at least date recorded, of 16 September 1987, College library)

A brief, single-page study, of which my recollections have faded to essentially nothing, with the book of a junior audience. The book is not in my possession, nor was the page photocopied of the day. Some minor tiling, albeit still of interest.


Burns, Marilyn. The I Hate Mathematics! Book. Cambridge University Press 1987. (not stated, guess 2000)


Burrett, Anthony. Mathematics in Time and Space. Peter Haddock Ltd. 1973. Project Club Booklet (25 January 1997)

Small (square) format paperback, 66 pp. From the 200-book Project Book series, introducing all kinds of pastimes from brass rubbing to building a home museum, with here mathematics of book No. 110. A handful of titles blossomed into a range of two hundred as the lavishly illustrated booklets caught on. An emphasis is on the practical aspect. The series is aimed at school children, c. 12 years of age. Mostly about time per se. Polyhedra pp. 4647, Minor tilings pp. 4849. However the treatment is so basic as to be  of no consequence.

Also see book No. 101, Mathematics for the 1970s, not seen.




Cadwell, J. H. Topics in Recreational Mathematics. Cambridge University Press 1966 (13 October 2006) First saw in Grimsby central library September 1987.

Occasional aspects of interest, largely of a popular level; Chapter 1 Regular Polyhedra, Chapter 9 Dissection Problems in Two and Three Dimensions, but mostly too advanced. Tessellation only in passing. Studied in September 1987, very much of the day, and somewhat  excessively, given the content.


Cain, John et al. Mathematics Miscellany. A source book for teachers. British Broadcasting Corporation 1966. (19 February 1998)

Flatters to deceive as to recreational maths aspects. Typical 1960s book. Of most interest Chapter 7 Geometry, Chapter 8, Three Dimensions, with tessellations. Escher is mentioned briefly, p. 64.


Callender, Jane. 2000 Pattern Combinations. A step-by-step guide to creating pattern. Batsford 2011 (7 April 2012) Grimsby library

Mistakenly states that there are ‘20 demi-regular tilings’; page 9; a howler, as noted as by Helmer Aslaksen in his Bridges paper.


Calvert, Albert F. Moorish Remains in Spain. Being a Brief Record of the Arabian Conquest of the Peninsula with a Particular Account of the Mohammedan Architecture and Decoration in Cordoba, Seville, &Toledo. London: John Lane, Bodley Head. 1904 (Downloaded from Internet 5 May 2015)
Of note is the length of this book, 586 pages! On mostly, Cordoba, Seville and Toledo, which concerns architecture, and so of limited appeal, although a most interesting chapter on Moorish ornaments is on page 479 onwards, with many ‘simple’ tessellations I was unaware of. One to study.


————. The Alhambra. George Philip & Son London 1904 (Downloaded from Internet 6 May 2015)

From a reference in Grünbaum. As such, this is very much like any other book on the Alhambra of the day; you seen one, and you’ve seen then all. Again, another weighty tome, of 464 pages. That said, was as unfamiliar with the tiling on p. 303 (521). P. 341 (558) could arguably be interpreted as a forerunner of Escher’s Other World print, likely he would have seen this viewpoint on his Alhambra visits, of which in later years may have rekindled in his print.

Campbell, Cyndie. M. C. Escher. Letters to Canada, 19581972. National Gallery of Canada Library and Archives Occasional paper No. 9. 2013 (10 December 2013)

A collection of letters from M.C. Escher to his son, George. Full of interest, with many new names not previously known. Padded out a little with commonly seen photographs and prints of Escher, though that said, there are the occasional photograph not having been seen. Introduction by George Escher.

Cantor, Moritz. Mathematischen Beiträge zum Kulturleben der Völker. Halle 1863 (Downloaded from Internet 27 April 2015)
From a reference in Bradley. Book quoted on p. 12. Somewhat of a let down; the book does not have a single diagram!

————. Vorlesungen über Geschichte der Mathematik. Second edition Leipsig 1894 (Downloaded from Internet 27 April 2015)
From a reference in Bradley. Somewhat of a let down; the book does not have a single diagram!

Carraher, Ronald G. and Jacqueline B. Thurston. Optical Illusions and the Visual Arts. Van Nostrand Reinhold Company New York (30 January 2015). First saw September 1987, Louth library

Although not strictly a mathematical book, this is included here as it was a book I studied right at the beginning in of my interest in tessellations, in 1987. This was first seen in Louth library in September 1987, and briefly ‘studied’ there, taking tracings of the pages of most interest.

As part of a concerted effort of eventually returning to old material that requires original material for updating, I decided to obtain such books from the period. Also, I note that Locher includes a reference to this book in regards of Escher, and so there was also the prospect of an Escher piece as well, although upon receiving the book this is a decided let down, of a single picture, Relativity, p.95, with minor commentary.

Inevitably, my memories of the book had dimmed. As such, it is not of a great deal of importance. Interesting, yes, and indeed with the occasional new aspect (such as a Dali sketch), but not in any way fundamental to tessellation studies.


Chamber, W. R; Murray, John. Shape and Size. Book 2. Nuffield Mathematics Project. Newgate Press Ltd 1968 (9 June 1996)

An arbitrary part series of a uncertain series, possibly of a series of four books. Juvenile, with instances of their work from the book. Occasional tessellation 3645. In relative terms, of more interest is Book 3, Shape and Size, confusingly of the same title. Of limited use in terms of innovation/usefulness, which is to be expected give its intended audience.


Chamber, W. R; Murray, John. Shape and Size. Book 3. Nuffield Mathematics Project. Newgate Press Ltd 1968). (2 June 1995)

Juvenile. Tessellations front and back covers. Chapter 5 Tile patterns - Tessellations 27-28; 32-41, Chapter 7 More about polygons and tessellations 3242. Includes studies of irregular pentagons! Of limited use in terms of innovation/usefulness, which is to be expected give its intended audience.


Chamber, W. R; Murray. Environmental Geometry. Nuffield Mathematics Project. Newgate Press Ltd 1969. (Teachers’ Guide).

Juvenile. This seems related in someway to the Shape and Size books above, although there are indeed differences. loosely on a premise of architecture. Whatever, of limited use in terms of innovation/usefulness, which is to be expected give its intended audience.


Chauvan, Sumi Krishna. Delhi, Agra & Jaipur. The Golden Triangle. First published in 1982 by Roloi Books International. 1988 (19 July 2014)

Although not a maths book, included on account of it containing some geometries of India, notably a possible Cairo tile sighting (now known not to be so) at Fatehpur Sikri at the Panch Mahal or Wind Tower, p. 65.


Christie, Archibald H. Pattern Designing. Oxford at the Clarendon Press. 1909? (6 August 1994)

The full title inside reads ‘Traditional Methods of Pattern Designing An introduction to the study of decorative art by Archibald H. Christie with numerous examples drawn by the author and other illustrations’. The majority of the book is of ornament and patterns per se, rather than of tessellations. A whole chapter refers to counterchanges, Chapter 13, pp. 282298. ‘Pólya’s ‘Do3’ tiling is shown, p. 296, Christies’ predating this, and Meyer of 1888 thereof. Page 133 gives the derivation of ‘Cosmati’, from Laurentius Cosma, of the thirteenth century.

Checked for any references to Cairo pentagon and par hexagon; none.


Clegg, Brian. A Brief History of Infinity. The Quest to Think the Unthinkable. Robinson, 2003 (6 May 2015)

Has an Escher print on the front cover, Knots.


Cook, L. H. Longley-. New Math Puzzle Book. Van Nostrand Reinhold 1970 (14 January 2017)

A ‘favoured chance’ finding whilst web searching. A relatively lengthy, although a little lightweight chapter on tessellation, Chapter 7, 109131. This includes minor Escher-like aspects, pp. 112, 117, 120. Incidentally, the related diagram on p. 127, titled as a ‘gingerbread man’ showed up upon a search, of which, although not stated, this is likely taken from MacMahon, p 108, of the utmost significance to me, as it underpins one of my own favourite human figures. On this diagram alone, I decided to pursue the book, with the chapter on tessellation a pleasing bonus. Cook seems to be a keen promoter of recreational mathematics, although no bibliography or index is given. Escher (incorrectly spelt) is mentioned in passing.


Coen, Enrico. The Art of Genes. How organisms make themselves. Oxford University Press, 2000. C. 2005-2008? - Date has faded; I have had this for many years; it’s certainly not in the last couple or so, say.

As such, this is not a maths book, but as it includes ‘occasional Escher’ I include for the sake of ‘everything Escher’. Escher aspects, 1-2, 137, 312313. Drawing Hands 2, Circle Limit I 137, Balcony 313.


Coffin, Stewart T. The Puzzling World of Polyhedral Dissections. Oxford University Press 1991. (3 June 1993)

Delightful throughout. Also, two-dimensional puzzles and dissections are briefly discussed, Chapters 1 and 2.


Cohen, Jack; Stewart, Ian. The Collapse of Chaos. Penguin Books 2000 (12 May 2002). 

Somewhat advanced.


Colby, Averil. Patchwork. B. T. Batsford, London. 1987 First published 1958 (18 November 2001)

Although obviously not a book on mathematics per se, it is nonetheless included in this listing as it has interesting pattern aspects, and indeed, it is one of the better books there is on the interrelation between the two, and indeed led to extensive studies of the day (1987, as indeed with other patchwork books). However, now, and for some considerable time, the nature of the material is considered hardly worthy the time originally devoted to it.


Cole, Alison. Perspective. Dorling Kindersley 1993.

Includes Escher’s Impossible World, page **.


Cole, Drusilla (General Ed). 1000 Patterns. London: A&C Black 2003.


Conway, J. H. On Numbers and Games. Academic Press Inc. (London) Ltd. 1976 (14 September 1996)

Of limited interest, mostly advanced maths.

Conway, J. H. and H. S. M. Coxeter. ‘Triangulated Polygons and Frieze Patterns’. The Mathematical Gazette Vol. 57, No. 400, June 1973, pp. 87-94. (8 July 2019)

Academic. Despite an ostensibly popular account by the title, too advanced for me. No diagrams. The article continues into the next issue.


————.. ‘Triangulated Polygons and Frieze Patterns (Continued)’

The Mathematical Gazette Vol. 57, No. 401, October 1973 pp. 175-183.  (8 July 2019). NOT SEEN

Academic. A follow-up to the above.


Conway, J. H. et al. The Symmetries of Things. A. K. Peters Ltd 2008 (19 March 2010).

Decidedly advanced for me! Escher plane tilings 67 Horseman, 22 Bird and Fish, 70 Butterflies, Circle Limit IV, pp. 134135, 152153, 224

Scholarly discussion of Angels and Devils 224. Cairo tiling apparently projected on a sphere, front cover and repeated page 74. First saw this book, briefly, at Bridges Leeuwarden, 2008, with a false first impression at the time that it would be suitable/useful for me.

Cook, Jill. Ice Age art: arrival of the modern mind. British Museum Press 2013, pp. 108-109, 134 (17 August 2019, but known a few years previously)

Of historical tiling interest. Upon seeing a post by Robert Fathauer on Twitter of 10 August 2019 on a 2,000-year-old (regular) hexagonal paving at the Great Temple, Petra, I resolved to revisit an old hexagonal tiling sighting myself (a few years back), details (title, author, date) long forgotten, in a book at Cleethorpes library, as above. I now see that this is on the Eliseevichi, Russia, tusk artefact, with a c. 12,000-15,000 BC dating, now in the Peter the Great Museum of Anthropology and Ethnography (the Kunstkamera), Russia.

This concentrates on the historical significance of the tiling without any mathematical aspect. Upon seeing this book again, and following up the sighting up, of the museum, and articles by (medics) Geoffrey Schott and Clare Caldwell, I posted to the Google tiling group on the potential of an exciting discovery, this has not been discussed in the mathematical literature as a tiling, as far as I was aware. However, this met with only lukewarm response, with only two replies, of an additional earlier artefact, the Blombos Cave, South Africa.

Corbalán, Fernando. The Golden Ratio. The Beautiful Language of Mathematics. Published by RBA Coleccionables, S. A, 2012. An English translation of a Spanish work (7 June 2014). In association with National Geographic.

One of a series of popular maths books, originating from Spain. The background is that a group of Spanish Mathematicians have written a comprehensive set of popular maths books, which have proved so successful that they're being translated into other languages, including English. The team behind the series have joined up with The Times newspaper and Marcus du Sautoy to present the series to the British audience.

As such, a nice treatment on the Golden Ratio, although I do indeed have, on occasion, serious concerns. Unsurprisingly, given a pentagon underlay, there is much of interest here.

Has occasional tiling aspects. Section on periodic and aperiodic tiles, pp. 76–87. Escher aspects: Spiral, p. 65 and two bird motifs p. 81.

On occasions shows bizarre golden ratio overlays, such as pp. 12–13, 107.

Beforehand, Corbalán was a new name to me. Amazon: Corbalán is a mathematician expert in this subject having written several books on the golden ratio or related themes.


Cordova, Chris De. The Tessellations File. Tarquin Publications. 1983 (3 June 1993)

Juvenile, for classroom work. Very basic indeed, pp. 16 are given largely to explanations, the rest of the book is of tilings on single pages, without any apparent structure. One instance of Escher-like tessellation, page 6, a human figure drawn without understanding of the issues, and which is particularly poor.


Costello, Matthew J. The Greatest Games of all Time. John Wiley & Sons Inc. 1991. (27 August 1997)


Cotterill, Rodney. The Cambridge Guide to the Material World. Cambridge University Press 1989. (Date has irretrievably faded, c. 1995).

Although not a maths book per se, included as it has Escher aspects. Page 63 E97 Bulldogs, E85, Lizard Fish Bat; 81 Print Gallery.


Courant, Richard and Herbert Robbins. What Is Mathematics An Elementary Approach To Ideas And Methods. Oxford University Press, U.S.A.  Second Edition 1996, revised by Ian Stewart (26 May 2017) Internet download


Cowen, Painton. Rose Windows. Thames & Hudson 1990 (21 May 1994, Sheffield)

144 pp. A4 paperback. Popular account, with diagrams and photos. Although I have no particular interest in rose widows per se, there is a decided geometrical element to this, and so thus likely chose to purchase, in Sheffield, on the basis of ‘buy or lose’ of the day, this preceding ready availability on the internet. However, I do not recall any direct study from this. Time forbids a dedicated re-reading. Pages of interest include p. 93 on Honnecourt with tilings. Geometrically of interest is ‘Divine Geometry’, pp. 121127. Incidentally, suitably recalling Honnecourt, I had a look on the web for any more tilings of his, but it can quickly be seen that tiling was only a minor interest.

Also see a more substantial publication, The Rose Window Spendour & Symbol by the same author.


————. The Rose Window. Splendour & Symbol. Thames & Hudson 2005, Oversize. (26 May 2014)

Although a most pleasingly produced book, this is somewhat of a disappointment mathematically. A single chapter is devoted to the geometry, but this is most brief indeed, of pp. 241263, and with most simple constructions given, such as bisecting an angle! Many references to local cathedral, at Lincoln.


Coxeter, H. S. M; M. Emmer, R. Penrose, and M. L. Teuber, Eds. M.C. Escher: Art and Science. Amsterdam: North-Holland 1986. (30 April 1994)

A collection of essays; indispensable.


Coxeter, H. S. M. Regular Polytopes. Dover Publications Inc., New York, Third edition 1973. The first edition is 1947, the second edition is 1963 (30 April 1994)

An earlier edition, of 1963? has the Cairo tiling featured on the front cover. As a broad statement, the book is too far advanced for me. Chapter 4, p. 5873 is on tessellations and honeycombs, albeit there is nothing here that I can use in any meaningful way. Other chapter on related interests, Chapter 1 Polygons and Polyhedra, p. 113 and Chapter 2 Regular and Quasi-Regular solids, p. 1530 and Chapter 6, Star-Polyhedra p. 93114 are all of a similar nature. One aspect of interest that I can follow is that each chapter ends with ‘historical notes’. Finally, the book has an excellent bibliography, full of obscure books.


————. Introduction to Geometry. John Wiley & Sons, Inc. Third Printing, 1963 (24 August 1996)

Academic. Escher pp. 57 (Horseman E67) - 59 (Beetles E91), 63. Very brief text. First studied, or at least recorded 25 January, and 5 February 1988 upon ordering from the library. Unsurprisingly, very little is accessible to me.


Coxeter, H. S. M. and S. L. Greitzer. Geometry Revisited. Of entire book! 1967, Fifth printing (30 January 2012)


Cracknell, A. G. and G. F. Perrott. Intermediate Geometry. University Tutorial Press Ld [sic]. Third impression 1940, when it was first published is oddly not stated (23 September 2001)

Typical generic geometry text book of the day, one of many that I have; simply, one would have sufficed. I do not believe that I have used this in any way; the material being mostly beyond my understanding. Very much of its day. Chapter 10 is on polyhedra. Some nice renditions of polyhedra pp. 147150


Cracknell, Arthur P. Crystals and their Structures. Oxford: Pergamon Press, 1969 (First saw, or at least recorded, 24 September 1987, at college library. Not in possession)

A minor study, in which the crystal studies are shared with other books of a like nature


Craig, Diana. The Life and Works of Acimboldo. Parragon Book Service Limited, 1996 (15 Sepember 2018)

Although nothing whatsoever on maths, included here as a tenuous interest as regards cluster puzzles, specifically of pp. 4243. I might just add that Arcimboldo is an artist I much admire, with his work of an imaginative nature, although I have not made any special effort to study his work. This book here, of a popular nature, of a 54-book series (including Escher), found by a casual browse in a charity shop, at least serves as an introduction pending a more detailed work being found.


Crane, Walter. The Bases of Design. London George Bell & Sons 1902. (Read online at Project Gutenberg, 9 June 2015)

Minor tessellation pp. 89, 128. Arabic designs pp. 213217, otherwise mostly of ornament. Nothing of any significance.


Crilly, Tony. 50 mathematical ideas you really need to know. Quercus, 2007 (13 May 2012)

Popular account from across the spectrum of mathematics, 1. Zero, 2. Number Systems, 3. Fractions etc. However, there is no tiling.


Critchlow, Keith. Order in Space. A Design Source Book. Thames & Hudson. A date of 1969 is given but it is unclear if this was when first published. The published date is apparently given as 1987, reprinted in 2000 (22 September 2007)

Barely readable, in that Critchlow has a belief in mystic, Eastern, philosophical leanings that permeates the book. Buckminster Fuller has heavily influenced him. Has Cairo diagram p. 49. Interestingly, in the bibliography, he quotes D. G. Wood, of indirect Cairo tile fame, perhaps he borrowed from him. This also has an interesting series of diagrams p. 83, best described as ‘variations’ with Cairo-like properties, with ‘par hexagon pentagons’ combined in tilings with regular hexagons, similar to Frank Morgan’s work. I am not totally sure of the originality of Critchlow’s work here. Repeats the fallacy of 14 demi-regular tilings, p. 60.


————. Islamic Patterns. An Analytical and Cosmological Approach. Thames and Hudson. Reprinted 2004. First paperback edition 1983. (17 May 2013) Kaplan gives a 1976 edition

Somewhat quirky; Islamic patterns interspersed with nonsensical cosmological and philosophical speculations thereof.


Cromwell, Peter R. Polyhedra. Cambridge University Press 1997 (10 August 2006)

Escher pp. 2, 171172 (sketch of a cutaway view of small stellated dodecahedron), 239, 251, 258. Mostly minor text, in conjunction with polyhedra.


Crook, Diana (editor). Mrs Henry Dudeney. A Lewes Diary 1916-1944. Tarturus Press 1988 (20 February 2018)

Alice Dudeney’s diaries, with Henry Dudeney puzzle interest. Quite how this publication came to my attention is not clear, but likely from Federickson’s writings; it is mentioned in Hinged Dissections. Typically, I did not pursue this immediately. Of biographical interest concerning Henry. No puzzle discussions as such, but of the man in the round. As such, I was initially planning only to read that of Henry himself, but upon reading the book in the round for a better appreciation decided to read all the way through. A delightful coffee-time read, a chapter a day, of which I approached each day with vigor. I warm to Alice. Her devotion to her dogs is admirable. Also the privations of war-time stories.


Crowell, Robert A. Intersight One. State University of New York at Buffalo 1990. 10. Students' work from the Basic Design Studios of William S. Huff 80-85. (8 May 2003)

Parquet deformations. Delightful. Works by Jacqueline Damino Right, Right, Left, Left; Fred Watts, Fylfot Flipflop; Rodney Watkins, In Two Movements; Darren Moritz, Enlarging on Four Points; Aleaandria Gelencsear, Hex-baton; Muarizio Sabini, Venetian Net


Cruys, Sander Van de, and Johan Wagemans. ‘Putting Reward in Art: A Tentative Prediction Error Account of Visual Art’. i-Perception, vol. 2, 9: pp. 1035-1062. 2011.

Non-tessellating article, with a one-line mention of Escher, p. 1042, illustration with Day and Night.


Cundy, H. Martyn and A. P. Rollett. Mathematical Models. Oxford University Press 1977 (?) First published 1951 (First saw 1986 or 1987, college library, studied  beginning 21 September 1987)

Of a mixed degree of relevancy to me; some parts are of the utmost interest, whilst others are way beyond me. ‘Models’ is used in the broad term; it contains much recreational aspects of tenuous connection to the term, such as geometric dissections, although naturally polyhedra are indeed to the fore. Of most note is that of Plane Tessellations, Chapter 2.9, pp. 5965, largely on semi-regular tilings. Also has a Cairo tiling diagram but naturally without the attribution, page 63. Note that this is not original with Cundy and Rollett, but is rather taken from MacMahon’s work, as they state themselves. A strong chapter on dissections, pp. 1926. (Arthur Percy Rollett)

Curbera, Guillermo P. Mathematicians of the Word, Unite! A. K. Peters, Ltd. Wellesley, Massachusetts First edition 2009. (12 November 2020, PDF) 

A history of the International Congress of Mathematicians, from the beginning to the present day.

Escher p. 138 (broadly in passing), and pp. 252253, in-depth, of his role in his exhibit at the Stedelijk Museum, 1954.

Of particular interest is the following passage in which a new name to me is mentioned, one J. J. Seidel, but without further detail:

The organizing committee of the congress—in particular, some of its members such as N. G. de Bruijn and J. J. Seidel—had the idea of having an Escher exhibition as an adjunct to the congress…

Upon research (Wikipedia), this is Johan Jacob "Jaap" Seidel, a Dutch mathematician who dealt with geometry and graph theory. It is stated:

...organized an exhibition by MC Escher.

Upon an initial search, there is little else on the Escher-Seidel connection. De Bruijn mentions this again in Nieuw Archief voor Wiskunde (New Archive for Mathematics) 5/2 nr. 3 September 2001 and is basically repeated in his tribute ‘Jaap Seidel 80’ in Designs, Codes and Cryptography, 2000, 7
10. Seidel also reviewed M. C. Escher: Art and Science in The Mathematical Intelligencer 10, no. 1 (1988), 69–70, but I don't seem to have noticed/saved this when I had free access to MI! It certainly does not justify the high cost now. There does not appear to be anything else on their interaction or his Escher interest elsewhere.




Daintith, John and R. D. Nelson. (editors, with ten contributors). The Penguin Dictionary of Mathematics. Penguin Books, First published 1989 (4 November 2000)

A more advanced treatment, aimed at ‘… first-year university students’, with over 2800 entries and more than 200 short biographies, although nothing, surprisingly, on tessellation! Primarily of text, has few illusions, with much beyond my understanding and interest. A useful occasional reference guide, but nothing more.


Dantzic, Cynthia Maris. Design Dimensions. An Introduction to the Visual Surface. Prentice-Hall Englewood Cliffs, New Jersey 1990 (18 April 1998? The date has faded somewhat).

Brief looks at design aspects. Much of interest. Leonardo quote p. 308. Numerous Escher pp. 49, 57, 60, 88-89, 103, 137, 252-253.

Paving stone with overlapping circle tessellation, of c. 700 BC, p. 48.

Mention of Your Hidden Skeleton, with ink blots designs, of 1900, p 53. Off hand I can’t recall an earlier instance.


Darton, Lawrence. The Dartons: An Annotated Check-list of Children's Books Issued by Two London. WANTED

Davey, Wheeler P. ‘A Study of Crystal Structure and Its Applications’. First Edition New York, London, McGraw-Hill Book Co., 1934. 712 pp. Seen (not downloaded) on Internet Archive (7 January 2020)

Of a Donald G. Wood reference (Space Enclosure Systems). Examined thumbnails (in the hope of pentagon tiling), but no tiling as such, with only the most superficial relevance on occasion.

Day, Lewis, F. Pattern Design. London, B. T. Batsford 1979. First saw 27 January 1988, art school library (18 February 2011) First impression 1903, and 1915 and 1923

Similar is style to Archibald H. Christie’s Traditional Methods of Pattern Designing, being of ornament and patterns per se, rather than of tessellations. Of interest, historically, is Erwin Puchinger’s tessellation-like designs, p. 271. Chapter 6, ‘The Evolution of Pattern’ is perhaps the most interesting, as it concerns tessellation, rather than pattern as implied by the title. Nonetheless, there are many other instances of tessellation throughout the book.


————. Textbooks of ornamental design. The Application of Ornament. B. T. Batsford 1898 (20 May 2016, seen previously) PDF
Has houndstooth-like basket weave p. 25. Of next to no tessellation, which only appears loosely.

Part of a trilogy, The Anatomy of Pattern, Planning of Ornament, The Application of Ornament.


————. Ornament and its Application (17 August 2017 Internet archive download)

I am more than a little confused here, with a somewhat convoluted series of titles and editions! 

Has houndstooth-like basket weave plate 11, p. 25, in the subsection ‘1. The Application of Ornament’ in further subsection 3, ‘Where to Stop in Ornament’, simply described as ‘African’, with only the most minimal of explanatory text. First edition, Plate 10 p. 40, as it was plaited and Fourth edition… Would any more pretentious form of art, be so entirely satisfactory for the purpose of basketwork as the ingeniously plaited pattern of Plate 11?’ Of next to no tessellation, which only appears loosely. No further details are given as to provenance. The book is dated 1895, or at least the PDF version I have, of the fourth edition. 

Part of a trilogy, The Anatomy of Pattern, Planning of Ornament, The Application of Ornament.

The Anatomy of Pattern (1887), The Planning of Ornament (1887),Pattern Design (1903), Ornament and its Application (1904), and Nature and Ornament (1908–9). He published in many journals, including the Magazine of Art, the Art Journal and the Journal of Decorative Art. Other books were Windows(1897),[3] Stained Glass (1903), Alphabets Old and New (1898) and Lettering in Ornament (1902).[4]


Davies, Linda and John Hardingham (designers, no author stated). Leapfrogs Poster Notes. 1986.


Davis, Adam-Hart. Mathematical Eye. Unwin Hyman. 1989 (12 April 1997 and 24 October 1998)

Tessellations pp. 96-97. ‘After Escher’ picture of birds and fish, No. 34, page 97. Juvenile.

Davis, C. (Editor), B. Grünbaum (Editor), F. A. Sherk (Editor). The Geometric Vein: The Coxeter Festschrift. Springer, 1981 First edition. (16 June PDF)

From a sub reference in Tilings and Patterns. Skim read the whole book. As to be expected, overwhelmingly academic. Occasional tiling articles referenced in the book, notably ‘Uniform Tilings with Hollow Tiles’ by B. Grünbaum, J. C. P. Miller and G. C. Shephard, ‘Spherical Tilings with Transitivity Properties’ by B. Grünbaum and G. C. Shephard, and ‘The Geometry of African Art III. The Smoking Pipes of Begho’, by Donald W. Crowe. Each of these is detailed by the author/s in my listing. 

Although tiling is discussed, this is typically academic, way beyond my understanding. Only snippets are ‘useful’.

The analysis was wholly disproportionate as to any benefits to be gained (as was admittedly expected), but nonetheless, there may have been something, and so I don't unduly begrudge the time spent on this. I now at least know, and can thus rest easy.

Note that I checked the (48) author listing with the references in Tilings and Patterns, but there are no others than that given as above.

Davies, Paul. God and the New Physics. Penguin Books1990, first published J. M. Dent & Sons Ltd 1983

Although on physics, included here as it has occasional recreational maths. Brief mention of Escher p. 93, within a discussion of Hofstadter ‘s Gödel, Escher, Bach. Brief discussion of Conway’s Life pp. 226-227. Although of a popular level, most of the text is beyond my understanding (or interest).


Davis, Philip J. and Reuben Hersh. The Mathematical Experience. Penguin Books Ltd 1988. (19 February 1998)

On mathematical philosophy, loosely of a popular level. Although widely mentioned in the literature, of limited value to me; there is no tiling or geometry at my level. Although there may be the odd snippet of interest, it would be disproportionate as to worth in time, of 400+ pages in re-reading/re-evaluating the book. I believe Martin Gardner criticised this book.

Day, Lewis F. Preamble
Lewis Foreman Day (18451910), a man of many and varied artistic interests, has many books (and brief articles) on pattern/ornament (‘tiling’) that relate to tiling, and furthermore are of undoubted interest. However, it is not a particularly straightforward task to differentiate these; each one is of a similar nature, with some reuse of diagrams. Further, these are based on earlier (short) articles, and with different editions of the books, documenting all this is fraught with difficulty. Pattern Design was the first book of his I saw, in 1988, of which being suitably impressed, decided to, in more recent years 2017, and again in 2021, investigate his other publications on the possibility of like interest, caused by his ‘forerunners’ (arguably generously stated) to parquet deformation.
His most commonly cited (and easily available) book appears to be Pattern Design, although this has much borrowing from his earlier books and articles, not realised until recently. Furthermore, he has also written other books of an art/craft theme, but not on pattern/ornament, such as Some Principles of Every-Day Art (1890), Windows (1897), Alphabets Old and New (1898), Art in Needlework (1900), Lettering in Ornament (1902), Stained Glass (1903), Moot Points, with Walter Crane (1903), and Enamelling 1907). Pleasingly, with the books out of copyright, these are to be readily found on the Internet Archive, Project Gutenberg and the Smithsonian. However, although I would at least like to skim read/view all, there is so much here that, at least of the books likely of effectively no interest (non tiling), I have largely overlooked these. His major articles include: ‘The Application of Ornament’ (I-VII) 1890, and ‘Nature in Ornament’ (I-III), 1982. These are available on JSTOR. Again, largely skim read/viewed.
An excellent all round piece on Day is by Elizabeth Rycroft, ‘Lewis Foreman Day (1845-1910) and the Society of Arts’ in the RSA Journal, April 1992.
Interestingly, in Pattern Design at least, he does not effectively differentiate between pattern and tiling/tessellation as we now know it. Rather ‘pattern’ is used to cover both. Further tessellation (or of his day ‘tesselate’ is used on but one occasion, p. 49 with ‘tesselated’.
On each entry of the books I have included the preface for general interest and to set out the context better.

Day, Lewis F. Text Books of Ornamental Design I. The Anatomy of Pattern. London B. T. Batsford 1887, 55 pp., with 35 plates (PDF, Internet Book Archive, 19 August 2021)
Skim read/viewed. The first in a series of books on pattern in the round by Day. Has pattern, tiling and ornament. Very nice treatment. Of note are Plate 11, of a precursor to parquet deformations, generously stated. Plate 15 is of clamshells. Much else is of interest.
There was a time in my own struggling for artistic existence, when I should have been so grateful for any practical teaching in ornarrient, that I fancy there must be students who will find it helpful to have set plainly before them what I have had to puzzle out for myself. Hence this series of Text Books of Ornamental Design; in which I have amplified and illustrated the substance of a series of Cantor Lectures delivered in December last before the Society of Arts.
I have assumed no great amount of technical or artistic knowledge on the part of the reader—only that he wants to know. And, elementary as my subject is, I have taken some pains to save him all unnecessary effort in following my meaning.
The illustrations are to be taken literally as illustrations, and not by any means considered as ornamental addenda to the book.
It is only as diagrams that they have any claim to insertion; although, as an ornamentist, I have naturally made the necessary diagrams as interesting as under the circumstances was feasible.
I have tried to make each one of the plates, as far as possible, explanatory in itself; so that from the study of them alone, apart from what I have to say, a fair idea of the construction of pattern might be gained.
Lewis F. Day
13, Mecklenburg Square, London, W.C. March 30th, 1887.

————. Text Books of Ornamental Design II. The Planning of Ornament. London: B. T. Batsford second edition 1887, 49 pp., 38 plates (PDF, Internet Book Archive, 19 August 2021)
Skim read/viewed. The second in the series, as above. Here, the title is indeed suggestive of the contents, namely ornament. Any tiling is very much secondary.
The second of a series of Text Books stands scarcely in need of preface. The aim and scope, as well as the origin, of this series was duly set forth in 'The Anatomy of Pattern.' What was there said applies for the most part to the present volume.
It was not possible in this case to make the plates speak quite so plainly for themselves as in the former handbook; but I have made a point of referring to them specifically at every turn, at the risk even of tiresome iteration. They are arranged strictly in the order in which mention is made of them, and placed as near as possible to the illusion to them in the text.
The fact that on the publication of 'The Anatomy of Pattern,' I was invited by the Department of Science and Art to deliver a short course of lectures on the subject at South Kensington, leads me to hope that these Text Books are likely to fulfil the educational purpose I had in view in undertaking them.
Lewis F. Day
13, Mecklenburg Square, London, W.C. March 30th, 1887.

————. Ornament and its Application. A book for students treating in a practical way of the relation of design to material, tools and methods of work by Lewis F. Day. London: B. T. Batsford. New York Scribner’s Sons 1904, 319 pp. (17 August 2017 Internet archive download)

Skim read/viewed.

Has houndstooth-like basket weave plate 11, p. 25, in the subsection ‘1. The Application of Ornament’ in further subsection 3, ‘Where to Stop in Ornament’, simply described as ‘African’, with only the most minimal of explanatory text. First edition, Plate 10 p. 40, as it was plaited and Fourth edition… Would any more pretentious form of art, be so entirely satisfactory for the purpose of basketwork as the ingeniously plaited pattern of Plate 11?’ Of next to no tessellation, which only appears loosely. No further details are given as to provenance. The book is dated 1895, or at least the PDF version I have, of the fourth edition. Preface

THIS book is based (like "Pattern Design") upon the foundation of an earlier volume. But, though it covers the ground of "the Application of Ornament," now out of print, it covers a larger area. It is really a new book. Here and there a fragment of the earlier one is incorporated in it; but even that has been shaped anew; for it seemed, looking back upon the work of fifteen years ago, there was little in it which could not be more simply said. The aim of "Ornament and its Application" is throughout practical. It appeals, however, less exclusively than some of my books to students of design ; in fact, it is addressed to all who are really interested in ornament. To those not practically acquainted with the subject, it may serve as introduction to that quality of workmanlikeness which to a workman is of the very essence of design.
What I have endeavoured to do is, to how the clear and close relation of design to workmanship; to arouse interest in a side of art which, regarding it in the rather forbidding light of "technique," lovers of art are accustomed to dismiss from their minds as no concern of theirs; and so to open their eyes to what is indeed a never-failing source of interest in art....
Skim read/viewed. As the title suggests, with only minimal mathematics in occasional places. Nothing of  tiling worth recording. Would likely make for a pleasant read, but I have no time!

Dearling, Alan and Howard Armstrong. The Youth Games Book. Third edition, Published by I. T. Resource Centre, Glasgow. 1985 First published 1980 (12 July 1998)

Paperback, 10 chapters, 247 pp. General puzzles and games of all types (serious and fun) for youths, seemingly intended for youth clubs. The format is a title and general discussion. Martin Gardner gets a mention. Largely a rehash of existing games, from a youth club perspective, but still a welcome contribution in that context. That said, there is little here that is original. I have no plans to re-read.


Deboys, Mary and Pitt, Eunice. Lines of Development in Primary Mathematics. Blackstaff Press 1986. (9 June 2002).

First seen as a library book, October 1993. Tessellations: cover, pp. 158-160, 278-286. Juvenile


Dedron, P and J. Itard. Mathematics and Mathematicians. Vols. 1 and 2 Methods and Problems. 1973 (translated from French by J. V. Field) (3 April 1993 and 28 October 1993)

Eclectic account, slim volume. Kepler plate from Harmonice Mundi, page 53, Vol. 1.


Degrazia, Joseph. Maths is Fun. First Four Square Edition. 1965 First published 1949. (15 July 1995)

Small format paperback, 159 pp. Gardeneresque. Mostly on number/arithmetic puzzles. No tiling or anything of a geometrical nature.


Deledicq, Andre and Raoul Raba. Zoo mathématique, ACL-Les Éditions  du Kangourou, Paris, 1997, 1998, 2002, 4th edition, 2009  (15 December 2017 edition 2002)

A little lightweight, of just 64 pages.


Devi, Shakuntala. Figuring. The Joy of Numbers. Andre Deutsch, First published 1977 (18 October 1992)

By the calculating prodigy, a throwback to bygone days of human calculators. On number calculations, and how she achieved such prodigious feats of stupendous calculation. One can only stand back and admire. Really of general interest only.

Dietz, Ada K. ‘Algebraic Expressions in Handwoven Textiles’. Louisville, Kentucky: The Little Loomhouse. 1949 monograph

Advanced! Although there is nothing directly houndstooth here (in whatever capacity), included as this is a landmark work, as ‘seen and noted’.


Dismore, Julian, Compiler. The Fun and Games Puzzle Book. First Published Boxtree Ltd, 1990 (25 April 1999)

Described on the cover as ‘65 brain-teasers from the popular [Yorkshire] TV series’, although when is not stated, and I’ve never heard of it. Or perhaps I have forgotten! Whatever, a small format paperback, with puzzles seemingly taken from existing instances, framed with Dudeneyesque storylines, but typically much shorter. Dismore is an unknown name to me. The book states that he is an economist among other lighter interests. Simply stated, the book is like one of many of the compilation genre, lacking in originality, and so is inconsequential.


Dixon, Robert. Mathographics. Dover Publications 1991 (10 August 2006)


Dolan, Daniel T. and James Wilkinson. Teaching Problem Solving Strategies (7 May 1998, Hull)

A partial PC of a library book. A few pages on polyominoes, nothing of any significance or substance.

Don, Sarah. Traditional Samplers. Viking, 1986. (16 November 2019)

Of peripheral tessellation interest. A general purpose book on samplers, of history to the modern day, although of course tessellation is not generally the premise! However, I have noticed on occasion that samplers do make use of tessellation and so therefore of possible or potential interest.

Of occasional historical interest as regards tessellation at the beginning, albeit in general is not to the fore, but rather geometric ornament and bands.

Of note is the historical interest, with one sampler of 1598 given (the earliest English surviving instance). The pages refer to pattern books of the 1500s, which I will follow up.

Much here is new to me in a general sense. I previously thought a stitch was a stitch, but no! There's Running Stitch, and more exotic, such as Algerian Eye, to name a few! To investigate…

As such, investigating the field of samplers as regards tessellation flatters to deceive and seems very much out of proportion as to worth in terms of time expanded. What I have investigated on the subject shows next to nothing, and of which I am not planning to investigate further, at least in a concerted effort. I may still have an occasional look.

Donovan, Johnston A. Curves. Exploring Mathematics on Your Own 14. 1966 (22 October 2005)


Dörrie, Heinrich (translated by David Antin). 100 Great Problems of Elementary Mathematics: Their History and Solution. Dover Publications, Inc. 1965 (24 August 1996). Originally 1958

‘Elementary’ here is relative; the problems are of a quite advanced nature! Only with a few of these do I even understand the premise, let alone the mathematics! No tiling as such. Minor MacMahon references, pp. 9 and 27.


Dresser, Christopher. Principles of Decorative Design. Cassell, Petter [sic] Galpin & Co. Fourth Edition. (Downloaded from Project Guttenberg 9 June 2015)

No tessellation as such, mostly of ornament in various forms.


Dudeney, Henry Ernest (edited by Martin Gardner). 536 Puzzles & Curious Problems. Souvenir Press London.1968 (7 June 1997) and second edition 1919 (26 August 2001)

An absolute classic in the field, but no tessellation as such! Dissection puzzles pp. 114-125.


————. Amusements in Mathematics. Thomas Nelson and Sons Ltd. 1947 (5 February 1994) and Dover Publications, Inc. 1958, 1970 (11 September 2000). First published in 1917. Numerous reprints.

Loosely 15 chapters, with in particular of interest a chapter on ‘Geometrical Problems’, pp. 27-55, with Dissection Puzzles, Greek Cross Various Dissection Puzzles, Patchwork Puzzles and Various Geometrical Puzzles. The book is full of interest; however, there is no tessellation whatsoever!


————. A Puzzle-Mine. Subtitled ‘Puzzles Collected From The Works Of The Late Henry Ernest Dudeney’, by J. Travers. Thomas Nelson and Sons Ltd. Date of publication surprisingly not stated. However, Frederickson gives this as 1931. (11 October 1997)

An editorial note states that the puzzles in this book were originally published in serial form in the magazine Blighty and after the war of 1914-1918….’

Four chapters of classic Dudeney fayre. Although all of interest, of most note is Chapter III, dissection puzzles, pp. 81-85. Likely these repeat others in his books. As ever, no tessellation as such.


————. (Edited by Martin Gardner) More Puzzles and Curious Problems. More than 250 tantalising brain teasers by the puzzle king. Collins Fontana Books. Small format paperback. (19 July 1992)

Essentially the same as immediately below, with ‘more’ added to the title, and the same contents, although of a three-page increase, the reason of which I refrain from investigating.


————. (Edited by Martin Gardner) Puzzles and Curious Problems. More than 250 tantalising brain teasers by the puzzle king. Small format paperback. Collins Fontana Books 1970. First published in Great Britain by Souvenir Press (qv) under the title ‘536 Puzzles and Curious Problems’. (8 August 1993)

This is the first part, of 258 puzzles, with answers (I do not have the second part). Oddly, within the same contents framework, and so would appear that the books are the ‘same’, the puzzles are different, and bear no direct correlation to each other!


————. The Canterbury Puzzles. Thomas Nelson and Sons, Ltd. Second Edition 1919 Fourth edition 1932 (with some fuller solutions and additional notes). (16 November 1996)

114 puzzles in nine chapters, with solutions. Occasional references to tiling and dissections: 19, The Puzzle of the Prioress asymmetric cross to square; 26, ‘The Haberdasher’s Puzzle’, dissection, triangle to square; 37, ‘The Crescent and the Cross’ (on dissection), 77 ‘Making a Flag’; 84 ‘The Japanese Ladies and the Carpet’, and of course much else of interest in a generalised sense.

Dürer, Albrecht. Underweysung der Messung mit dem Zirckel un Richtscheit, 1525. (The Four Books of Measurement with Compass and Ruler). Translated as The Painter's Manual. Abaris, New York, 1977, 472 pp. (First saw as The Painter’s Manual in 1993?; not in possession)

From the Introduction: "not only for painters, but also for goldsmiths, sculptors, stonemasons, carpenters, and all those for whom using measurement is useful."

Anon: The treatise synthesized a number of classical and contemporary mathematical texts with the knowledge of geometry Dürer had accumulated over a lifetime of artistic practice, in order to train German artists in precision drawing and, by extension, precision thinking.

Likely from a bibliographic reference, now long forgotten. I seem to recall ordering The Painter's Manual from British library, in 1993. As such, tiling forms but a small part of the book, in Book II, but nonetheless is of decided interest, especially with my interest in pentagon tilings, Dürer's work being the first known representation. As such, the book was briefly studied of the day (1993), but with other aspects being of lesser direct interest, was effectively put aside. As I recall, I was slightly disappointed with it, with more on tiling being expected. However, upon reading (June 2019) Noam Andrews’ 2016 paper, ‘Albrecht Dürer personal Underweysung der Messung’, my interest was once again piqued, upon finding that in his own first edition copy he made a series of corrections, and with further tilings. These are available in the digital collections of the German Bayerische Staatsbibliothek:

However, the tilings are relatively minor, really only of interest due to the historical aspect. There are no further pentagon tilings.


Dye, Daniel Sheets. The New Book of Chinese Lattice Designs. 372 Designs. Dover Publications Inc, New York 1981 (first published). Edited and with an introduction by Nancy Balderston Conrad (9 April 1993)

The introduction states that these are designs that were not included in his earlier book Chinese Lattice Designs. The book is diagram heavy and text light (only the barest of descriptions are given for classifications), of which the later is sorely missed; these are crying out for background details. I considerer this book to be very much the poor relation to the other. Balderston, mentioned in the dedication, is a relative of Dye in some way. His wife has Balderston as her middle name.

No Cairo tilings. Of occasional interest: p. 69, with a par hexagon divided into unequal kites, with a secondary feature of squares or vice versa. P. 103, of a curious two-tile tiling of a common arc of an underlying square tessellation worthy of study.


————. Chinese Lattice Designs. 1200 Designs. Dover Publications Inc, New York 1981. (9 April 1993)

This apparently first appeared in 1937 titled as A Grammar of Chinese Lattice. Checked entire book for Cairo type tilings May 2011. Only ‘faux’ instance is of p. 340, a Greek cross with a ‘x’ in centre. Page 420 has a Chinese parquet likeness source from Gardner’s 1983 article.

The above is stated by Peter Hilton  and Jean Pederson of a reprint of A Grammar of Chinese Lattice  Harvard-Yenching Institute Monograph Series, Vol .VI, Harvard University Press, Cambridge, Mass, 1937.




Eastaway, Robert, editor. Enigmas. The World’s Most Puzzling Book. Arlington Books (Publishers) Ltd 1982. (31 October 1993). First saw in Scartho library 21 December 1987, or at least the first recorded study.

A compilation from the pages of New Scientist’s weekly Enigma column, described as ‘… 69 brain-teasers…’, in nine sections. The style is Gardneresque. Curiously, it would appear from the acknowledgements that Eastaway had nothing to do with the puzzles per se! The puzzles are mostly  in the form of ‘stories’, rather than of abstract mathematical problems. Of most interest is ‘Shapes and Sizes’, pp. 83-90 of six geometrical figures. The first, ‘grid halving’, on geometrical dissections, of cutting a figure into two equal parts, was the only one ‘studied’. Another dissection problem, ‘Two-Square Dissection’ was not studied. Note that there are no tessellation problems. As such, a minor, inconsequential study was made.


Eastaway, Rob and John Haigh. How to take a Penalty. The Hidden Mathematics of Sport Robson books, an imprint of Chrysalis books group PLC (6 August 2011)



Eckler, Ross. Making the Alphabet Dance. Recreational Wordplay. Macmillan 2001. First published 1997. (21 February 2015)

Chance finding. Although out of my direct interest, with many notables named here, such as Martin Gardner, it was judged worth a look. I must say that I am surprised that the book’s author, Ross Eckler, and indeed the book itself, first published in 1997, had escaped my orbit.


Edwards, Cyril and Phil Boorman. Geometry. Macdonald Educational Colour Units 1976 (5 June 1994 and c.2000?)

Do I have two copies? Note that although this is a book in its own right, it is also part on a series of Mathematics by the Macdonald Educational, Colour Units with other titles: Sets and Religion, Trigonometry, Statistics*, Number and Patterns, Groups and Finite Arithmetic*, Matrices, Calculating Aids, Vectors, Graphs, and Algebra. * In possession. Statistics is by Lynn Jones. The level is fairly basic (and of relatively few pages, just 24), with simple geometric constructions. Although there is some advanced maths here and there, the tenure is one of for beginners. Save for one instance, more or less in passing on page 24, there is no tiling here.


Edwards, Cyril. Groups and Finite Arithmetic. Macdonald Educational Colour Units 1974 (12 November 1995)

Although of repeat patterns and symmetry there is nothing of any real interest.

Egleston, T. ‘Diagrams to illustrate the lectures on crystallography, delivered at the School of mines of Columbia college.’ Second Edition New York: School of Mines, Columbia College, 1872. Available on Hathitrust (7 January 2020)

Of a Donald G. Wood reference (Space Enclosure Systems). Examined thumbnails (in the hope of pentagon tiling). Replete with crystal diagrams, but no tiling.

Elffers, Joost. Translated by R. J. Hollingdale. Tangram. The Ancient Chinese Shapes Game First published Verlag M. Dumont Schauberg 1973 (19 May 1995). Penguin Books 1984.

Small format paperback, c. 200 pp. Perhaps more of a collaboration  than the single author given suggests, with a Introduction (Jost Elffers with Erik van Grieken). History (by Jan van der Waals) pp. 9-27, and bibliography by (Jan van der Waals), pp. 29-31 pp. 123-124, an essay ‘Counting and Classifying Tangrams’ (by Michel Dekking with Jaap Goudsmit). Gives a most interesting and useful tangram history. Then gives a series of tangram diagrams without further commentary. Of general geometrical interest, nothing more. Quite what the background of Elffers is went unresolved.


Elffers, Joost and Michael Schuyt. Tangram. The Ancient Chinese Shape Game. Barnes & Noble Books 2001 (26 April 2010).

A boxed set of tangrams and book.


Elliot, Marion. The Tile Decorating Book. Lorenz Books 1997. (19 October 2008)


El-Said, Issam; Ayse Parman. Geometric Concepts in Islamic Art. World of Islam Festival Publishing Company Ltd. 1976 (2009)

Many references to ‘tomb towers’ re Carol Biers’ interest.

Elwes, Richard. The Maths Handbook. Everday Maths Made Simple. Quercus, 2011 (6 July 2019)

'Back to basics', and of its type most impressive, one of the better (if not one of the best) books. No tessellation. 


Engel, Peter. Origami: from Angelfish to Zen. Dover Publications Inc. 1989 (26 May 2008).
Occasional reference to Escher’s tessellations and prints, pp. 2-5, 69. Cover has an adapted ‘Drawing Hands’, in relation to the origami premise of the book.


Ernst, Bruno. The Magic Mirror of M. C. Escher. Tarquin Publications 1985 (first published 1972. (19 August 1988) First saw in 1987, and ordered 19 August 1988

A major work on Escher, one of the ‘core value’ books; Indispensable! However, of note is just how little tessellation there is! The book is primarily on spatial structures. And what little there is, this is subsumed by the above premise.


————. Adventures with Impossible Figures. Tarquin Publications 1986. (9 April 1993)

Popular account.


————. The Eye Beguiled. Optical Illusions. Benedikt Taschen 1992 (10 August 1993)

Although not strictly a tessellation book, included here as there is a certain amount of crossover. More of impossible objects, Ernst’s forte, rather than a generic optical illusion book. Has a scholarly bibliography. Escher prints Concave and Convex p. 27, Belvedere p. 77. Small section on Escher per se, pp. 74-80. Escher Belvedere model by Shigeo Fukada pp. 92-93.


Escher, M. C. Grafiek en tekeningen M. C. Escher. Contribution by P. Terpstra. Zwolle: J. J. Tijl, 1960 (first printing 1959). (21 October 2016)

Gift of Peter Raedschelders. In Dutch. One of the core value, ‘must have’ books on Escher. In brief, an eclectic selection of 39 of his works (later expanded to 76 in a subsequent edition), divided into nine (and later 10) classifications. Shows 13 plane tilings. Each entry is accompanied by a brief commentary, albeit in Dutch, of which I discuss this is the English translation.

Of note here is P. Terpstra’s essay, pp.11-13, ‘Its over de wiskundige achtergrond van regelamatige vlakverdelingen’ not shown in subsequent editions. Also has a catalogue not in subsequent editions.


Escher, M. C. The Graphic Work of M. C. Escher. Oldbourne, London 1970. (8 August 2004) and Taschen (10 August 1993) (First saw in September 1987, Louth library)

One of the core value, ‘must have’ books on Escher. Expanded edition of the 1960 first published. In brief, an eclectic selection of 76 of his works, divided into ten classifications: 1. Early prints, 2. Regular division of a plane, 3. Unlimited spaces, 4. Spatial rings and spirals 5. Mirror images, 6. Inversion, 7. Polyhedrons 8. Relativities, 9. Conflict flat-spatial and 10. Impossible buildings. Each entry is accompanied by a brief commentary, albeit this is generally lightweight, and of  which shows little new insight.


————. M. C. Escher 29 Master Prints. Harry N. Abrams, Inc. Publishers New York 1983. Edited by Darlene Geis. First saw 18 July 1992. (9 April 1993)

Very large format book, 64 pp. In addition to the 29 Master Prints (by and large a fair description, given the inclusion of Day and Night and Sky and Water I, although no Verbum), both tessellation and others, the book includes an essay by Escher, 'On Being a Graphic Artist' and with commentaries on the prints, mostly by Escher, and additionally, in most a most minor way, by C. H. A. Broos, J. L. Locher, Bruno Ernst and H. S. M. Coxeter. However, none of this text appears to be original; it appearing in other sources, as according to the book.

According to a reference on an old ring (cardboard) binder cover, I first saw this in Grimsby central library  on 18 July 1992, but have since completely forgotten about this! Whether this was as on the shelves or was ordered I do not recall. Whatever, it was not significant in that no new studies arose from this.


Espy, Willard R. The Game of Words. Wolfe Publishing Ltd. 1971 (two books, one obviously forgotten upon purchasing, one book not dated, one 7 June 1997)

Although not strictly mathematical per se, being of word play, of interest to the mathematical mind, and so hence included here.




Falletta, Nicholas. The Paradoxicon. A Collection of Contradictory Challenges Problematical Puzzles and Impossible Illustrations. Turnstone Press 1985 (29 November 1992). First published by Doubleday and Company, New York, 1983. First saw (or at least studied) 15 September 1987, Scartho, Grimsby library

The 1983 edition has a front cover picture (among others) of Escher’s Drawing Hands. Has much of interest in a generalised sense, albeit some I have no interest in. However, it is more of a compilation nature, rather than of original research. Many references to Escher, notably with a dedicated chapter, ‘M. C. Escher’s Paradoxes’ pp. 24-34, and illustrations throughout; pp. 54, 98-99, 101, 157, 190. Other chapters of note not pertaining to Escher include ‘Geometric Vanishes’, pp. 35-40.


Falkener, Edward. Games Ancient And Oriental And How To Play Them. Dover Publications, Inc., New York 1961.


Farnworth, Warren. Techniques and Designs in Pin and Thread Craft. B T Batsford Ltd, London 1977. First saw c. 23 June 1987. (26 April 1998)

Although not strictly on mathematics, included as it was studied among my early’ mathematical’ studies of 1987.


Farrell, Margaret A. (ed). Imaginative Ideas for the Teacher of Mathematics, Grades K-12. Ranucci’s Reservoir. National Council of Teachers of Mathematics (NCTM) 1988 (30 April 1994)

A compilation by Farrell of 21 articles, in five parts, of Ernest Ranucci’s works. Of most interest is Part 4: Inventiveness in Geometry, with tessellation articles: ‘A Tiny Treasury of Tessellations’ and ‘Master of Tessellations: M. C. Escher, 1998-1972’.

Such a ‘type’ of book was more interest in the ‘old days’ (pre-internet), where easy access to published journals was not widely available. The book also contains an excellent bibliography of his works and biography. Of the two papers, of note is that the Cairo-esque diagram. I had though that this was unique to his book Tessellation and Dissection.

Has Fish and Scales on front cover.


Fathauer, Robert. Designing and Drawing Tessellations. Tessellations (Fathauer’s own company). 2008 (18 July 2009)

First, I consider the title a little misleading, given that the premise is overwhelmingly one of creating, or of studies leading up to, of Escher-like tessellations, rather than generic tessellations per se as the title would otherwise suggest. Be that as it may, pleasingly, this is one of the few books to broach the topic of Escher-like tessellations in depth, and so is warmly welcomed, although I have various issues in places. As a generalised statement, the advice given by Fathauer is very good indeed, with many useful hints and tips as to the ways and means of creating Escher-like motifs. Although primarily aimed at a school-age level (12-16), anyone with an interest in creating Escher-like tessellations will find it advantageous, as it broadly addresses the all-important understanding of the underlying issues concerning the creation of life-like motif, of which this aspect is generally disregarded. Very few books concern themselves with this matter, even in passing, and of which lies at the crux of the problem of designing high-quality motifs, and so this aspect in particular is warmly welcomed. As such, one could argue that Chapters 1-4 of history and background matters, could very well be excluded on the grounds of familiarity, as previous books have covered the same ground. Still, for an easy, and convenient, ‘basics to hand’ covering of such matters, there is nothing to fault here. However, for those familiar with such matters, the book only really begins much later, with Chapter 5, this concerning the Escher-like aspect, with a series of tips on drawing and designing. Chapter 6 then gives a series of techniques, in effect attempts at improving upon the initial tessellation. Chapters 7-11 are concerned by more specific matters, with tessellations based on specific tiles that frequently occur. Broadly, upon the reader having absorbed each chapter, a series of activities are then given, suitable for the above age range.

The whole book is in black and white, without colour and without Escher’s works.

Escher references: Preface, pp. 1, 5, 7, 42, 47-51, 54-56, 66, 69, 71, 89, 93, 102, 129, 137, and 140. Note that these are mostly in passing, with only p. 5 of a true discussion. Note also that there are no pictures of his works. 

Cairo tiling references: P. 2, with a brief discussion, and p. 27. 

Also see my review of the book for a more detailed, specific analysis.

Féblien, André. Traite de l’Architecture, 1676. (2 December 2019)

From a reference in The Sense of Order, by Gombrich. Tilings as such form a small part of the book, as glass panels. As occasioned by a revisit to Gombrich, of ongoing Michio Kubo interest, of which a plate from the book precedes that page. The book has * plates of tiling. Of historical interest, of innovation, albeit nothing truly outstanding. Available on the Internet Archive.

Fellows, Miranda. The Life and Works of Escher. Parragon Book Service Limited, 1995 (14 May 2016)

Small format hardback. Fellows comments on a selection of Escher’s works. Seen (where is long forgotten) many years ago, but (I think) judged so lightweight as to not worthy of pursuing, perhaps a little unfairly in retrospect.

Does not have a Escher bibliography, as might have been thought.


Fenn, Amor. Abstract Design and How to Create It. Dover Publications Inc 1993. Republication of the original of 1930, with a new introduction by Richard M. Proctor (21 September 2012)

The premise is of design, with stripes, wall papers, rather than tessellation per se. This is very much as in the style of another book of the time, Pattern Design, by Lewis F. Day. Houndstooth tiling p. 129, albeit nor sourced in the text. Nothing particularly innovative here, certainly as regards tessellation.


Feravolo, Rocco. Wonders of Mathematics. A Wheaton & Co. Ltd. 1964 (not dated, c.10 years ago)



Ferris, Timothy (ed.) The World Treasury of Physics, Astronomy, and Mathematics. Little, Brown and Company 1991. (3 September 1998?; The last digit has faded).

Anthologies by sixty leading authors; G.H Hardy, Benoit Mandelbrot etc. (Mathematics, Chapter 4).


Fetter, Nancy, Nancy Eckert, Ann Fetter, Doris Schattschneider, Cindy Schmalzried, Eugene Klotz. The Platonic Solids Activity Book. Backline Masters. Key Curriculum Press, Berkeley, CA. 1991 (30 April 1994)

Cairo reference and line drawing page 21, and repeated page 96, the reason for this being teachers and student questions. The quotation repeats Gardner’s Scientific American assertion re ‘ … seen in Moorish buildings…’ (and is likely taken from that reference; Schattschneider’s contribution?). Minor Escher-like art, a bird, page 20.

Also see companion book The Stella Octangula Activity Book, Hilary Brest et al.


Field, Michael and Martin Golubitsky. Symmetry in Chaos. A Search for Pattern in Mathematics, Art and Nature. Oxford University Press 1992.

Decidedly advanced, very little of which is accessible to me. Mostly of pattern using advanced equations rather than tiling. Escher's Horsemen p. 59.


Field, J. V. Kepler's Geometrical Cosmology. The University of Chicago Press, 1988. (19 November 1994)

Also see her article on ‘Kepler’s star polyhedra’.


Field, June. Creative Patchwork. Pan Books 1976. First edition 1974 by Sir Isaac Pitman and Sons Ltd. (30 September 2000)

Although obviously not a book on mathematics per se, it is nonetheless included in this listing as it has crossovers, However, now, and for some considerable time, the nature of the material is considered hardly worthy the time originally devoted to it. as such, I seem to recall this book from my 1987 studies, although there is no documented connection.


Field, Robert. Geometric Patterns from Roman Mosaics and how to draw them. Tarquin Publications 1988. (3 June 1993)

Small booklet, 64 pages. Note that Field has a like format five-book series with the title ‘Geometric Patterns’, with a variation. Tiles and Brickwork, Islamic Art and Architecture, Churches & Cathedrals, From Patchwork Quilts, and one outlier, Mazes Ancient & Modern. No Cairo pentagon.

————. Geometric Patterns from Islamic Art & Architecture. Tarquin Publications 1998 (First edition). Small format paperback, 63 pp. Downloaded from Internet Archive 31 December 2019

Very nice indeed! Primarily an image book, without pretence as to overt mathematics, with captions/text likely kept in the background. Of especial note:

Whirling kites/Pythagoras tiling pp. 8-9

Breath of Compassionate pattern pp. 22-23

Bow tie tiling p. 40, at Brighton Museum and Art Gallery, of which I was unaware of!

Other titles include:

Geometric Patterns in Churches and Cathedrals, Geometric Patterns from Tiles and Brickwork, Geometric Patterns from Roman Mosaics: and How to Draw Them, Geometric Patterns for Patchwork Quilts

Amazon: I have written six books on geometric patterns which are used by many people as a source of creative design in subjects as diverse as embroidery, patio paving, knitwear and quilt making. I also published and illustrated an anthology of poems about mice - I Think Mice are Rather Nice (Triplecat). I have also illustrated two books by my partner, the author Roderick Grant, Clap Hands for the Singing Molecatcher (Triplecat & Birlinn editions) and Strathalder - A Highland Estate (Triplecat & Birlinn editions).

————. Mazes Ancient & Modern. Tarquin Publications 2001. (Date not stated)

Fletcher, David and Joseph Ibbotson. Geometry Two. Holmes McDougall Limited 1967 (25 October 1998 year is semi legible)

Pitched at a 8-12-year-age level. Note that this is a three-book series, of which I only have book 2. Tilings pp. 20-21, but only of the most simplest investigation of the ‘angle proof’. Gives ‘new’ means of drawing octagons, p. 44.


Fletcher, Harold. Mathematics for Schools. Teacher’s Research Book. Level II Books 1 and 2.

Addison–Wesley Publishers Limited 1971 (3 September 2006).

Juvenile. No real interest, primary maths. Symmetry pp. 50-54, no tessellation.


Fletcher, Alan. The art of looking sideways (Sic). Phaidon. No bibliography detail! (Grimsby library, 5 May 2012, although seen many years ago)

Although not a maths book per se, included as it has a few pages on tilings, notably p. 255 and next three pages – pages are not ‘truly’ numbered here! Although the book is indeed light on tiling, the tilings it does contain are of significance, containing new material. These are taken from a page in Mathematical Models, page 64, itself taken from an earlier source, Daily Telegraph in 1955 (the exact issue is uncertain, regrettably, no other details are given, and so have not been able to obtain). Fletcher apparently builds on this, with further tiling. I say apparently, perhaps these first appeared in the Telegraph? He credits the Telegraph article.


Ford, Karin (translator) and Janet Wilson, editor. English Language version. Escher on Escher. Exploring the Infinite. Harry N. Abrams, Inc. 1989. (29 May 1991). With a contribution by J. W. Vermeulen. Compiled by W. J. van Hoorn and F. Wierda. Originally published under the title Het oneindige

Small format paperback. A series of translated essays of Escher's own writings and previously unpublished speeches in Dutch, and so warmly welcomed. These include:

1. Newsletter of the Dutch Circle of Graphic Artists and Illustrators, No. 5, December, 1950. The Craft. 10-12. Dear Oey…

2. Newsletter of the Dutch Circle of Graphic Artists and Illustrators, No. 3, June, 1950. Our Brother 13-15. Dear Oey…

3. De Grafische (The Graphic Arts), No. 13, September, 1951. White-Grey-Black 16-18

4. Acceptance Speech by M. C. Escher upon receiving the Culture Prize of the City of Hilversum on March 5, 1965 19-22

5. Prepared lecture for Lexington, Massachusetts, US not given by Escher due to ill health - The Regular Division of the Plane 24-53 (part 1); Other Themes 54-80 (part 2)

6. How Did You as a Graphic Designer Come to Make Designs for Wall Decorations? De Delver (the Digger), xiv, No. 6, 1941 83-88

7. The Regular Division of the Plane 90-122 (also published in M. C. Escher The Complete Graphic Work)

8. Approaches to Infinity (no context or date given). 123-127 (as given in Locher)

9. Perspective (no context or date given). 128-134

10. The Impossible (no context or date given). 135-136

11. I’m Walking All Round All By Myself Here, by J. W. Vermeulen 139-153. A portrait of Escher, by his accountant.

A notable aid in Escher scholarship, with numerous Dutch texts made readily available. Has a ‘serious’ bibliography, p. 154, albeit brief, under the title ‘selected bibliography’. This is best described as partial, taken from ?


Forty, S. M C Escher. Taj Books 2003 (11 October 2009)

Oversize. The premise is of a ‘grand picture book’ per se, with 74 works, of prints (mostly) and drawings. There does not appear to be any new research, with the brief introductory text apparently assembled from existing sources. Shortcomings and faults abound here. Ideally each print or drawing would be accompanied with some text; however there is no individual commentary whatsoever, a major shortcoming. There is no formal introduction per se. The text that serves for the introduction, pp. 5-11, as ‘Maurits Cornelis Escher 1898-1972’ is of an overall guide. However this is riddled with errors, of basic English and story. Apostrophes are used both incorrectly and correctly, with ‘the Escher's first…’ p. 7 and ‘the Eschers’. Also apostrophes are omitted (purposefully?) in the plates 1, 7, yet are used correctly elsewhere, plates 44, 59. Such slapdash work is inexcusable, given that (a) the author is a graduate of London University, and so should know better, and (b) the text, of just seven pages is hardly of such a length that it would be overlooked as would a piece in say a 300-page work feasibly would. Some text is just plain wrong: ‘failed all his exams except mathematics’’.

At high school in Arnhem, I was extremely poor at arithmetic and algebra because I had, and still have, great difficulty with the abstractions of numbers and letters. When, later, in stereometry [solid geometry], an appeal was made to my imagination, it went a bit better, but in school I never excelled in that subject. But our path through life can take strange turns.

Other statements need checking for veracity. Given the above shortcomings and errors I am not sure how much the text can be relied on, but I lack the time to investigate as I would like. Useful for seeing Escher’s prints at a larger size than in most books, but not much more. No bibliography, although the nature of the book does not lead to this.


Foster, Leslie. Rainbow Mathematics Encyclopedia. Grisewood & Dempsey Ltd. 1985 W.H. Smith edition (19 March 2005)



Foster, Richard. Patterns of Thought. The hidden meaning of the great pavement of Westminster abbey. Jonathan Cape, London 1991 (12 February 1994, York)

A general account of the pavement. Chapter 6, pp. 111-130 concerns the aspect most of interest, from a geometrician point of view.

Fourrey, E. Curiosités Géométriques. Paris 1907 (downloaded from internet archive 28 April 2015).
From a reference in Bradley. A little disappointing, in that tiling is only mentioned briefly pp. 363-371 on the PDF numbering.

————. Recreations Mathématiques. Paris 1899
From a reference in MacMahon. Strictly number recreations, of which although of interest again disappointing, as I was hoping for tiling.

Franke, Herbert W. Computer Graphics Computer Art. Phaidon, 1971. Other, later editions of 1985. Originally published as Computergraphik-Computerkunst (First saw November 1987?) (30 December 2016)

Purchased for foreseen forthcoming review purposes, having previously last studied in November 1987. As such, I only had dim and distant memories of this book, having last seen it nearly 30 years ago! Indeed, I couldn’t really picture or remember the contents. Be all that as it may, the book has next to no connection with tessellation. Indeed, there is not a single instance! As such, a useful guide to computer graphics of the day, but now a little dated, but still of interest as to historical matters. Has an excellent biography and bibliography sections, albeit I simply do not have the time to pursue these, as much as I would like to.

My brief studies of this, of just four sheets, are dated 23 November and 2 December 1987, of  p. 96. Pages of interest include, frontispiece, of a dragon space-filing curve, p.18, 30, of a metamorphosis, of a loose parquet deformation nature; op art by A. Michael Noll, p. 67.

Francis, Daryl. Puzzles & Teasers for Everyone. Paperfronts. Elliot Right Way Books c. 1980 (10 August 1991)
Usual repeated fare.

Freaker, Daniel and Alan Parsons. Series consultant Harry Smith. Revise BTEC National Art and Design. Revision Guide. Pearson 2018. (19 June 2018)

For 11-16 age children. Features my bird and fish tessellation, p. 85. The text is Freaker and Parsons' own, of which I do not entirely agree with .


Freeman, Mae and Ira Freeman. Fun with Geometry. Kaye & Ward, London 1969. First published in 1958 (24 October 1998)

28 different two-page essays on ‘popular geometry’ both ‘theoretical’ and ‘applied’, aimed at a juvenile audience. That said, some aspects are new to me here! Measuring distances, pp. 24-25 and the three tags, pp. 50-51. Much of this is Gardneresque nature, albeit pitched at youth. Geometric dissections pp. 52-53, but no tiling as such.


Freese, Ernest Irving. Perspective Projection: A Simple and Exact Method of Making Perspective Drawings. New York Pecil Points Press, 1930 (15 March 2018)

From a link on Greg Frederickson’s update page, viewed (but not downloaded) at The Hathi Trust. As the title suggests, on perspective, with no foreshadowing of his work in geometric dissections, or anything on tiling per se.


Frederickson, Greg N. Dissections: Plane & Fancy. Cambridge University Press 1997. (28 February 1998)

An absolute delight! Highlight after highlight, too many to list here, although I am merely an ‘interested bystander’ in the field. 24 chapters and an excellent bibliography. Speculations as to who ‘A. E. Hill’ was, pp. 157-158, 290-291. Has many interesting brief biographies of the main people in the field, past and present, including Dudeney, p. 81. For me at least, and I suspect most other people, this is the more important of his three books on the theme, the other two, Hinged  Dissections and Piano-Hinged Dissections are more of a specialised nature.


————. Hinged Dissections: Swinging & Twisting. Cambridge University Press. 2002 (?)

Perhaps somewhat out of mainstream interest, of a specialised branch of dissections. Nonetheless, it remains full of interest. Has asides in the form of ‘Curious Case’ and ‘Turnabout’, with much on Dudeney.


————. Piano-Hinged Dissections: Time to Fold. A. K. Peters, Ltd. 2006. Not date stamped. A receipt states ‘processed September 9 2008’ gives indication as to obtaining.

Perhaps somewhat out of mainstream interest, of a specialised branch of dissections. Nonetheless full of interest. Has asides in the form of Ernest Irving Freese’s lost manuscript and ‘Folderol’ (of which such term I was unfamiliar with; the dictionary gives it ‘anything trifling’).


————. Ernest Irving Freese's Geometric Transformations: The Man, The Manuscript, the Magnificent Dissections! World Scientific 2018

(2 February 2018)                                        

At last, after no less than 60 years, Freese’s work is shown in its entirety!


————. ‘Hugo Hadwiger’s influence on geometric dissections with special properties’. Elemente der Mathematik. 65 (2010) 154 –164 (2 September 2016)


Freebury, H. A. A History of Mathematics. For Secondary Schools. Cassell & Co. Ltd. 1958 (8 July 1995)


French, P. Introducing Polyhedra. McGraw-Hill Publishing Company Limited 1966. First published 1964 by the House of Grant. (24 October 1998).

Slim volume paperback, 39 pp. Juvenile, Junior, 8-11 years. Part of the 13-book series on the title ‘Exploring Mathematics’, by P. French and R. J. Rickard, under the general editor J. B. Palframan. Gives compass construction nets and brief history. although not explicitly stated for children, it is clear that this is the indeed the intended audience. Too simplistic to be of any use.


Friedhoff, Richard Mark and William Benzon. Visualization. The second computer revolution W. H. Freeman and Company New York 1991. (11 September 1994)

A pleasing read, largely accessible, although there is only subsidiary discussions on related mathematical aspects, such as fractals. No tessellation, no Escher.


Frisby, John P. Seeing. Illusion, Brain and Mind. Oxford University Press, 1979. First saw 1987. (13 February 2017)

Although a book on seeing in the broad context and so not on maths, it is included in this listing as it features Escher’s ‘seeing-related’ prints (tessellation is not mentioned), as well of interest in a variety of ways in a generalised sense. Upon the ongoing (2017) review process of all 1987 studies, specially purchased. Although ostensibly of a popular nature, the text nonetheless remains is in general of a forbidding nature. Features two of Escher’s prints in Chapter 1 ‘Pictures in our Heads’, pp. 22-23, Ascending and Descending and Waterfall. However, there is only minor commentary, p. 19. However, one pleasing nuance is that Frisby astutely observes the fine distinction of ‘Monk’s work’ in the discussion of Ascending and Descending as useless labour of which most other commentators do not, missing the ‘useless’ point.

Further to the book, I happened to notice on the dust jacket the following intriguing quote:

One of his special interests is in the art of M. C. Escher.

Upon following up with him (mail, February 2017), he told me:

… When I published the first edition of Seeing the publishers suddenly sprang on me a request for ‘special interests’ and in a bit of a rush I mentioned Escher whose work at that time (around 1978-79) I was using to illustrate some lectures. In fact, while an admirer, I have no deep interest in Escher.

Therefore, it wasn’t a ‘special interest’ after all, but rather just a passing interest! But at least I know now.


Fuller, Buckminster R. Utopia or Oblivion. The Prospects for Humanity. Penguin Books 1972 (20 September 1992). First published in the USA 1969

Small format paperback, 416 pp. Text heavy. The cover has  sphere packing diagrams, which my have attracted my attention. I simply don’t have the time to read this. It may come in useful for reference purposes. Chapter 3, Prevailing Conditions in the Arts’, on geometry, is the only aspect of direct interest. I read somewhere (Coxeter?) that Fuller was somewhat crankish (or overstated) in many ways. Whatever, the book was never studied per se.



Gale, Howard et al. The Times Tournament of the Mind. Times Books Limited 1988. (not dated, c 10 years +)


Gardiner, Anthony. Mathematical Puzzling. Oxford University Press 1987 (15 September 1989)

Popular account from an academic, 153 pp. Not in possession, saw at Grant Thorald library on 15 September 1989 and made a minor study of that date on a shared paper of A. Racinet and Michele Byam  of 10 and 17 January 1989 respectively. The book has long been deleted from stock, and ideally I would obtain again, if only to aid in a review. However, even of the lowest price on Amazon I am not planning on doing so. Pages studied 49, of circle packing and 73, of indeterminate means. Gardiner is a notable mathematician, with 15 books to his name; possibly this was the first.

See Chapter 11, ‘Circles and Spheres’, pp. 49-51. and Chapter 16 Polygons, pp. 71-74. although ostensibly Chapter 11 is on circle packing, this is not so, or at least as seen at first impression such a fitting circles into a square. Rather the premise is of fitting the largest circles inside the central part hole of each configuration.


Gardner, Martin. 1. Mathematical Puzzles and Diversions. Penguin Books (original edition 1959). (30 August 1993). Also London G. Bell and Sons Limited 1963. Hardback (19 November 1994) and Pelican (30 August 1993)

First, regarding the listing of columns in Gardner’s compilations in books below, the entries in bold are of extra special interest, primarily of tiling matters, although drawing hard and fast lines is an invidious task at times. Quite simply his collection is indispensable! However, annoying, and infuriatingly, these do not always reflect the original title and so correlating like articles is not a straightforward task as it may otherwise appear to be. As a preamble to Gardner’s collection of columns over 25 years in 15 books, these are a fresh delight time and again, as due to such an extensive compilation one simply forgets, save for core value articles! Further to the core values, for each book, where appropriate I list such instances, primarily involving tessellation and/or Escher aspects, although at times there is no firm boundary. For each book I list each chapter, although these do not always tally with the original article in Scientific American (a bone of contention).

1 Hexaflexagons, 2 Magic with a Matrix, 3 Nine Problems, 4 Ticktacktoe, or Noughts and Crosses, 5 Probability Paradoxes, 6 The Icosian Game and the Tower of Hanoi, 7 Curious Topological Models, 8 The Game of Hex, 9 Sam Loyd: America’s Greatest Puzzlist, 10 Mathematical Card Tricks, 11 Memorizing Numbers, 12 Nine More Problems, 13 Polyominoes, 14 Fallacies, 15 Nim and Tac Tix, 16 Left or Right? References for Further Reading


————. 2. More Mathematical Puzzles and Diversions. Penguin Books 1966. First edition 1962 (19 November 1994)

1 The Five Platonic Solids, 2 Tetraflexagons, 3 Henry Ernest Dudeney: England’s Greatest Puzzlist, 4 Digital Roots, 5 Nine Problems, 6 The Soma Cube, 7 Recreational Topology, 8 Phi: The Golden Ratio, 9 The Monkey and the Coconuts, 10 Mazes, 11 Recreational Logic, 12 Magic Squares, 13 James Hugh Riley Shows, Inc., 14 Nine More Problems, 15 Eleusis: The Induction Game, 16 Origami, 17 Squaring the Square 18 Mechanical Puzzles, 19 Probability and Ambiguity, 20 References for Further Reading

Of note is the Dudeney reference, of June 1958.


————. 3. New Mathematical Diversions from Scientific American (1966) London George Allen and Unwin Ltd., 1969 (12 October 2012)

(Full title is Martin Gardner’s New Mathematical Diversions from Scientific American; cover and title page differ)

1 The Binary System, 2 Group Theory and Braids, 3 Eight Problems, 4 The Games and Puzzles of Lewis Carroll, 5 Paper Cutting, 6 Board Games, 7 Packing Spheres, 8 The Transcendental Number Pi, 9 Victor Eigen: Mathemagician, 10 The Four-Color Map Problem, 11 Mr. Apollinax Visits New York, 12 Nine Problems, 13 Polyominoes and Fault-Free Rectangles, 14 Euler’s Spoilers: The Discovery of an Order-10 Graeco-Latin Square, 15 The Ellipse, 16 The 24 Color Squares and the 30 Color Cubes, 17 H.S.M. Coxeter, 18 Bridg-it and Other Games, 19 Nine More Problems, 20 The Calculus of Finite Differences

Of core interest: 17 H.S.M. Coxeter, with use of Escher's works: Horseman, Two Birds, Verbum


————. 4. The Numerology of Dr. Matrix (columns 1-7, 1967; expanded 1976 with columns 8-18 as The Incredible Dr. Matrix; expanded 1985 with columns 19-22 as The Magic Numbers of Dr. Matrix) Charles Scribner’s and Sons, 1976 (7 November 2012)

1 New York, 2 Los Angeles, 3 Sing Sing, 4 Lincoln and Kennedy, 5 Chicago, 6 Miami Beach, 7 Philadelphia, 8 Pi, 9 Wordsmith College, 10 Squaresville, 11 Left Versus Right, 13 Fifth Avenue, 14 The Moon, 15 Honolulu, 16 Houston, 17 Clairvoyance Test, 18 Pyramid Lake, [and later, 1985 edition] 19 The King James Bible, 20 Calcutta, 21 Stanford, 22 Chautauqua, 23 Istanbul, Answers and Commentary

All on numerology; a major disappointment! I was expecting other articles with the Dr Matrix columns, as with other books in which an initial  title is insinuated. I have the second edition, The Incredible Dr. Matrix.


————. 5. The Unexpected Hanging and Other Mathematical Diversions. (1969; UK Further Mathematical Diversions) Simon and Shuster 1969 (14 June 2011)

1 The Paradox of the Unexpected Hanging, 2 Knots and Borromean Rings, 3 The Transcendental Number e, 4 Geometric Dissections, 5 Scarne on Gambling, 6 The Church of the Fourth Dimension, 7 Eight Problems, 8 A Matchbox Game-Learning Machine, 9 Spirals, 10 Rotations and Reflections, 11 Peg Solitaire, 12 Flatlands, 13 Chicago Magic Conventions, 14 Tests of Divisibility, 15 Nine Problems, 16 The Eight Queens and Other Chessboard Diversions, 17 A Loop of String, 18 Curves of Constant Width, 19 Rep-Tiles: Replicating Figures on the Plane, 20 Thirty-Seven Catch Questions, Bibliography

Of most interest: Geometric Dissections, pp. 43-51 and Rep-tiles Replicating Figures on the Plane, pp. 222-233


————. 6. Martin Gardner's Sixth Book of Mathematical Diversions from Scientific American (W. H. Freeman and Co, 1971) (24 December 2011)

1 The Helix, 2 Klein Bottles and Other Surfaces, 3 Combinatorial Theory, 4 Bouncing Balls in Polygons and Polyhedrons, 5 Four Unusual Board Games, 6 The Rigid Square and Eight Other Problems, 7 Sliding-Block Puzzles, 8 Parity Checks, 9 Patterns and Primes, 10 Graph Theory, 11 The Ternary System, 12 The Trip around the Moon and Seven Other Problems, 13 The Cycloid: Helen of Geometry, 14 Mathematical Magic Trick, 15 Word Play, 16 The Pythagorean Theorem, 17 Limits of Infinite Series, 18 Polyiamonds, 19 Tetrahedrons, 20 Coleridge's Apples and Eight Other Problems, 21 The Lattice of Integers, 22 Infinite Regress, 23 O'Gara, the Mathematical Mailman, 24 Op Art, 25 Extraterrestrial Communication

22 Infinite Regress has Escher’s ‘Drawing Hands’ print p. 224, and is mentioned in passing, p. 223


————. 7. Mathematical Carnival. Pelican Books (1977) 1978. (Undated c. late 1990?) Hardback

1 Sprouts and Brussels Sprouts, 2 Penny Puzzles, 3 Aleph-Null and Aleph-One, 4 Hypercubes, 5 Magic Stars and Polyhedrons, 6 Calculating Prodigies, 7 Tricks of Lightning Calculators, 8 The Art of M.C. Escher, 9 The Red-Faced Cube and Other Problems, 10 Card Shuffles, 11 Mrs Perkins' Quilt and Other Square-Packing Problems, 12 The Numerology of Dr. Fliess, 13 Random Numbers, 14 The Rising Hourglass and Other Physics Puzzles, 15 Pascal's Triangle, 16 Jam, Hot and Other Games, 17 Cooks and Quibble-Cooks, 18 Piet Hein's Superellipse, 19 How to Trisect an Angle, Bibliography


————. 8. Mathematical Magic Show. Viking (1977) 1984 (26 May 2001)

1 Nothing, 2 More Ado About Nothing, 3 Game Theory, Guess It, Foxholes, 4 Factorial Oddities, 5 The Cocktail Cherry and Other Problems, 6 Double Acrostics, 7 Playing Cards, 8 Finger Arithmetic, 9 Möbius Bands, 10 Ridiculous Questions, 11 Polyhexes and Polyaboloes, 12 Perfect, Amicable, Sociable, 13 Polyominoes and Rectification, 14 Knights of the Square Table, 15 The Dragon Curve and Other Problems, 16 Colored Triangles and Cubes, 17 Trees, 18 Dice, 19 Everything, Bibliography

16 is on MacMahon.


————. 9. Mathematical Circus. Optical illusions! Games, puzzles, paradoxes. (1979) Hardback (23 December 1995)

1 Optical Illusions, 2 Matches, 3 Spheres and Hyperspheres, 4 Patterns of Induction, 5 Elegant Triangles, 6 Random Walks and Gambling, 7 Random Walks on the Plane and in Space, 8 Boolean Algebra, 9 Can Machines Think?, 10 Cyclic Numbers, 11 Eccentric Chess and Other Problems, 12 Dominoes, 13 Fibonacci and Lucas Numbers, 14 Simplicity, 15 The Rotating Round Table and Other Problems, 16 Solar System Oddities, 17 Mascheroni Constructions, 18 The Abacus, 19 Palindromes: Words and Numbers, 20 Dollar Bills, Bibliography


————. 10. Wheels, Life and Other Mathematical Amusements (1983). W. H. Freeman and Company. Hardback (18 August 2011)

1 Wheels, 2 Diophantine Analysis and Fermat's Last Theorem, 3 The Knotted Molecule and Other Problems, 4 Alephs and Supertasks, 5 Nontransitive Dice and Other Probability Paradoxes, 5 Geometrical Fallacies, 6 The Combinatorics of Paper Folding, 7 A Set of Quickies, 8 Ticktacktoe Games, 9 Plaiting Polyhedrons, 10 The Game of Halma, 11 Advertising Premiums, 12 Salmon on Austin's Dog, 13 Nim and Hackenbush, 14 Golomb's Graceful Graphs, 15 Charles Addams' Skier and Other Problems, 16 Chess Tasks, 17 Slither, 3X+1, and Other Curious Questions 18 Mathematical Tricks with Cards, 19 The Game of Life, Part I, 20 The Game of Life, Part II, 21 The Game of Life, Part III


————. 11. Knotted Doughnuts and Other Mathematical Entertainments 1986. W. H. Freeman and Company. (8 January 2013)

1 Coincidence, 2 The Binary Gray Code, 3 Polycubes, 4 Bacon's Cipher, 5 Doughnuts: Linked and Knotted, 6 The Tour of the Arrows and Other Problems, 7 Napier's Bones, 8 Napier's Abacus, 9 Sim, Chomp and Racetrack, 10 Elevators, 11 Crossing Numbers, 12 Point Sets on the Sphere, 13 Newcomb's Paradox, 14 Reflections on Newcomb's Paradox, 15 Reverse the Fish and Other Problems, 16 Look-See Proofs, 17 Worm Paths, 18 Waring's Problems, 19 Cram, Bynum and Quadraphage, 20 The I Ching, 21 The Laffer Curve


————. 12 Time Travel and Other Mathematical Bewilderments 1988 (11 October 2011)

1 Time Travel, 2 Hexes and Stars, 3 Tangrams, Part 1, 4 Tangrams, Part 2, 5 Nontransitive Paradoxes, 6 Combinatorial Card Problems, 7 Melody-Making Machines, 8 Anamorphic Art, 9 The Rubber Rope and Other Problems, 10 Six Sensational Discoveries, 11 The Császár Polyhedron, 12 Dodgem and Other Simple Games, 13 Tiling with Convex Polygons, 14 Tiling with Polyominoes, Polyiamonds, and Polyhexes, 15 Curious Maps, 16 The Sixth Symbol and Other Problems, 17 Magic Squares and Cubes, 18 Block Packing, 19 Induction and Probability, 20 Catalan Numbers, 21 Fun with a Pocket Calculator, 22 Tree-Plant Problems

Of note is that this highlighted contains extended Cairo references p.176, and includes a little extra to the text per se , with It underlies… p. 171 (the original article in Scientific American contained just three) and Gardner’s enigmatic quote of street tiling and unsubstantiated claim of mosaics of Moorish building. Dunn’s reference was included, from which he is likely taking from.

Also of interest is his Chapter 7 on speculations as to ‘melody making machines’, of a mechanical procedure of producing music, that can in theory be applied to tiling life-like tessellations.


————. 13. Penrose Tiles to Trapdoor Ciphers. W. H. Freeman and Company 1989 First edition 1989 (10 November 2007)

1 Penrose Tiling, 2 Penrose Tiling II, 3 Mandelbrot's Fractals, 4 Conway's Surreal Numbers, 5 Back from the Klondike and Other Problems, 6 The Oulipo, 7 The Oulipo II, 8 Wythoff's Nim, 9 Pool-Ball Triangles and Other Problems, 10 Mathematical Induction and Colored Hats, 11 Negative Numbers, 12 Cutting Shapes into N Congruent Parts, 13 Trapdoor Ciphers, 14 Trapdoor Ciphers II, 15 Hyperbolas, 16 The New Eleusis, 17 Ramsey Theory, 18 From Burrs to Berrocal, 19 Sicherman Dice, the Kruskal Count and Other Curiosities, 20 Raymond Smullyan's Logic Puzzles, 21 The Return of Dr. Matrix, Name Index


————. 14. Fractal Music, Hypercards and More…. W. H. Freeman and Company 1992 (7 February 2013)

1 White, Brown and Fractal Music, 2 The Tinkly Temple Bells, 3 Mathematical Zoo, 4 Charles Sanders Peirce, 5 Twisted Prismatic Rings, 6 The Thirty Color Cubes, 7 Egyptian Fractions, 8 Minimal Sculpture, 9 Minimal Sculpture II, 10 Tangent Circles, 11 The Rotating Table and Other Problems, 12 Does Time Ever Stop? Can the Past Be Altered? 13 Generalized Ticktacktoe, 14 Psychic Wonders and Probability, 15 Mathematical Chess Problems, 16 Douglas Hofstader's Gödel, Escher, Bach, 17 Imaginary Numbers, 18 Pi and Poetry: Some Accidental Patterns 19 More on Poetry, 20 Packing Squares, 21 Chaitin's Omega

6 is on MacMahon and his cube puzzles.


————. 15. The Last Recreations. Copernicus An imprint of Springer-Verlag 1997 (26 March 2013)

1 The Wonders of a Planiverse, 2 Bulgarian Solitaire and Other Seemingly Endless Tasks, 3 Fun with Eggs, Part I, 4 Fun with Eggs, Part II, 5 The Topology of Knots, 6 M-Pire Maps, 7 Directed Graphs and Cannibals, 8 Dinner Guests, Schoolgirls, and Handcuffed Prisoners, 9 The Monster and Other Sporadic Groups, 10 Taxicab Geometry, 11 The Power of the Pigeonhole, 12 Strong Laws of Small Primes, 13 Checker Recreations, Part I, 14 Checker Recreations, Part II, 15 Modulo Arithmetic and Hummer's Wicked Witch, 16 Lavinia Seeks a Room and Other Problems, 17 The Symmetry Creations of Scott Kim, 18 Parabolas, 19 Non-Euclidean Geometry, 20 Voting Mathematics, 21 A Toroidal Paradox and Other Problems, 22 Minimal Steiner Trees, 23 Trivalent Graphs, Snarks, and Boojums


————. (editor.) Mathematical Puzzles of Sam Loyd. Dover Publications, Inc., New York 1959. (30 April 1994). Selected (from Loyd’s 1914 work Cyclopaedia of Puzzles), and edited by Martin Gardner, with his own introduction.

Typical Loyd fayre.


————. (editor) More Puzzles and Curious Problems. Henry E. Dudeney. Fontana Books 1970. First published in Great Britain by Souvenir Press under the title ‘536 Puzzles and Curious Problems’. (19 July 1992)


————. (editor) Puzzles and Curious Problems. Henry E. Dudeney Fontana Books 1970. (First published in Great Britain by Souvenir Press under the title ‘536 Puzzles and Curious Problems’) (8 August 1993)


————. The Ambidextrous Universe. Left, Right, and the Fall of Parity. Penguin Books 1970. (14 June 1995)

Many aspects of interest (albeit largely outside of tessellation), too numerous to list. Especially see Chapter 4, Magic, of a wordplay nature.


————. The Annotated Alice. Penguin Books 1970 revised edition. First published 1960

(5 June 2013)


————. Codes, Ciphers and Secret Writing. Dover Publications Inc, New York. 1984, Unabridged and unaltered republication of the work first published by Simon & Shuster, Inc, New York, 1972 (23 August 1994)

Popular account. Of note is Thomas Jefferson’s wheel cipher invention, p. 59.


————. Puzzles from Other Worlds. Fantastical brainteasers from Isaac Asimov’s Science Fiction Magazine. Oxford University Press. 1989 (24 October 1998)


Gardner, Martin. Gardner’s Whys & Wherefores. Oxford University Press 1990 (5 October 1996).

Mostly philosophical speculations. Pentominoes pp. 92-93.


————. More Mathematical Puzzles of Sam Loyd. Dover Publications, Inc., New York 1960. (30 April 1994)


————. Science Magic. Martin Gardner’s Tricks & Puzzles. Sterling Publishing Co., Inc. 1997 (not dated, c. 5 years ago)



————. Undiluted Hocus-Pocus. The Autobiography of Martin Gardner. With a foreword by Persi Diaconis and afterword by James Randi. Princeton University Press, 2013 (16 June 2018)

Pleasant coffee time reading. Gives a good story of the great man in the round. A few snippets that I was unaware of: his association with Salvador Dali. No Escher, perhaps surprisingly. Mathematically, of most interest is Chapter 15, pp. 134-149, with his association with Scientific American. I was less than enamoured with the practical joke supposed humour of his friend Bob Murray.


Garfunkel, Solomon. For all Practical Purposes. Introduction to Contemporary Mathematics. (COMAP) W. H. Freeman and Company Third edition1994 (First edition 1988). (30 April 1994)

Various aspects of mathematics, most outside of my interest (and understanding). Popular level, of 16-year-old. Probably best described as a compilation from other sources. However, scattered throughout are various tiling matters and ‘spotlights’/biographies, such as Angels and Devils. pp. 642-643. Stanford teapot p. 647. Reference to par hexagon, pp. 701, 716. Of most interest are Chapter 21, on Symmetry and patterns, and Chapter 22, Tilings pp. 693-722. Includes Escher-like tilings, Marjorie Rice, Penrose tiles, Quasicrystals. Various colour plates with a tiling theme, Penrose, Escher’s works, Hyperbolic tilings, Marjorie Rice.


Geary, A. and H. V. Lowry, H. A. Hayden. Mathematics for Technical Students Part One. Longman, Green and Co. 1954. First published 1938 (21 June 1992)

Typical generic maths text book of the day, one of many that I have; simply, one would have sufficed. I do not believe that I have used this in any way; the material being mostly beyond my understanding. Very much of its day, with calculation to the fore, with chapters on arithmetic, algebra, geometry, mensuration and trigonometry. Reference to the dissection of square to rectangle paradox of 64 and 65 units, p. 167.


Gellart, W et al. The VNR Concise Encyclopedia of Mathematics. Van Nostrand Reinhold Company 1977 First saw c. 1986 (27 August 1997)

Principia-esque! Ironically, one of the first maths books I ‘studied’! c. 1986.


Gerston, Judith (Series Editor) The Human Body [series] The Eye Window to the World. Torstar Books Inc. 1984 (2 August 2014).

Although obviously not strictly a maths book (A part work on the human body, with here eye), included here as Escher is featured p. 125 Other World, and pp. 140-141, Convex and Concave and with an essay (author unknown) ‘M. C. Escher Impossible Worlds’, albeit nothing of significance. Escher print is also featured in ‘Brain’ in the series, not obtained.


Gettings, Fred. The Meaning and Magic of Art. Paul Hamlyn Ltd. 1963 (18 April 2015)

Although a book on art rather than mathematics, included as it has many crossover references on mathematical matters, such to the golden section, notably pp. 36-43, but of the usual nonsensical type. Snowflakes, spirals and curves pp. 64-65. Also, analysis of pictures by overlaying of lines without any foundation whatsoever. Ideally requires rebutting, but I have no time for now.

Of note is that it can be seen that Mottershead shamefully appropriates Gettings’ diagrams on (p. 128 of Sources…’) without any mention of Gettings!


Ghyka, Matila. The Geometry of Art and Life. Dover Publications, Inc. New York 1977. First published 1946 (30 April 1994)

A brief chapter on tiling, Chapter 5, of which a mistake is made re demi regular tilings, as noticed by Grünbaum. The book is somewhat curious, with many instances of picture analysis based on the golden section. I remain to be convinced (as with other books, such as Mario Livio, pp. 167-168) that the artist set out with this intention (and of other ‘harmonic division’, e.g. plate LXX). Far too much wishful thinking is involved, with lines chosen as to the artists’ interpretation as regards ‘best fit’ (or none at all as far as I can see in plate LXX!). Of no real interest.


Gibbons, Stanley. Stanley Gibbons Stamp Catalogue Part 4 Benelux. 5th edition, 2003. (7 December 2013)

Although this cannot in any way be described as a maths book, and indeed a book itself, being of a catalogue, I nonetheless include here. The reason for its inclusion is that two of Escher’s stamps are shown, on pp. 309 and 371, of the Netherlands Antilles and Suriname respectively. However, there is little else by means of detail, albeit an exact date of issue is given i.e. day and month, which was previously unknown, although in itself this is of no consequence. Note that Part 4 reference to a 22-volume set; and is not of a series of the Benelux as might otherwise be imagined by the title.


Gibbons G. W., E. P. S. Shellard, S. J. Rankin. The Future of Theoretical Physics and Cosmology: Celebrating Stephen Hawking (Google excerpt) (15 June 2015)

Occasional Escher illustrations; 55-?


Gibson, Walter B. Magic with Science. William Collins & Co. Ltd, 1970. (9 January 2016)

Although of a science premise of a children’s book, is included here as it has a small chapter on recreational mathematics: ‘Geometrix (sic) ‘Tricks Involving Geometrical Principles’, pp. 107-114. Included are Mobius strips and lost line, and Hooper’s cut. (Hooper is mentioned in the introduction).


Gill, George; publisher. (Author and date published oddly not stated; c. 1900?). Gill’s New School of Geometry. George Gill and Sons, Minerva House, Warwick Lane, E.C. (9 July 1994)

Subtitled practical plane and solid geometry. Typical generic geometry text book of the day, one of many that I have; simply, one would have sufficed. The reason for obtaining the book was to be able to look up any geometric construction as and if required, but I do not believe that I have used this in any way. Geometrical tracery, pp. 111-115. Minor tilings p. 111.


Gjerde, Eric. Origami Tessellations. Awe-Inspiring Geometric Designs. A. K. Peters Ltd, 2009

Complimentary copy from A. K. Peters for using my Pólya bird bird tessellation, as an ‘overview’, p. 2. (From the Leeuwarden 2008 Bridges art exhibit)


Glendinning, Paul. Maths in Minutes. 200 Key Concepts Explained in an Instant. Quercus, 2012 (20 August 2017)

Small-format book, of a pleasing, coffee-time reading nature. However, most of the concepts are beyond my understanding. Disarmingly, for someone of Glendinning’s stature, a professor of applied mathematics, he is one of many with a fallacious belief of the Golden Ratio appearing in the Parthenon, p. 37. His own example is particularly excruciating. Further, the often seen Nautilus shell associated with the Golden Ratio features on the cover, of which this is seemingly implied, although, oddly, is not discussed in the book. Has a chapter on Geometry, pp.108-162, with tessellations pp. 148-149, Penrose tilings pp.150-151.


Glenn, Robert. Foundation Maths. For GCSE and Standard Grade. Heinemann Educational Books Ltd 1988 First saw c. December 1990, a date of study (15 January 2001)

Textbook,12-year- old target audience. Escher's swan outline used p. 49, unaccredited. Pattern, tessellation pp. 115, 117, 197-198 barely worth mentioning.


Glenn, William H. and Donovan A. Johnson. The Theorem of Pythagoras. Exploring Mathematics on Your Own. Volume 4. John Murray, First published in Great Britain 1964, Reprinted in 1965 (24 October 1998 and 22 October 2005) Two books

Small format hardback, 48 pp. Volume 4 of a 17-book series (none ostensibly on tessellation) of what is clearly intended for juvenile audience, although not stated as such. Without a doubt, readable, and ideally I would re-read, as for me at least it gives a splendid introduction to the Pythagoras Theorem, with numerous illustrations, although at times the mathematics is beyond me (and to an extent of interest too). Obtained in conjunction with volume 3, at a sale.


————. Number Patterns Exploring Mathematics on Your Own 3. John Murray 1964 (24 October 2005)

Juvenile, advanced.


Gleick, James. Chaos. Making a New Science. Sphere Books Ltd. 1990 (21 July 1996)


Goldberg, Kenneth P. Learning Commodore 64 Logo Together. An Activity Book for Creative Parents, Teachers, and Kids. Microsoft Press 1984. (21 February (1998?)

Early days of computing, and so all rather dated. Nothing of any interest now. Of most note (relatively) is a small subchapter on ‘Polygon Patterns’, pp. 142-152, with simple geometrical drawings and occasional tilings pp.150, 152.


Goodstein, R. L. Fundamental Concepts of Modern Mathematics. Pergamon Press 1964 (31 October 1996)

Of very limited interest. Chapter 5, Networks and maps (topology) pp. 241-268.


Golomb, Solomon W. Polyominoes. Puzzles, Patterns, Problems, and Packings. Revised and expanded second edition. Princeton University Press 1994 (2 February 1998). Original edition 1965

The bible of polyominoes; not that I’ve done much with it!

Gombrich, E. H. The Story of Art. First Published 1950. 1972 Phaidon Press (9 October 2005) 498 pp.

Gombrich, a noted art historian, has written many landmark books, including some of direct interest as to mathematics and Escher, of which it is thus not always easy to recall specifics. Therefore, I include 'all' here, even though not all bear any relation to mathematics and Escher, as indeed this is an instance. I checked the contents and index for any possible direct interest, of which there is essentially nothing. Much time could be wasted upon a casual browse through so many  pages, and so I thus refrain from further investigation.

————. Art and illusion: a study in the psychology of pictorial representation: the A. W. Mellon lectures in the fine arts, 1956, National Gallery of Art, Washington by Gombrich, E. H. (Ernst Hans), 1909-

I can’t remember if I have this; perhaps I am confusing it with other books by Gombrich.


————. Meditations on a Hobby Horse and Other Essays on the Theory of Art. New York: Phaidon, 1963

Illusion and Visual Deadlock, pp. 151-158. Many Escher references and illustrations in the chapter. Originally published under the title ‘How to Read a Painting’ in the Adventures of the Mind, series of the Saturday Evening Post, July 29, 1961. Note that Escher’s Horseman tessellation is used for the cover of a subsequent later edition.


————. The Sense of Order: A Study in the Psychology of Decorative Art. Second edition, Phaidon Press Limited, Second Printing 1980 (Date unknown, c. 2005) First published 1979 (First saw 1988, Grimsby Art School library)

Has occasional tessellations aspects, but this book continually flatters to deceive; it’s more of ornament in the broader sense than tessellation. Many aspects of interest. Has Escher boat and fish p. 89, Escher-like tessellation by an unknown Japanese, Michio Kubo, dated 1968 on p. 91. Frequent occurrence is the term ‘counterchange’ applied to any black and white tilings. I much prefer my own usage! The book has an excellent bibliography, with many books not commonly mentioned, most of which are worthy of following up. Gombrich is seemingly the popularizes, following up Stuart Durant (the circumstances of which is not detailed), of Douat’s Truchet tiling follow-up, pp.70-72.


Goodfellow, Caroline. Games & Puzzles. The Collector’s Guide to Indoor Games from The 1700s to the Present Day. Eagle Editions Ltd, First  edition 2002 (24 November 2018)

A most pleasing scholarly approach, of a popular level. 128 pages. Of most interest is Chapter 7 ‘The Early Jigsaw’, pp. 110-117. Other chapters remain of interest, indirectly.

Previously, I was unaware of Goodfellow. I see that she has written a variety of bygone games, dolls, toys type books, and of which I see that she was previously curator of dolls and toys at the Victoria and Albert Museum, and a member of Board Game Studies, an international society of experts on board games, a body of which, again, I was unfamiliar! Likely now that I am familiar with her name I will chance upon it upon game book reading.


Gorini, Catherine A. The Facts on File Geometry Handbook. 2003, 2009 revised edition. Facts on File Inc, and imprint of Infobase publishing

Cairo tiling illustrated p. 22, equilateral. Gives the following definition: Cairo tessellation: A tessellation of the plane by congruent convex equilateral pentagons that have two nonadjacent right angles; so called because it can be found on streets in Cairo.


Graham, Duncan; Graham, Christine. Mathematics GCSE. 1987 Revision book.

Tessellation barely mentioned; just one line.


Green, Patrick. Seeing is Believing. Vineyard Books 1996 (27 January 2007)

Juvenile. Escher’s House of Stairs p. 34. The Escher reference, a single picture with no text is so unimportant to be barely worth mentioning. Indeed, ‘Escher’ per se does not get a mention; the book shows just his print!


Gregory, Richard L. The intelligent eye. Weidenfeld & Nicolson, 1970 (18 August 2015)

Minor use of Escher's pictures, Waterfall and Belvedere, pp. 52-53 to illustrate paradoxes of depth, with a brief commentary, of no particular insight. Of note is that this book was first seen in 1987, likely in college library.


Gregory, Richard L & E. H. Gombrich (eds.). Illusion in Nature and Art. Duckworth, First published 1973 (4 May 2017) First saw 1987

Schattsneider reference in Visions. Of general interest. Six scholarly psychology-led chapters, with contributions from Colin Blakemore, R. L. Gregory, H. E. Hinton, Jan B. Deregowski, E. H. Gombrich and Roland Penrose. The essays are a little obscure, of which time spent studying ‘in depth’ would be disproportionate as to worth.

Purposefully latterly obtained (2017) as part on my ongoing 1987 review, as this was studied in 1987, the essence of the book being long forgotten. Minor, inconsequential Escher references, as regards impossible objects rather then tessellations, of just a few lines (no illustrations) pp. 86, 280. Skim read.

The book has as its origin the setting up an exhibition initiated by Sir Roland Penrose, at the Institute of Contemporary Arts, London. Of note is p. 207 and Bust of Voltaire by Houdon, and the ‘projecting eye’ ruse.


Gregory, Richard L. Eye and Brain.

Small format paperback.


Gribbin, John and Mary Gribbin. Big Numbers. Wizard Books, 2003 (no date)

On numbers in science, rather than a mathematics book per se. Popular account. Occasional browse. P. 68 explains why toadstools have long stalks, the reasons of which I didn’t know before!


Grünbaum, B. and G. C. Shephard. Tilings and Patterns. W. H. Freeman and Company New York, 1987 (11 January 1993; first saw in 1989)

The bible of tiling and mind boggling in its depth! Indispensable, although much is way too advanced for me. Largely, indeed overwhelmingly, academic, but still accessible on occasion. Cairo-esque p. 480, as part of the 24 polygonal isohedral types of proper tilings by pentagons. And much more beside!


Greer, A. A Complete GCSE Mathematics Higher Course. Stanley Thornes (Publishers) Ltd. 1989 (15 October 1995).

Textbook. Tessellation pp. 297-300, very basic, barely worth mentioning.

Guinness Word Records. Guinness World Records Limited. 2002. (17 December 2016)
Although not a maths book per se, obtained as Götz-Peter Reichelt’s cluster puzzle Noah’s Ark is mentioned (and illustrated), although oddly no reference is made to the interlocking premise.

Gullberg, Jan. Mathematics From the Birth of Numbers. W. W. Norton & Company, New York London (7 August 2016, first saw in Grimsby library, at least 2001)
A weighty tome, of 1093 pages! Minor reference to tessellation, p. 395 (albeit with poor quality diagrams) and Escher, p. 375.

Gunther, S. Vermischte Untersuchungen zur Geshichte der Mathematischen Wisssenschaft. Leipzig, 1876. (Downloaded from GDZ site 29 April 2015)
From a reference in Bradley. Somewhat of a let-down; no tiling. Mostly of an academic nature, text heavy, with occasional geometric diagrams throughout the first part of book, and polyhedra pp. 36-37, with Kepler references. Of no practical use.

Guy, Richard K. and Robert W. Woodrow (Editors). The Lighter Side of Mathematics. Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics, 1984.

MAA Spectrum, 1994. (18 January 2012)

In three main parts: 1 Tiling and Colouring, 2 Games and Puzzles, 3 People and Pursuits. Many aspects referring to tiling and Escher in Part 1. Of special note:

Escher: A Mathematician In Spite of Himself, Doris Schattschneider (first appeared in Structural Topology, 1988)

Fun with tessellations, John Rigby

Escheresch, Athelstan Spilhaus

Henry Ernest Dudeney: Britain’s Greatest Puzzlist, Angela Newing (has much detail on Dudeney not previously published)

The Utility of Recreational Mathematics, David Singmaster

Puzzles Old & New: Some historical Notes

Has Escher bird tiling on front cover Locher 361A, April 1949




Hall. Dorothea (ed). Memories of Childhood. Chartwell Books Inc 1990 (26 June 2016)


Hambidge, Jay. The Elements of Dynamic Symmetry. Dover Publications, Inc. New York first published by Dover 1967, a reprint of 1926 edition (30 April 1994)

I don’t quite know what to make of this book. It gives a lot of ‘dynamic symmetrical’ constructions involving squares and rectangles, but I largely remain to be convinced of its efficacy. I recall someone somewhere describe Hambidge as a crank. Indeed, Mario Livio for one is of this opinion, see p. 171 in which he largely discredits his work, or at least implies this. Whatever, the book is of limited appeal. No tessellation.

Hagen, Victor W von. The Roads That Led To Rome. Weidenfeld and Nicolson, 1967 (15 June 2019)

Of pavement interest. A dedicated, authoritative account, 288 pp. Very nice indeed, of which a re-read would be ideal, given that much here was new to me. Has references to Ostia, but no pictures of the ‘gorseciki’ paving (I forget where I first became aware of this)

Wikipedia: Victor Wolfgang von Hagen (St-Louis, Missouri, United States, February 29, 1908 – Italy, March 8, 1985) was an American explorer, archaeological historian, anthropologist, and travel writer who traveled in South America with his wife (Christine, later Sylvia). Mainly between 1940 and 1965, he published a large number of widely acclaimed books about the ancient people of the Inca, Maya, and Aztecs.

Hanby, G. A. Geometry I. (First saw 19-20 March 1986)


Hand, William. ‘Scientific Mysticism’ in Rosicrucian Heritage No. 1 2005. (9 June 2015)

Use of Escher's print Print Gallery, p. 21; no other mention of Escher in article.

Hann, Michael A. and Briony G. Thomas. Patterns of Culture: Decorative Weaving Techniques. No. 36 in the Ars Textrina series, published in association with the University of Leeds International Textiles Archive (ULITA) as an accompaniment to the exhibition ‘Patterns of Culture - Decorative Weaving Techniques’. Foreword by D. Holdcroft. 80 pp. (25 June 2019) PDF

Hann and Thomas have written many books/monographs together. Of weave interest. Skimmed read the PDF. Herringbone in Cairo, 4 BCE, p. 47. To what extent there is original research here is unclear. Relatively sparsely illustrated. No houndstooth. A detailed bibliography.

Hannas, Linda (Introduction). Two Hundred Years of Jigsaw Puzzles. Exhibition catalogue of 1968 at the London Museum, 40 pages, with introduction by Linda Hannas (19 November 2016)

Jigsaw Puzzle interest. Slim booklet of 40 pages, written for the London exhibition of 1968, with much input by Hannas. A speculative purchase, being a commonly-quoted book in jigsaw puzzle circles in the hope of detail of direct cluster puzzle interest, of which there is indeed one of note, namely an entry for Mrs Elspeth Eagle-Clarke, p. 37, albeit without a picture. However, disarmingly, two mistakes are made in the text, with ‘Miss’ rather than Mrs and ‘Clark’ rather than Clarke. Another mistake is on p. 10, where the previous eagle-eyed buyer had noticed an incorrect date on a John Wallis publication and duly corrected, not 1768 but rather 1788. Of note is the caption:

66. Dragon’s Land 1934 Manufactured by Chad Valley Co Ltd.

17 x 15½ in. Colour print of design by Miss [sic] Elspeth Eagle-Clark [sic]. Each piece is a picture in itself dovetailing into a complete design London Museum 67.92/2. An original picture of 1930 mounted on plywood in 1967 by the same craftsman at the Chad Valley Works who cut the prototype in 1934’.

The last sentence is full of ambiguity as to meaning. I tried to resolve this with Anne Williams, but to no avail.

Eagle-Clarke aspects aside, the book is full of historical aspects of interest.


————. The English Jigsaw Puzzle 1760-1890. Wayland Publishers, London, 1972 (22 October 2014)

Jigsaw puzzle interest. Obtained primarily in relation to possible interest regarding cluster puzzles, this being a commonly-quoted book in jigsaw puzzle circles. As such, for my specific purposes, somewhat of a let down; there is nothing cluster puzzle-related, not that I was really expected anything in this field. But one never knows…. As such, it consists mostly of text, with relatively few pictures. I think it would have been improved by more. However, as regards its true purpose, of a historical account, then it is indeed ideal, and indeed a pioneering work of outstanding scholarship. Indeed, Hannas must be lauded for her quite outstanding research. That of John Spilsbury is quite outstanding.

As an aside, perhaps of most note is an illusion, plate 14, titled ‘Before and after Marriage’ of 1789, of two heads that when turned upside down resemble another picture. This needs investigating the historical aspect; I cannot recall having seen this before.

The site shows a later version of this, of 1884.


————. The Jigsaw Book. Celebrating two centuries of jigsaw-puzzling round the world. Bellew & Higton Publishers, 1981. (16 January 2016)

Jigsaw puzzle interest. Although not of a mathematical nature, included as regards my investigations into cluster puzzles, and the author being of note per se in the jigsaw community. Likely a purposefully, more popular account than her more serious books. Relatively lightweight, of just 91 pages. Nothing at all in the way of cluster puzzles. However, of sight interest is p. 91, where a puzzle has been cut into a tessellation premise of a broad single tile. Also of indirect interest is a generic Hamley Brothers puzzle ‘Society Dissected Picture Puzzle’ label, p. 18, although this does not appear to have been captioned or discussed. Also has ‘Before and after Marriage’ of 1789, p.11.


Harbin, Robert. Origami 1. The Art of Paper-Folding. Coronet Books, 1974. First printed 1968 by Teach Yourself Books as Teach Yourself Origami (15 August 1993)

The first of a three-book series, all of a like nature, with a brief introductory discussion of a few pages, followed by diagrammatic instructions. Only of minor interest, in passing, and not studied as such. As such, there is nothing here overtly mathematical, but I include here nonetheless, as paper folding can loosely be regarded as ‘mathematical’ in nature.


————. Origami 2. The Art of Paper-Folding. Previously published as More Origami. Coronet Books, Eleventh impression 1975 (26 June 1994)

As detailed above.


————. Origami 3. The Art of Paper-Folding. Coronet Books, Fourth impression 1975 (26 June 1994)

As detailed above.


Hargittai, István; Hargittai, Magdolna. Symmetry A Unifying Concept. Shelter Publications Inc. 1994 (10 August 2006)

Popular account of symmetry, very pleasing. Escher pp. 191-192, 207. Fish and Boats, E113; Bird and Fish E115; Bat, Bird, Bee, Butterfly 81; Bulldogs E97; Pegasus E105. ‘Japanese Cairo’ tiling p. 174.

Harpe, Pierre de la. Quelques Problèmes Non Résolus en Géométrie Plane. L’Enseignement Mathématique, t 35 (1989), p. 227-243 (in French)

Cairo tiling page 232, likely taken from George Martin’s work, given that it is the same ‘unusual’ configuration.

c. late 2011?

Harris, Ella & Caroline Christin (eds). Puzzle Chest. Barnes & Noble Books, Sterling 2003. (5 July 2015)

A compilation of a series of Sterling books on a puzzle theme, of a juvenile audience. See p. 177 for ‘Jockeys on Ponies’, of a Loyd premise, p. 221 for possible Sam Loyd source of two donkeys, and p. 229 for a discussion of Schuster’s ‘three-stick clevis’. The book largely flatters to deceive. For instance, the Penrose tribar is used in many different trivial forms throughout. A typical illusion book in that well-known illusions are repeated without any fresh insight, or indeed novelty.


Harrison, E. P. Scottish Estate Tweeds. Johnstons of Elgin, First Edition, 1995 (14 December 2020)

Undoubtedly of the utmost importance.

Hatton, Richard G. Design. An Exposition of the Principles and Practice of the Making of Patterns. London Chapman and Hall Ld, 1902 (Internet book archive, 20 October 2015)

Downloaded upon a general search on the off chance that it may possibly contain tiling in some form. As such, not really; although it has loose elements, but nothing is entirely satisfactory. Probably the best chapter is pp. 149-165, of Lewis Day-esque, but I am not planning on revisiting this.


Hayman, Margaret. Essential Mathematics: A Modern Approach to CSE. Macmillan Education. 1979. (13 December 2000)



Heesch, H. and O. Kienzle. Flächenschluss. System der Formen lückenlos aneinanderschliessender Flachteile. Springer-Verlag, Berlin 1963 (in German) (2010) PDF

In German, 135 pages, somewhat hindered by a lack of translation. Seems so many diagrams of interest, but understanding them in a foreign language is the difficulty. Tilings pp. 1-3, 34-36, 52, 64-77, 80, 85-89, 98-107, 114-115, 120-129. No ‘true’ Cairo or pentagon studies, at least as far as I can make out. Quoted by Schattschneider. Schattschneider, p. 326, focuses on p? where Heesch shows his Set of 28, including the Wikipedia paving, no. 9, although not exact….. P. 68 shows the tile in detail.

Heesch, Heinrich. Reguläres Parkettierungsproblem (Regular tiling problem)1968 WANTED

————. Gesammelte Abhandlungen 1986. Translated: Heesch, Heinrich: Collected essays WANTED

Hemmings, Ray and Dick Tahta. Images of Infinity. Tarquin Publications 1992 (First pubished 1984, of Leapfrogs Group)  (3 June 1993)

Popular account of infinity, perhaps of a school-age audience, 96 pages. Oddly, the book does not have a contents or introduction, or indeed, any structure at all! Escher’s Circle Limit I, p. 14, albeit without explanation or caption! Pentagons p.57, Dart tessellation (in context of quadrilaterals) p. 72. Escher inspired ‘hand drawing hand’  pp. 3, 46.  Liberally illustrated, in black and white pen drawing, and early computer drawings, now somewhat dated. As such, the book is lacking, for reasons as outlined above. A pleasant enough read, but there is nothing of any real substance here.

Henderson, P. 'Functional geometry'. In Conference Record of the 1982 ACM Symposium on Lisp and Functional Programming, D.P. Friedman and D.S. Wise (Eds.). Pittsburgh, Pennsylvania, ACM Press, 1982, pp. 179–187. (2020)

On aspects of Square Limit procedures. Most impressive.


Hendler, Muncie. Infinite Design Allover Patterns. Dover Publications, Inc. New York. 1985 (15 October 1995)

Various tessellations, of 46 plates, in outline form. Of no consequence, being unstructured. Would appear to be intended as a child’s colouring-in book. Trivial.


Hendricks, Gordon. Eadweard Muybridge: The Father of the Motion Picture, Grossman Publishers, 1975 (First saw, or at least recorded, 9 August 1988, Grimsby central library)

A minor study, of 9 and 23 August 1988, of an indirect manner of Escher-like tessellation, in which I studied various animal’s outlines, as with this book, featuring horses, of a five-sheet study. However, this type of study is no longer active, and of which my interest essentially peaked and ended in 1988. The study, such as it was, consisting of photocopying horse motion photographs of interest and then assembling for easier viewing.
The book is largely long forgotten and although this is available on the internet archive, and at bookstores of an accessible price, as this particular study is no longer active I shall not pursue this.


Heritage, R. Learning Maths Book 1 (first saw 14 January 1988)

Has minor tessellation, with a novel design method, not fully understood, and two Escher-like tessellations of a cat’s head with gaps and a fish? showing no understanding of the issues. Much to my annoyance, I cannot now find details of this book online, at Bookfinder, or elsewhere. Likely this was a primary or secondary school oriented.

Hessemer, Friedrich Maximilian. Arabische und Alt-Italianische Bau-Verzierungen. Berlin, G. Reimer, 1842. Translated: 'Arabic and Old Italian Construction Embellishments'. (2013).

Has many plates of decided interest. Especially see fused pentagon. First drawn to my attention by Pail Tucker, 2013. Available on Internet Archive.

Heyden Van der, A. The Glory of Egypt. Wunderbares Ägyten / Les Splendeurs d'Egypte Amsterdam, Elsevier u. Kairo, Al Ahram, 4th printing, 1982. (19 September 2015)

English, German, French book, on ‘sights’ of old Egypt, rather than of modern-day street scenes. Cairo tiling at the Old Cataract hotel seen from afar, diagram 39 (book is unpaginated!), although the sighting is strictly not discernible, with foreknowledge required, albeit this can only indeed be the paving. This is now the earliest recorded sighting at the Old Cataract Hotel, and likely of 1974, in a earlier edition, but not seen.


Hicks, G. A. Modern Technical Drawing Vol. 2 1971 (from a c. 1987 study)

A minor geometrical construction study of no consequence. This was a library book and is not in my possession. I do not recall the book in any way.


Higgins, Muriel. New Designs from Machine Patchwork. Charles Scribner’s Sons, New York, 1980 (23 September 2017)

Chance finding. Has much of tessellation interest than others of its type, hence its purchase. Of particular note is an tiling based upon the well known eight pointed star and pointed cross inspired by Islamic geometry, p. 123 with an additional tile. Although of a most simple nature indeed, I do not recall having seen this previously. Upon research, I see that the eight pointed star and pointed cross is known as the ‘Breath of the Compassionate’, a seemingly new term to me. However, upon yet more research, I see that it is mentioned in Chorbaci’s paper, but had been forgotten! Also see Abbas, where this is named ‘Khatem Sulemanii’.


Highland, Ester Harris. The How and Why Wonder Book of Mathematics. Transworld publishers London 1961 (21 June 1997)

Juvenile, with a leaning towards historic aspects. Minor recreational aspects: Three utilities problem, map colouring theorem, no tiling.


Hilbert, D. and S. Cohn-Vossen. Geometry and the Imagination. Chelsea Publishing Company, New York. 1952.

An English translation of the German edition. A bitter disappointment, in that it is far too complex for me (as I suspected), given the main author, but I saw it recommended somewhere as being ‘recreational’!


Hill, Francis S. Jr. Computer Graphics. Macmillan Publishing Company New York, 1990. (16 June 2011)

The content is now hopelessly dated, only obtained due to a known Cairo tiling reference, p. 145. Escher tilings: p. 143 Horseman, Birds and fish p. 143, with a small tessellation article. Chapter 2 heading has a line drawing of Escher ‘Drawing Hands’ Chapter 5, p. 141, is concerned with tiling, despite a perhaps less than accurate title ‘Approaches to Infinity’; no other chapter heading has Escher's use. High and Low p. 403, Ascending and Descending p. 408.

See p. 256 for famous graphics teapot (although there is no apparent reference, save for bibliography, with F. C. Crow) Snowflake p. 171. Dudeney dissection p. 382, although not credited.


Hillman, David. Pentagames. A colourful collection of classic games designed by Pentagram. Guild Publishing 1990 (not dated, c 10+ years).

Largely a ‘coffee table’ book. Puzzles, nothing per se specifically of pentagon theme as indicated by the title. ‘Pentagames’ is a brand name for a company


Hiner, Mark. Up-Pops. Paper Engineering With Elastic Bands. Tarquin Publications 1996 (21 August 2011)


————. Phantasmagrams. A collection of visual and optical illusions designed by Pentagram Ebury Press 1992 (not dated, c 10+ years).


Hitchcock, Henry-Russell. World Architecture: An illustrated History. Bookplan, 1960. Oversize (22 June 2019) 

The subject is decidedly peripheral to my interests, but has occasional tiling. The text is by a gathering of experts.

Hocking, Martyn. Tiling. Crescent Books, 1994. (9 November 2019)

Wholly on the practical aspect of tiling, rather than the mathematics. It contains much useful advice, such as starting a tiling from the centre; intuitively, I would have begun from the side! And other useful hints and tips too, which I do not list here. As such, I have neglected this aspect of tiling, of which it only became of interest in 2019, upon a general side interest as tiles per se arising from the Cairo tiling studies as floors and wall tiles. Small format, 48 pp. hardback.

Hoffa, Alan; Koss, Roberta. Focus on Geometry. Addison Wesley Secondary Math. 1998 (15 October 2005)

Tessellations pp. 242, 247, 253, 404-415. All inconsequential. 16-year-age.


Hoffman, Paul. The Man Who Loved Only Numbers. The story of Paul Erdos and the search for mathematical truth. Fourth Estate, London 1999. First published in 1998 (23 September 2006)

Accessible account of Erdos’ life.


Hofstadter, Douglas R. Gödel, Escher, Bach: An Eternal Golden Braid. Metaphorical fugue on minds and machines in the spirit of Lewis Carroll. Penguin Books 1979 (First saw 21 December 1988 (recorded on a menu card), finally obtained 3 December 2006)

Many uses of Escher’s prints, too numerous to mention here. Book is a bit quirky, if not downright odd. Indeed, in a general sense, all of Hofstadter’s writings are quirky, to me at least, but likely it’s just Hofstadter’s advanced nature that’s way beyond me! As such, I do not believe, that, unlike other Escher references, this was studied in any way.


————. Metamagical Themas: Questing for the Essence of Mind and Pattern. Basic Books; New edition) 1996 First Printing edition 1985) (21 November 2016) PDF

Upon researching for parquet deformation, as I do at random, in 2016 I stumbled across the work of David Oleson, in which by circuitous means I found was featured in Hofstadter’s book. This was a total surprise; I was under the impression that this was a simple facsimile replication of his columns in Scientific American, and of which as I had the more important ones and so did not bother to pursue. However, this is not so, as evidenced by the Oleson finding! How infuriating! And much time lost too. This is now re-titled ‘Parquet Deformations: A Subtle, Intricate Art Form’ July, 1983 pp. 190-199.


————. Fluid Concepts and Creative Concepts. Computer Models of the Fundamental Mechanisms of Thought. Allen Lane The Penguin Press 1997. (N. B. The date has faded, 10 April 1999?).

A heavyweight tome, of 500+ pages, of largely of an academic nature, although readable, but obscure, with numerous essays, albeit invariably of limited interest. A single page discussion on Parquet Deformations, albeit without diagrams, p. 477. Scott Kim p. 403. Nothing on Escher.


Hogben, Lancelot. Mathematics for the Million. First Published 1936 George Allen. * 1940

and Pan Books Limited, 1967 paperback (7 March 1993 hardback; 16 April 1995 paperback)

Small format paperback, of 649 pages! From the title, seemingly of a popular level, although still notably advanced for a general readership. Liberally illustrated. Lots of equations. The approach is indeed thorough, but largely beyond my interests and understanding. There may indeed be some aspects of interest, but finding these in such a lengthy book is not easy. Nothing really recreational, despite the title. No tessellation, polyhedra, Escher. Pressures of time forbids a re-read.

The 1967 edition is described as ‘extensively revised with additional material and completely re-illustrated’.


————. Man Must Measure. The Wonderful World of Mathematics. Rathbone Books, London 1955 (4 August 1996)

Oversize, Juvenile.


Holden, Alan. Shapes, Space, and Symmetry. New York Dover Publications 1991 (earlier edition 1971). (19 November 1994, York)

Delightful, a popular account, readily accessible, from the basics on onwards.


Holderness, Jean. GCSE Maths Foundation Level. Causeway Books 1987 (4 November 1995).

Textbook. Tessellation pp. 315-316, simple, barely worth mentioning.


Holiday, Ensor. Altair Creative Colouring Books. Book 3. (9 March 1996 (year semi legible))

Juvenile colouring book on tilings, of insignificant worth detailing here. A major drawback is that it lacks an index, making finding references awkward.


Hollands, Roy. A Dictionary of Mathematics. Longman 1980 (not date stamped, c.10+ years)

Tessellation p. 151, inconsequential.


Holme, Audun. Geometry Our Cultural Heritage. Second edition, 2010, Springer (4 February 2017)

Academic in tone, of a chance finding at a bargain price, and so bought. Oddly for a Springer book, riddled with typos and minor errors in English, likely due to the translation from the author’s native Norwegian to English. Overwhelmingly too advanced for me, albeit with occasional recreational aspects, along with readable histories that may be referred to as and when required. Minor referral to tessellations, of Archimedean pp. 233-239, and symmetry of plane ornaments p. 445.


Holt, Michael and Ronald Ridout. Illustrated by Peter Edwards. The Big Book of Puzzles. Puffin Books 1976 (12 September 1993).

Small format paperback, 142 pp. Usual  introductory puzzle fare of all types, stated as ‘something for all the family’. The title is misleading; it’s a standard size paperback! Stated as a compilation of puzzles old and new, although of a scan I do not see anything new here. 153 puzzles with answers, but is not listed as such as contents, which is not given. Has Escher-inspired Relativity and Penrose staircase front and cover. No tessellation. In short, just a ‘fun book’ on puzzles, and there’s nothing wrong with that, and not intended to be for scholarly reference. One of many of its type.


Holt, Michael. What is the New Maths? First published in 1967 by Anthony Blond Ltd. First saw and studied 18 Sep 1986 (23 September 2000)

A small format hardback, of just 99 pp. New maths subjects, with sets etc, in a recreational style aimed at the parent with a child. Various aspects of recreational maths of mild interest, but nothing more. No tessellation.

Wikipedia: Michael Holt (born 1929) is a UK author of puzzle and quiz books for children, including several Doctor Who related quiz books and Crisis In Space in the Make Your Own Adventure with Doctor Who series. He was also the co-author of Puffin Books' Big Book of Puzzles series. He taught mathematics and geometry in London schools in the 1960s and 1970s.


————. Mathematics in Art. Studio Vista: London, New York: Van Nostrand Reinhold Company, 1971 (25 August 2016).

From a reference in Schattschneider and Locher. A small format paperback, of just 96 pp, of six chapters, of a popular level. Escher frontispiece, pp. 42, 46, 49-50, 77-78, 83. Most of the Escher references are in passing only, and in when ‘in detail’ are brief. Illustrated with Ascending and Descending, p. 46, and Horseman Mobius band, p. 77. Aside from Escher, has topics of general interest, such as Penrose tribar, but nothing too important. as an aside, this is typical of many of the books in Schattschneider’s listing of ‘Escher appearances in books’, namely they are all relatively minor, mentioned/illustrated almost in passing.


Hooper, Alfred. Makers of Mathematics. Faber and Faber Limited. (24 August 1996)

Historical account. Newton, Leibniz, Gauss. Some mathematics beyond me.


Hooper, W. Rational recreations in which the principles and numbers of natural philosophy are clearly copiously elucidated by a series of easy entertaining interesting experiments. Vol II The second edition, corrected. London 1774. (downloaded from Internet, 8 May 2015)
From a reference in MacMahon. No mention is made of different volumes; the one I have, Vol. II, is purely on general science, with a leaning towards optics; certainly, there is no mathematics here at all.


Hopkins, C. H. Project Mathematics Stage four (sic) Longmans 1967 (17 August 1997)

No tessellation.


Hornung, Clarence P. Handbook of Designs and Devices. 1836 basic designs and their variations. Dover Publications, Inc. New York 1959 (28 March 1998). Note that this is a revision of a 1932 work

As such, no tessellating designs at all; but that said, still of interests due to the geometric aspects. The book leans towards the designs themselves, and although they are indeed discussed, this is very much of a secondary aspect.


Hovanec, Helene. The Puzzler’s Paradise. Paddington Press New York & London 1978 (16 March 1996)


Huberich, Paul G. The Master System of Short Method Arithmetic and Mechanical Calculations Simplified. Max Stein & Company, 1951. First published  by Joe Bond, 1924

Small format (square) paperback, American, 128 pp. The cover states ‘Methods used by the world’s foremost experts. Adapted for home study’. Gives guidance on arithmetic, some of which I have concerns with. p. 3 Arabic number beginning. Some seem decidedly obscure, beyond reason. e.g p. 21, to multiply any number by  16 2/3.  The merits of the procedures given I leave to others. Date stamping has faded to point of illegibility. c. 1998? Also has other mathematical aspects, perhaps  peripheral to the title. In short, of no practical benefit..


Huff, Darrell. How to Lie with Statistics. Penguin Books 1988. (11 July 1998)

Popular small format paperback account of statistics. However, of limited interest in the extreme, as I am not really interested in the subject. Lacks an index, which would help to find terms as given on the back cover. The seemingly basic ‘samples’ and ‘errors’ subcategories were unfamiliar to me.




Irving, Washington. Treasures of the Alhambra. Geocolor, 1979 (6 August 1994, Lincoln)

Although not strictly a maths book per se, included for its tiling aspect.


Isenberg, Cyril. Soap Film Experiments with Kubic (sic] Bubbles. 'Manufacturer’s Guide Booklet' (no ISBN), not dated. (13 July 1995)

Juvenile. Obtained at the height of my interest in soap film bubbles (1995), of which although I still have an interest, I have moved away from. Booklet, of 13 pp, from a self-assembled kit, for school children, to make two sets of four frameworks (tetrahedron, cube, prism and octahedron) and two plastic plates with joiners to view film in two dimensions. Begins with a historical discussion of Joseph Plateau, then advice on the assembly of the kit, and then references and further reading. Although not dated as such the back cover gives copyright ‘Advanced Educational Toys, 1974’. They are Knotts Lane, Canterbury (of Isenberg’s city).


Cyril Isenberg MBE is an English physicist at the University of Kent, where he is an Honorary Lecturer. Isenberg is known for pioneering the analog computing possibilities of soap bubbles; in 2012, his 1976 article on the subject was one of a set of "classic articles" selected by American Scientist to celebrate their centennial. He has also frequently given physics lectures to schoolchildren and appeared in television shows, and is the organizer of the British Physics Olympiad. He is the author of books The Science of Soap Films and Soap Bubbles (Dover, 1978) and Physics Experiments and Projects for Students (with S. Chomet, Taylor & Francis, 1989 and 1996).




Jacobs, Harold R. Mathematics A Human Endeavour. W. H. Freeman and Company 1970 (18 June 2015)

Semi-popular, semi-text book. The book is described as ‘a textbook for those who think they don’t like the subject’, with a foreword by Martin Gardner, more or less aimed at a sixteen-year-old school age. I’m not entirely sure quite what to make of this; as such, it is in-between a recreational and textbook. Ten chapters, with of note Chapter 3, Mathematical Mosaics, pp. 202-208 and Chapter 4, The Regular Polyhedra, pp. 209-244. Certainly, there is nothing ‘new’ here for me. Uses three of Escher's artworks: Horsemen, on cover, Waterfall p. 19, Horseman again p. 207 and Möbius Band p. 478.


————. Geometry. W. H. Freeman and Company 1974 (25 August 2007) First saw in Grimsby reference library c. 30 Jul 1987

Semi-popular, semi-text book with 16 chapters, with each chapter subdivided into a series of lessons.

Many instances of Escher use throughout the book (although oddly not indexed), on the cover, Ascending and Descending, Man with Cuboid pp. 128, 227, Periodic Drawing 25 (Reptiles), 148 (and 153), Periodic drawings of Beetles, Birds, Flatfish, Bulldogs all 227, 300 Birds and Fish, Other World, p. 315, Circle Limit III (of Angels and Devils) 469, Circle Limit III p. 662. Oddly, aside from the first page, Escher is not mentioned elsewhere in any of the other credits (and of which makes for finding references most inconvenient).

Full of interesting bits of geometry, at a largely accessible level. The book was first studied in 30-31 July 1987, albeit somewhat chaotically, of which the study has dated badly, and can be considered as next to worthless.

Jacoby, Oswald with William H. Benson. Mathematics For Pleasure. Victor Golanz Ltd 1962 Not dated, c. 2000
Compilation of popular puzzles in the style of Dudeney. Usage is made of Hubert Phillips (Caliban) work. Next to nothing of a geometrical nature.

Jackson, John. Rational amusements for winter evenings. London 1821 (downloaded from Internet, 11 May 2015)
From a reference in MacMahon, 200 pages. No tiling or polyhedra, despite a promising lead with ‘geometric puzzle’ chapter, pp. 22-32 (31-40). Has geometric dissections. Overall the puzzles are relatively simple, with the sub title refers to ‘young people’ in mind.

Jackson, Tom (ed.). Mathematics. An Illustrated History of Numbers. Shelter Harbour Press, 2012 (16 December 2017)

A popular accent, liberally illustrated under four main groupings: prehistory to the middle ages; the renaissance and the age of enlightenment; new number, new theories and modern mathematics. A most pleasing read. However, perhaps surprisingly, nothing on tessellation or Escher!


Jackson, Valerie (ed.). The Complete Book on Patchwork and Quilting. WI Books limited 1985 (11 June 2013)

Although obviously not a book on mathematics per se, it is nonetheless included in this listing as it has interesting pattern aspects, and indeed, it is one of the better books there is on the interrelation between the two, and indeed led to extensive studies of the day (1987, as indeed with other patchwork books). However, now, and for some considerable time, the nature of the material is considered hardly worthy the time originally devoted to it.


James, E. J. Modern School Mathematics Books 1-4. Oxford University Press, 1959 (First seen September 1987) A date of 16 September 1987 is recorded on a menu card

Seemingly part of a four-book series, albeit of which the one I saw was not recorded. From a reference on a shared sheet in Cundy and Rollett. The book is long forgotten, I cannot picture it in any way, although the title seems vaguely familiar. However, this above is not necessarily of the book, as James has at least three other book to his name, but the title does indeed seem the most likely.


————. Curve Stitching. Oxford University Press, 1962. From the series Mathematical Topics for Secondary Schools. (Quoted in Murray-Rust) WANTED


Jamnitzer, Wentzal. Perspectiva Corporum Regularium. Nuremberg 1568. (Downloaded from internet 10 June 2015)

On polyhedra. Has five main sections, based on the platonic solids. Has many plates not commonly shown in books.


Jankel, Annabel and Rocky Morton. Creative Computer Graphics. Book Club Associates with Cambridge University Press, First published 1984 (13 June 2004). Oversize

Good popular account of the early days of computer graphics. Of its time, and still of  interest, albeit it has dated amazingly quickly.

Snippets of direct interest include: p. 95, on Robert Abel, with an Escher-like image that from memory I saw in the Appendix in Art and Science; without it I doubt very much I would have noticed the association. This is from ‘Changing Pictures, TRW’, of which upon an initial look (2018) I could not find freely available. Islamic computer generated tiling as background p. 16. Posted on Broug Ateliers as to the earliest possible instance (although unlikely), with disappointing results (two insignificant comments!)

Jarrow, Jastrow. Fact and Fable in Psychology. Houghton Mifflin Company, 1900. (Internet Download 21 April 2020)

Of impossible/ambiguous figure interest, from a reference in Shuster. Of most interest is Chapter 5, ‘The Mind’s Eye’, with a profusion of illustrations. Of special interest is a houndstooth type diagram, p. 283.


Jaworski, John & Ian Stewart. Nut-crackers. Puzzles and games to boggle the mind. Pan Books Ltd 1976 (14 October 2000)

Small format paperback, 125 pp. Typical children’s ‘fun’ paperback,  with 100 popular puzzles (with answers), albeit nothing more than a compendium of well-known puzzles really. Lacks an introduction and index. Escher-style illusions (no tessellations) are prominent, and indeed the cover is a stylized version of Waterfall, and the back cover shows a Penrose staircase. Pp. 48-49 also refers to Waterfall.

Of the authors, Stewart is too well known to document, although Jaworksi is decidedly less so. Upon research, he appears  to have been Stewart’s colleague at Warwick University, and editor of the university Manifold magazine. He also wrote a piece on the Alhambra.


Jeger, Max (edited by David Wheeler). Transformation Geometry. George Allen and Unwin Ltd. 1970. First published in England 1966. (Date not stated, as a guess, 2000)

Small format paperback, 143 pp. Part 1 of a five-book series. Translated from the  German, with an English version by A.W. Diecke and A. G. Howson. Advanced in nature, way beyond my understanding. Reference is made to Escher and Terpstra in the bibliography. Oddly, there is no reference to Escher in the text (I checked each page, 2018). Tessellation of sorts pp. 42-43, but in the context of vectors. Of limited interest and use in the extreme. I have no plans to re-read.


Jenkins, Gerald and Wild, Anne. Mathematical Curiosities, Books 1, 2 and 3. Tarquin Publications 1980, 1989 and 1990.


————. Make Shapes. Books 1, 2 and 3. Tarquin Publications. 1990, 1990 and ?


Jenkins, Gerald and Magdalen Bear. The Final Stellation of the Icosahedron. Tarquin Polyhedra No. 3. Tarquin Publications, 1985. (1 April 1993).

Nets to be assembled; disappointingly, no text is giving at all concerning the background to this.


————. Paper Polyhedra – in colour. A collection of 15 symmetrical mathematical models to cut out and glue together. (25 October 2014). 2004, first edition 1998. Tarquin Publications

A varied collection of polyhedra, to be assembled.


Jobbings, Andrew. Note 89.93. ‘Dissecting a triangle into rectangles’. Mathematical Gazette Vol. 89, 516 (November 2005) 501-502 (21 March 2013)

Popular account.


Johnson, Donovan A and William H. Glenn. The World of Measurement. John Murray. 1964 (24 October 1998). Volume 2 of the 12 book series ‘Exploring Mathematics on You Own’.

One of a series of five books I have of a 12-book series, pitched at a juvenile audience, 12-year-old. This is mostly of ‘simple’ measurement calculation, of little interest.


————. Invitation to Mathematics. Exploring Mathematics on Your Own. John Murray. 1964 (24 October 1998)


————. Understanding Numeration Systems John Murray. 1964 (24 October 1998)


Jones, Charles Booth-. More Brain Ticklers. Beaver Books 1978 (12 September 1993).

Standard fare.


Jones, Christine. Roman Mosaics. 1988. Not dated, c. 10 years+

This looks like a museum booklet, of just 12 small pages, rather than a book per se.


Jones, Tim Glynne-. The Book of Numbers. Arcturus 2007 (24 January 2015)

Various commentaries on numbers per se, albeit with many instances of numerology, and on occasion incorrect mathematics, such as with the Golden Section.


Jones, Mike and Bibby, John. Recreational Mathematics Resource Guide No. 5. (Year Unstated)


Jones, Lynn. Statistics. Macdonald Educational Colour Units 1974 (28 September 1997)

Note that this is not a book in its own right, but part on a series on mathematics by the Macdonald Educational, with other titles: Sets and Religion, Trigonometry, Statistics*, Number and Patterns, Groups and Finite Arithmetic*, Matrices, Calculating Aids, Vectors, Graphs, and Algebra. * In possession. Also see Edwards for other references in possession. Nothing of any real interest here.


Jones, Owen. The Grammar of Ornament. Studio Editions 1989. First published in 1856 (10 August 1993)

First seen (or at least as recorded) in 8 October 1987, where I undertook extensive studies of the day, albeit merely of ‘selective tracing’, and then larger, ‘freeform’ studies. As such, nothing remotely original emanated from this (a common complaint for such book-based studies of the day).

As such, a glorious, sumptuous book, deserving of greater study. ‘Paving of Diane’ Byzantine plate No. 3, Fig. 19. Also of note is a reference to what has become known as a houndstooth pattern, p. 15, of plaited straw from the Sandwich Islands. Lockwood and Macmillan in Geometric Symmetry, p. 90, refers to this, although not referenced directly.


Judson, Horace Freeland. The Search for Solutions. Hutchinson & Co. (Publishers) Ltd 1980 (28 February 2009).

General Science. See Chapter 2, Pattern, in the broader sense.




Kappraff, Jay. Connections. The Geometric Bridge Between Art and Science. McGraw-Hill Inc. 1991 (not date stamped)

Very nice indeed, full of interest, although that said it largely repeats existing research. Especially see Chapter 5, Tiling with Polygons. Many references and pictures relating to Escher, pp. 71, 134, 191, 248, 265. Many chapters on polyhedra. Cairo tiling featured as the dual of 32 .4. 3. 4, p. 181, although very carelessly drawn as regards accuracy. has an excellent bibliography. I also have a later edition of this book as a PDF, with minor extra material, with a supplement, and additional references.

Parquet deformation pp. 190-194, within the chapter 5, Tilings with Polygons, albeit this merely excepted from Huff’s article (1983), as the author credits. ‘Consternation’ is shown.


Kasner, Edward and James Newman. Mathematics and the Imagination. G. Bell and Sons, Ltd. 1970 (25 April 1999) Simon and Shuster 1940. British edition first published 1949.

From Wikipedia: Mathematics and the Imagination …  rapidly became a best-seller and received several glowing reviews. Special publicity has been awarded it since it introduced the term googol for 10100, and googolplex for 10googol. The book includes nine chapters, an annotated bibliography of 45 titles, and an index in its 380 pages. … According to I. Bernard Cohen, "it is the best account of modern mathematics that we have", and is "written in a graceful style, combining clarity of exposition with good humor". According to T. A. Ryan’s review, the book "is not as superficial as one might expect a book at the popular level to be. For instance, the description of the invention of the term googol... is a very serious attempt to show how misused is the term infinite when applied to large and finite numbers." By 1941 G. Waldo Dunnington could note the book had become a best-seller. "Apparently it has succeeded in communicating to the layman something of the pleasure experienced by the creative mathematician in difficult problem solving."

Edward Kasner (April 2, 1878 – January 7, 1955) was a prominent American mathematician who was appointed Tutor on Mathematics in the Columbia University Mathematics Department.

James Roy Newman (1907–1966) was an American mathematician and mathematical historian. He was also a lawyer, practicing in the state of New York from 1929 to 1941. During and after World War II, he held several positions in the United States government, including Chief Intelligence Officer at the US Embassy in London, Special Assistant to the Undersecretary of War, and Counsel to the US Senate Committee on Atomic Energy. In the latter capacity, he helped to draft the Atomic Energy Act of 1946. He became a member of the board of editors for Scientific American beginning in 1948.

Popular account. Eminently readable.  Has many snippets of interest, although no tessellation. Uses the term parhexagon pp 14-16. Space-filling curves pp. 343-355. Chapter IV Assorted Geometries – Plane and Fancy predates Frederickson’s use of the term.


Kay, Keith. Take A A Closer Look. Bright Intervals Books 1991 (3 June 1993)

On optical illusions. Escher tessellations Ascending and Descending, p. 36 and Belvedere, p. 42. No text worthy of the name. Also shows Otto van Eersel’s fish tessellation.


Keefe, John O’. and Phillip Rush. Weights and Measures. Methuen and Co Ltd. 1966 (12 October 2002)

Advanced juvenile.


Kelsey, Kenneth and David King. The Ultimate Book of Number Puzzles. Cresset 1992 (10 August 1993)


Kemp, Martin. The Science of Art. Optical Themes in Western Art from Brunelleschi to Seurat. Yale University Press New Haven and London. Second printing 1992.

A major work. As a broad statement, a series on perspective, of notable substance. Much of interest and accessible. For example, Vredeman de Vries, p.111, with a possible source of Escher’s ‘Other World’. Dürer’s geometrical designs, p.57. Many references to polyhedra, pp. 62-63. p. 159 shows two glass spheres, by J. M. W. Turner, with loose connection to Escher's Three Spheres II. Also has a substantial section on colour, of which I had forgotten about….

However, although largely a popular, albeit scholarly approach, much remains inaccessible, of which finding aspects that I can understand amidst more weighty material is few and far between.


Kenney, Margaret J. and Stanley J. Bezuszka. Tessellations Using Logo

Dale Seymour Publications, 1987 (8 March 1995). From Jonathan Press.

Somewhat dated, with blocky diagrams, likely as a consequent of Logo. Fused Cairo tile based on a square pp. 27-29. Has occasional ‘new’ tilings, such as p. 59, but not of any significance. Alphabet tessellations (L, W, T), pp. 66-68 Islamic designs pp. 69-74. Chapter on Escher type tessellation pp. 75-80 with ‘Fish’, Cat head, pecking pigeon, frog tessellation of no particular merit. All in all the book is of no consequence.


Kepes, Gyorgy (ed). Education of Vision. Studio Vista, London (24 September 2017)

Chance purchase at car boot sale. Broadly, on ‘basic design’, with 14 essays by the leading authorities in the field. Most I am unfamiliar with, but of the few I recognise includes Arnheim, Itten and Maldonaldo. However, there is next to nothing of any real interest here; the book is most wordy indeed, and I simply don’t have the time for an in-depth read, only skimming the pages.

P. 35 has a counterchange reference (although of no consequence) by W. Turnbull, of London Central School of Arts and Crafts. An admittedly brief look on Google for this proved fruitless. I seem to recall having seen this elsewhere, although I am far from certain.

I have no plans to re-visit this book.


Kepler, Johannes. Harmony of the World.

Available on-line from


Kepler Johannes. The Six-Cornered Snowflake. Oxford, UK: Clarendon Press, 1966. Trans. C. Hardie. (September 2016)


Kim, Scott. Inversions. W. H. Freeman and Company New York 1989. (30 April 1994)

Absolute delightful. Escher’s Sky and Water I p.112, commentary p. 113; Escher inversion p. 45. Parquet deformation pp. 14-15.


Kirkby, David and Peter Patilla. GCSE Maths Investigations. (7 May 1998, Hull)

A partial photocopy of relevant pages of interest. Very minor tessellation.


King, Elspeth. People’s Pictures: The story of tiles in Glasgow. Glasgow Museums 1991. PDF (13 September 2018)

Small booklet, of just 12 pages. Gives a history. No mathematical tiles as such. Of general interest.


Kinsey, L. Christine; Theresa E. Moore. Symmetry, Shape and Space with Geometer’s Sketchpad. Student Lab Manual. Key College Publishing 2004 (15 October 2009).

Tessellation pp. 57 onwards.


Klarner, David A. editor. The Mathematical Gardner. Wadsworth Inc. 1981 (24 March 2009)

A collection of articles in honour of Martin Gardner, with tiling featuring prominently. Especially see: In Praise of Amateurs, by Doris Schattschneider, pp. 140-166 re Marjorie Rice and pentagons; Some Problems on Plane Tilings, pp. 167-196, Branko Grünbaum; Angels and Devils. H.S. M Coxeter. pp. 197-209. Escher references Colour plate IV, Coxeter article p. 198 Angels and Devils, with typical Coxeteresque obscure text. Escher Sphere with Fish p. 201. Polyhedron with Flowers, p. 202.


Kline, Morris. Mathematics. An Introduction to its Spirit and Use. (Readings from Scientific American). W. H. Freeman and Company 1979.

Chapter 3 has an extensive series of articles by Martin Gardner of ‘geometric constructions’, from his columns. (book not date stamped)

(Oddly, the front cover has a Penrose tiling on the cover without any reference to this in the articles!).


————. Mathematics in Western Culture. The Scientific Book Guild 1954 (30 July 2002).

Of limited interest.


Kneale, Nicholas. The Tile Book (Fired Earth). Printed by The Artisan Press Leicester, June 1991 (14 September 1997)

Tile manufacturers’ 89 page catalogue/book with various aspects of actual floor tiles. Of general interest, but nothing of undue significance. Refers to a Mexican paver Saltillon p. 83 which I will follow up. No Cairo.


Knox, Gerald M. (editor). Better Homes and Gardens Treasury of Christmas Crafts and Foods. 1980, Meredith Corporation, Des Moines, Idaho pp. 6-7, 15, 19 (16 June 2014)

Although strictly a crafts book, included here as it has a cluster puzzle reference, of a nativity scene, apparently by David Ashe. However, there is no background detail here at all. An open question is to whether this is the first recorded instance of the type in print.


Knuth, Donald. The Art of Computer Programming.

Volume 1 Fundamental Algorithms, Third Edition (Reading, Massachusetts: Addison-Wesley, 1997), xx+650pp. ISBN 0-201-89683-4

Volume 3 Sorting and Searching, Second Edition (Reading, Massachusetts: Addison-Wesley, 1998), xiv+780pp.+foldout. 

Grunbaum reference. High-end, computer talk, way beyond my interest and understanding. Martin Gardner, Undiluted Hocus-Pocus also refers to Knuth, p. 145-146, as to recreational elements, hence my latter day search for this.


Kordemsky, Boris A. (edited by Martin Gardner) The Moscow Puzzles. Penguin Books 1976 First published in the United States and Canada 1972, and in Russian, 1956 (9 October 1993)
Dudeneyesque in style, and indeed most of the puzzles are derived from him. Occasional dissections, no tessellation as such.

Wikipedia Boris A. Kordemsky (1907 – 1999) was a Russian mathematician and educator. He is best known for his popular science books and mathematical puzzles. He is the author of over 70 books and popular mathematics articles. Kordemsky was born in Kiknur, Vyatka Governorate, Russian Empire. He received his Ph.D. in education in 1956 and taught mathematics at several Moscow colleges.

Kraitchik, Maurice. Mathematical Recreations. George Allen & Unwin, Ltd. First published 1943 (18 March 2000, Lincoln). When first saw is unclear. Recorded on a menu card is ‘Math Recreations College M. Kraitchik, again 17 September 1987’. The first studies are dated 21 September 1987

Twelve chapters on various aspects of recreational mathematics, with most of note Chapter 8, pp. 193-213 on tilings, with: 1. on Geometric Recreations, 2. Mosaics, pp. 199-207. Also see 3. Mosaic on the Sphere, pp. 208-209. Simple tiling diagrams, and ways of tiling with various regular polygons in combination. Mention of MacMahon p. 53 as regards Bachet. The preface mentions a French edition of sorts.

Kubo, Michio. Hidden Birds. Self Published, 1968 (29 November 2019)

A scan of the book, kindly undertaken by Makiya Torigoe. The otherwise obscure title refers to the figure ground aspect of Kubo’s black and white tessellations, seen throughout the book. In essence a picture book, no text, or even captions!

Kurajica, Stanislav. Rendgenska Difrakcija Na Prahu. HDKI/FKIT, 2020 (29 December 2020)  

 = X-ray Diffraction Powder 

Uses (with my permission) my Birds and Fish tessellation, p. 46. 

Academic, in Croatian. The book is on X-ray diffraction, way beyond my understanding, and of which there is no other tiling as such. Only skimmed.



Laithwaite, Eric. Engineer Through The Looking-Glass. British Broadcasting Corporation 1980. (11 October 1997, Lincoln)

Brief discussions on Mobius band, flexagons and polyominoes. pp. -31; 75-79.


————. An Inventor in the Garden of Eden. Cambridge University Press 1994 (22 January 2007)

Although more accurately a general science book, it also contains occasional mathematics, hence its placement here. See Von Koch snowflake curves pp. 23-25, Solid geometry pp. 91-94. Delightful reading. and worthy of a reread.


Land, Frank. The Language of Mathematics. John Murray 1960 (21 June 1992)


Langdon, John. Wordplay. Bantam Press. 2005 (3 March 2007)

Delightful. Langdon can be described as a master of his craft. Escher pp. 170 Sky and Water I, 181 Angels and Devils.


Langdon, Nigel and Janet Cook. Introduction to Maths. Usborne Publishing Limited 1984 (16 July 1994).

Juvenile. Usage is made of Escher’s Swans tessellation, p. 13, but without detail or credit!


Langdon, Nigel and Charles Snape. A way with maths  Cambridge University Press, 1984, 48 pages, juvenile. (First saw, or at least recorded date of study 15 July 1988)

Of note is a tessellation section, of an Islamic tiling, pentagons and Escher-like, not all of which I photocopied, with only p. 19 so copied.


Langdon, John. Practise Your Calculator Skills. Usborne. 1983 (20 July 199** - year has faded)


Larcher, Jean. Allover Patterns With Letter Forms. Dover Publications, Inc. 1985. (22 September 1993)

More inclined to pattern per se (with letters) than tessellation. The book lacks structure, seemingly of an ad hoc arrangement of letters (albeit of all the alphabet) in a symmetrical arrangement. Of limited interest.


Large, Tori. The Usborne Illustrated Dictionary of Maths. Usborne Publishing Limited, first published 2003. (16 May 2015)

Ostensibly for a juvenile audience, although some parts are decidedly advanced! 500 maths terms are explained, of which frequently served as a refresher for me. Has an extensive chapter on Shapes, space and measures , with tessellation featuring, p.36, although only of regular and semi regular tessellations. No Escher aspect.


Lasker, Edward. Go and Go-Moku the Oriental Board Games. Dover Publications Inc., Second revised edition 1960 (of a 1934 work) (23 August 1992)

Popular account. Never played the game though! Got on general interest.


Last, Derick (ed.) The Art Machine Pattern Book. Leapfrogs 1990. (30 April 1994).

Of interest is a Cairo pentagon-esque in combination with a kite, p. 5. Many computer drawn examples, badly dated. Tiling pp. 49-51, 54, the latter of Escher-like ‘gnomes’, by Richard Ladds.


Lanz, Sherlee. Trianglepoint. From Persian Pavilions to Op Art with One Stitch. The Viking Press 1976 (28 June 1998)

From a reference in Grünbaum. All of a triangular premise. Has many pleasing tessellation aspects throughout. Of note a truncated houndstooth tiling, titled ‘snowcaps’ colour plate 29 and p. 96 where it is stated ‘woven shawl, nineteenth century, the Sandwich Islands’, which I have seen quoted elsewhere.


Lea, Derek. Creative Photoshop. Digital Illustration and Art Techniques. Focal Press, 2007 (c. 2011)

Strictly a book on Photoshop rather than mathematics per se, and so its listing here is perhaps somewhat questionable. However, it justifies its inclusion here as it contains a tutorial on a composition based on Escher's premises of Bond of Union, page 195 and (primarily) Sky and Water I, pp. 340-349, and so I thus include here for the sake of convenience.


Leapfrogs. Curves. Leapfrogs 1982. Tarquin Publications (26 March 1994)


Leapfrogs. Poster notes. Tarquin Publications not dated (3 June 1993)

Some tessellation but treated in a lightweight manner. Written in conjunction with a series of posters produced by Leapfrogs.


Lemon, Don. Everybody’s Illustrated Book of Puzzles. London, Saxon and Co, 1890 PDF (Downloaded from internet 10 June 2014)

From a reference on Rob Steggman’s site. 794 puzzles. Very much alike in style to Dudeney’s later works. Whether Dudeney was aware, or was influenced remains conjecture; in his books he does not give a bibliography. Various geometric puzzles and dissections, pp. 8, 11-12, 35, 40, 46, 51, 55, 63, 67, 69, 77, 89. No tessellation or polyhedra.

Lemmen, Hans van. 5000 Years of Tiles. British Museum Press 2013, 304 pp. (28 August 2019)

Escher p. 252 (name check) and pp. 254-255, in Chapter 6, ‘The Century of Design’. A small paragraph on Escher, in the context of tile history, of his tile design made by De Porceleyne Fles in Delft. Illustrated with Swans. A good general history.

Lewis, Donald J. Introduction to Algebra. Harper and Row. 1965 (29 May 1994)

Academic. Illustrated with Escher’s prints: Cover, Preface, Three Spheres; introduction, Puddle Chapter 2, p.26 Three Worlds; Chapter 3, p. 77 Metamorphosis, Chapter 4, p. 138, Relativity; Chapter 5 p. 232, Reptiles.


Levy Joel? Optical Illusions. Dorling Kindersley Limited 2012. (30 May 2016)

A nicely produced book, of school-age level, of interactive nature, with various paper engineering pop-ups. However, there is nothing new or innovative here; it consists of illusions that are already known. Of perhaps most interest is that of the ‘Get of the Earth Puzzle’, p. 28, of which I have seen but not actually have a workable model to hand until now.

Note that I am unsure of the author; a whole list of people are given, of which who is most associated with the book is unclear. Joel is given above as ‘most likely’, albeit with the above in mind.


Lewis, K. Polyhedra. Further Experiments in Mathematics. Book 2. Longmans, Green and Co Ltd 1969. (24 October 1998).

Juvenile, but still of interest.

Leuvechon, Jean. Récréation Mathématique. no publisher. 1624 (Downloaded from Internet 7 May 2015). 200 pages
From a reference in MacMahon. No tiling or polyhedra. Occasional geometry, pp. 38-39 (58-59) and 73 (93). Mostly text, although indeed with many diagrams. Note that there is considerable debate at authorship of his book (see Singmaster), of which in itself is of historical importance, it being the first bearing the title of ‘recreational mathematics’. Albrecht Heeffer has written a scholarly article on this.

Libbrecht, Kenneth. The Snowflake Winter’s Secret Beauty. Colin Baxter Photography Ltd, 2004. First published 2003, US (22 May 2016)

Popular account, from a physicist.


Licks, H. E. Recreations in Mathematics. D. Van Nostrand Company, Inc. second printing 1916 PDF ((Downloaded from Internet 14 July 2014)

From a reference in Stegmann’s site. 15 puzzle pp. 20-21; magic squares pp. 39-43, geometric fallacies pp. 54-55, map colouring pp. 61-62, bees speculations pp. 91-99, 155 pages.


Liebeck, Pamela. How Children Learn Mathematics. Penguin Books 1988 (16 February 1995)

Tessellation, pp. 118-119 (includes a fish of no great merit). Basic, as to be expected.


Lindgren, Harry. Recreational Problems in Geometric Dissections & How to Solve Them. Revised and Enlarged by Greg Frederickson. Dover Publications, Inc. New York 1972. Originally published in 1964 as Geometric Dissections (1 September 1995)

Delightful! I went thorough the book at date unknown looking for anything ‘Cairo-like’, or of a par hexagon. As such, nothing. That said, a diagram on p 105 could have been made into a Cairo tile.


Livio, Mario. The Golden Ratio. Review, 2003, first published in 2002. (12 April 2014)

Much of interest (and accessible) throughout the book, but especially see re tiling Chapter 8, pp. 201-228 ‘ From the Tiles to the Heavens’

Escher p. 203, Penrose tiling pp. 203-206.

Note that the Livio here is not the same as a namesake, Livio Zuccha of tiling fame; it’s easy to mix them up.


Locher, J. L. (general editor). Escher The Complete Graphic Work. Thames and Hudson 1992. (9 April 1993). Note that this is an English edition translated (by Tony Langham and Plymm Peters) from the original Dutch Leven en Werk von M. C. Escher of 1981

Indispensable! One of the core books on Escher. Includes essays by M. C. Escher, with five joint author credits: F. H. Bool, Bruno Ernst, J. R. Kist, J. L. Locher and F. Wierda. Locher wrote the preface. However, the rest of the text is a combined effort; whether any one author is leading is not stated. Although not given as chapters, twelve can be identified, along with an extensive catalogue (the main part of the book) complied by F. H. Bool, J. Locher and F. Wierda. Invaluable are the ‘notes on illustrations’, pp. 329-343. Includes a one-page ‘selected bibliography’, p. 345, with misspelling of Maas. And to think I waited until 1993 to obtain this!


————. The World of M. C. Escher. Abradale Press Harry N. Abrams Publishers Inc. New York 1988 (9 April 1993) First Published 1971

Another core value book, indispensable. Has five essays: The World of M. C. Escher, J. L. Locher; Escher: Science and Fiction, C. H. A Broos, Approaches to Infinity, M.C. Escher. Structural Sensation G. W .Locher, The Mathematical Implications of Escher’s Prints. H. S. M. Coxeter, and a catalogue of the more important prints. Includes a excellent selected three-page bibliography, pp. 57-59, with misspelling of Maas.


————. The Infinite World of M. C. Escher. Abradale Press/Harry N. Abrams Inc. New York First published 1984 (First saw c. 14 December 1987) In possession 10 April 2018

The book is described as: ‘This 1984 edition is published  by Harry N. Abrams Inc. New York. It is a concise edition of Abrams’ The World of M.C. Escher, originally published in 1972…’.  151 pp as against 263 pp. Tihis concise edition has only two of the five essays of the earlier book (‘The work of M.C. Escher’ and ‘Approaches to Infinity’). It also lacks the ‘Selected Bibliography’ and ‘Exhibitions and Lectures’. Most of the plates are retained, albeit with minor rearrangements. The colour plates remain the same.  As such, I see little merit to this concise edition; there is nothing new here. indeed, with The World in my possession, I saw little need to purposefully pursue this subsequently. however, upon reviewing (2018), it became highly desirable, to refer page numbers  and  to check which exactly of the prints were shown, and so as it was available at a respectable price (£3.69)  I decided to obtain.


Locher, P. and C. Nodine. ‘The Perceptual value of symmetry. Computers and Mathematics With Applications 17, 4-6 475-484, 1989

From a Craig Kaplan thesis reference.


Locke, John. Isometric Perspective Designs and How to Create Them. Dover Publications, Inc. 1981. (22 September 1993)


Lockwood, E. H. A Book of Curves. Cambridge University Press 1963 (first printed 1961) (not date stamped).

A delightful book, although much is beyond my understanding. Gives history as well. One of the first books I ‘studied’, in 1987, from the college library. Quite when I later obtained it is decidedly unclear; I neglected to date stamp. At a guess, 1998, albeit with a five year leeway either side!


Lockwood, E. H. and R. H. Macmillan. Geometric symmetry. Cambridge University Press 1978, 2008. (21 December 2010)

Largely of an academic nature. ‘Indirect’ Cairo reference p. 88. Escher p. 4, Shells and Starfish, E42, Fish E41, p. 66 Lizards, E56.

Shows a houndstooth design p. 90 (on a small piece on making automatic reproductions), and of which although claiming to be from the Sandwich Islands (clearly derived from Owen Jones' account, with plaited straw) is not strictly so. Rather, for unclear reasons, this is a variation, indeed interesting in itself, but is not directly based on the Jones diagram.


Lodding, Ken. Byte. The small systems journal. 1979 Volume 4 No. 2 (February) 21 September 2016)

‘Escher inspiration’ on cover, of ‘Drawing Hands’, with minor acknowledgement to Escher pp. 3-4.


Logi, Angiolo. Text by Daniele Ravenna editorial coordinator Linda Fox. Australia Puzzle. Contemporary Silverware & Jewellery. Puzzle Pty Ltd 1994 (19 November 2016)

Gift of Lorenzo Logi. Many instances of cluster puzzles: pp 10-11 (the Discovery of Australia) The First Black Swan pp14-15; The Southern Cross pp.16-17; Escher mention c. 20. The Dreaming (Gatefold pull-out); Australian Land and Seas (1986); The Japan Puzzle (1989) pp. 42-43; Stevie Wonder with Australia Puzzle p. 54.


Loeb, Arthur, L. Color and Symmetry. Robert E. Krieger Publishing Company. Huntingdon, New York reprint 1978 (the original edition is 1971)

Occasional reference to Escher: pp. 65-66, 79, 102, 119-120, 162-169. Pictures include p. 66 Horseman, p. 120 Running man, p.163 Fish, p. 164 Lizards, p. 166 Butterflies.


————. Concepts & Images Visual Mathematics. Design Science Collection. Birkhäuser Boston 1993. (9 October 2014)

Found upon a Google book search, upon which I noticed some pentagon studies. Especially see Chapter 9, pp. 89-100 ‘Pentagonal Tessellations’, featuring a unaccredited Cairo tiling, and Chapter 10 pp. 101-105, ‘Hexagonal Tessellations’. Largely, save for the pentagon chapter in particular, the book is a disappointment, the concepts are too difficult for me to follow.


Loon, Borin van. Geodesic Domes. Tarquin Publications 1994 (30 April 1994)

Of peripheral interest. The book has cut out nets to assemble, but not undertaken. Commentary is given as to the domes.


Love, Brian. Play the Game. Book Club Associates, 1978. (29 January 2014)

Included despite there strictly being no mathematics here whatsoever. General board games of yesteryear, with each game over a two-page spread. Oversize. Checked for any jigsaw type puzzles/games but there are none.


Loveridge, Emma. Egypt. Country Fact Files. Macdonald Young Books, first published 1997. Children’s book (22 June 2014. First saw in Cleethorpes library c. 2013)

Although not a maths book per se, included as it has a picture of the Cairo tiling. Cairo tiling photo at the Old Cataract hotel pp. 8-9. However, this is only recognised with foreknowledge, as the picture is from afar that without cognizance of the tiling would otherwise pass unnoticed. The photographer credit is ‘The Image Bank, Kodansha Images’, but upon searching I could find no reference to the picture here.


Loyd, Sam [Jr]. Sam Loyd’s Cyclopedia of 5000 Tricks and Puzzles. New York. The Lamb Publishing Company 1914 PDF (17 May)

As compiled by his son, Sam. Impressive, even when due allowance is made for unaccredited borrowing from Dudeney. However, the book is not without fault. Gardner states (in Mathematical Puzzles of Sam Loyd) it is ‘riddled with mistakes, typographical errors, wrong answers and frequently no answers at all’. No tessellation of any note.


Lukas, Edouard. L’Arithmétique Amusante. In French, 1895. Gauthier Villars et fils, France PDF (25 June 2014)

From a reference in MacMahon (and others). As found on Rob Steggmann’s site. Nothing on tessellation, polyhedra and the barest minimum on geometry. Lots of playing card recreations.


Luckiesh, M. Visual Illusions. Their Causes, Characteristics & Applications. Dover publications Inc., New York 1965. Introduction to Dover edition by William H. Ittelson, 1965. Originally 1920 (18 September 1995)

Although strictly not a book on mathematics, included as it has certain crossovers. Maple leaf tessellation p.65, with a chapter on equivocal figures. Much of interest in a generalised sense.




MacGillavry, C. H. Symmetry Aspects of M. C. Escher's Periodic Drawings. Oosthoek, Utrecht 1965. (Reprinted as Fantasy & Symmetry. The Periodic Drawings of M. C. Escher. Harry N. Abrams, New York 1976.) (First saw April 1988 and again 18 August 2003)

41 plates of Escher tessellations, 12 in colour. Each plate is accompanied by text, with a crystallographic premise (this being MacGillavry’s background). Although these are broadly ‘readable’, the analysis strays into abstruse discussions, way beyond what Escher had in mind, and so consequently is of limited interest. Escher also wrote the preface. Many of the tessellations were not previously published of the day, but the book has since been put in the shade in this regard by Schattsneider’s inclusion of all the periodic drawings, in Visions of Symmetry of 1990.


MacMahon, P. A. New Mathematical Pastimes. Cambridge University Press 1921 and 1930. (Reprinted by Tarquin Books 2004) (31 March 2005)

Most impressive. Has considerable tessellation interest. Cairo diagram (but obviously not attributed) page 101, the first (1921) recorded instance in a book or article? (Moore's patent predates this). The only possible precursor to this is Haag (1911), as the others in Schattschneider’s list i.e. Laves et al are all after 1921.


Madachy, Joseph S. Madachy’s Mathematical Recreations. Dover Publications Inc, New York. 1979 (10 August 2006). Note that this is a re-titling of Mathematics on Vacation, Charles Scribner’s Sons, 1966, unabridged, with corrections

First, note the title change as above. Originally saw this (Mathematics on Vacation) in College library (and ‘studied’, or at least first recorded broadly stated in October 1987), but only in 2006 did I obtain. Of most interest is Chapter 1, Geometric Dissections pp. 15-33. Chapter 3, Fun with paper pp. 55-84, on flexagons. As such, the material is derived from Madachy’s Journal of Recreational Mathematics Magazine P. 8 states ‘Much of the material is taken from Recreational Mathematics Magazine (of 1960-1964). Upon an initial glance through the book, there is nothing original here; the material appears to have been taken from existing sources.


Malone, Maggie. 120 Patterns for Traditional Patchwork Quilts. Likely Published by HarperCollins Distribution Services 1983. NOT IN POSSESSION, FIRST RECORDED STUDY OF 2 July 1987

Although obviously not a book on mathematics per se, it is nonetheless included in this listing as it has interesting pattern aspects, and indeed led to extensive studies of the day (1987), as indeed with other patchwork books). However, now, and for some considerable time, the nature of the material is considered hardly worthy of the time originally devoted to it. note that Malone has published a whole series of books of numbered titles, with 115, 500, 1001 patterns, of which I only have the latter. Investigating the others seems hardly worth the bother.


Malone, Maggie. 1001 Patchwork Designs. Sterling Publishing Co Inc, New York 1982 (28 August 2001)

Essentially an illustrative book, with any text at a premium. Note that the 1001 designs are not discussed individually.


Mallinson, Phillip R. Geometry & its Applications Tessellations. Comap, 1996. (?) PDF

Note that I have this as a PDF rather than a book. Cairo tiling p. 17.


Mankiewicz, Richard. The Story of Mathematics. Cassell & Co 2000 (Grimsby Library). (c. 2011)

Escher, pp. 6, p. 125 Circle Limit IV, p.129 Mobius Strip II .


Maor, Eli. To Infinity and Beyond. A Cultural History of the Infinite. Princeton University Press 1991 (Grimsby Library).

Has tessellation articles: Tiling the Plane, pp. 102-106, which contains a Cairo diagram, albeit not original, the diagram taken from O’Daffer and Clements, and Maurits C. Escher – Master of the Infinite, pp.164-178 (16 October 2010).


Mandelbrot, Benoit B. Fractals: Form, Chance and Dimension.  W. H. Freeman & Co Ltd, 1977. First saw Grimsby library 11 May 1991

365 pp. From a reference on an old cardboard ring binder with a Grimsby Central library reference. The reference also mentioned the similarity of a hexagon to the outline of France. I now (2018) do not recall anything from this book. As such, it was likely seen out of possible interest in and nothing more. Certainly, no studies have emanated from it. I am not actively going to pursue this.


————. The Fractal Geometry of Nature. (updated and augmented) W. H. Freeman and Company, 1983? (10 October 2016). PDF.

A weighty tome of 468 pages. I have seen occasional references to this, although Escher and tessellation are mentioned essentially in passing, in regard of hyperbolic geometry, pp. 23, pp. 158-169, and bibliography.

The nature of an electronic copy prevents a pleasant reading, of which I have looked at just the first few pages.


Marjoram, D. T. E. Exercises in Modern Mathematics. Pergamon Press 1975 (18 September 1988?)

The only interest is in Chapter 10, Topology.


Holt, Michael and D. T. E. Marjoram. Mathematics Through Experience. Seemingly a five book series. 2 HarperCollins Distribution Services (March 1966

First seen, or at least recorded on a shared sheet of many different studies. E. H. Lockwood describes this as of CSE level, which book I saw is uncertain. A reference gives ‘No. 3’, but this may be association with a page number of the book to hand, not necessarily of Book 3


Marks, Robert W. The New Mathematics Dictionary and Handbook. Bantam Books 1967 (9 April 2007)

No entries for ‘Tessellation’ or ‘Tiling’! Be that as it may, still a handy reference guide.


Martin, George E. Polyominoes. A Guide to Puzzles and Problems in Tiling. Mathematical Association of America. 1991 (2 February 1998)

A general overview of the subject, with questions. Mostly of a popular level. Brief discussion on the Penrose loaded wheelbarrow p. 165, pp. 170-171.


Maxwell, E. A. Geometry For Advanced Pupils. Oxford at the Clarendon Press, 1966. First edition 1949 (11 October 1997)

Advanced it is indeed, of which despite claiming to be aimed at schools, is more properly described of a university level! Unfortunately it is far too advanced for me, of no practical use. Note that this is not a text book as such, in the spirit of Euclid, but rather a series of various aspects of Geometry, such as theorems of Menelaus and Ceva, to give an arbitrary instance.


McCann, Chris. Master Pieces: The Art History of Jigsaw Puzzles. Published by Collectors Press, Inc., 1998 (24 February 2017)

Although not a maths book, included as regards my jigsaw puzzle interest as I have seen this book quoted in various ‘serious’ jigsaw books, I obtained on the off chance that it may be useful to me in some way. However, as such, it is a relative disappointment, at least to my special interests in the field, although I was indeed prepared for this, given the title as the book is  indeed subtitled, this is of art history aspect of jigsaws, with biographies, and so there is indeed relatively little on jigsaws per se (Tuco is the best, p. 197, his special interest); certainly nothing on cluster puzzles! (or indeed any type of ‘innovation’). Williams critiques this (GRN?) for generally lacking the puzzle manufacturers names, of which I concur. Although occasionally some of the manufacturers are indeed mentioned, this is most scanty. The book also lacks an index. However, one should not perhaps be too critical here, as the title admirably describes the book! It is not McCann’s fault that our respective  interests are different. Although there is nothing of direct interest, there might have been, and so the matter is settled conclusively.


McCartin, Brian J. Mysteries of the Equilateral Triangle. Hikari Ltd 2012


McCleay, Heather. The Knots Puzzle Book. Tarquin Publications 1994 (7 November 1998)


McCloud, Scott. Understanding Comics. HarperCollins, 1993 (2009)

From a reference in Craig Kaplan’s thesis. Has many salient point indirectly as to Escher-like tessellation.


McCormack, Tony. Driveways, Paths and Patios. A Complete Guide to Design, Management and Construction. The Crowood Press Ltd, 2005. (2 July 2016, Cleethorpes library)

On in situ paving (having previously seen his most informative and interesting website on paving). Mostly of background matters as to the intricacies of paving; as such, there is next to nothing on pattern in the broad sense. Of little to no interest mathematically.


McGary, Debi. Wonderful Wood Puzzles. Plaid Enterprises Inc, Norcross, GA 1996


Although not a maths book, included as regards my cluster puzzle interest. Anne Williams reference.

Six wood-themed cluster-type puzzles, with the veracity varying considerably, from true tessellation to considerable vacant regions. Her work is inconsequential. McGary is oddly anonymous on the web.


McGregor, Jim and Alan Watt. The Art of Microcomputer Graphics for the BBC Micro/Electron (First saw college library 1987 (the day and month are uncertain, with the earliest reference being 24 January) Addison Wesley 1994

Despite being a book ostensibly on ‘microcomputer graphics’, it has notable tessellation aspects, and so hence my interest in it of the day. The book is notable for its plagiarism of Martin Gardner, with verbatim text.

Specific aspects of interest include: Chapter 5 Night and day – a journey through the world of tesselations (tesselations as spelt as in original)

Cairo pentagon references: text, p. 196, and picture, p. 197 Illustrated with a line drawing. (and p. 208?)

Text: ‘An example of a pentagon that will tesselate (sic) is the well-known Cairo tile, so called because many of the streets were paved in this pattern (Figure 5.2). The Cairo tile is equilateral but not regular because its angles are not the same’.

Moore pentagon, p. 198.


McLeish, John. Number. From cave people to computers, a revolutionary view of ourselves. Bloomsbury Publishing Limited, 1991 (17 December 2005)

Historical account. 18 Chapters.  Of little direct interest.


McMorris, Penny. Quilting. An Introduction to American Patchwork Design. British Broadcasting Corporation. First Published in 1981, US. UK edition with revisions first published in 1984 (13 October 2001) First saw c. 1987

Although obviously not a book on mathematics per se, it is nonetheless included in this listing as it has interesting pattern aspects, and led to minor studies (a dual sided sheet) of the day (1987), as indeed with other patchwork books. However, now, and for some considerable time, the nature of the material is considered hardly worthy the time originally devoted to it.


Meehan, Adrian. Celtic Design. Animal Patterns. Thames and Hudson 1995 (7 March 2009)

‘How to…’ book.


Meer, Ron van der. The Ultimate 3-D Pop-up Art Book. Dorling Kindersley, 1997. Originally published by Van der Meer publishing, 1995 (7 June 2014)

Although not a maths book per se, included as it has a Escher reference, of fish and frogs periodic drawing; pages are not listed. Many pages are of interest in a generalised sense, with aspects of ‘scientific art’.


Menkhoff, Inga. Optical Illusions. Amazing Deceptive Images - Where Seeing is Believing. Paragon Books Ltd 2007 (1 May 2011)

‘Ascending and Descending’ and ‘Relativity’, pp. 92-93. Minor text.


Meyer, Franz Sales. Handbook of Ornament. Dover Pub. Inc. 1957. First edition 1888 (30 October 1993, Sheffield) first saw 23 June 1990

The book is rather of ornament in its many forms rather than tessellations. However, there are indeed tilings here, notably pp. 10-12, albeit simple, of an arbitrary nature without structure. Of note in particular is of plate 6, diagram 11. This can be seen to be the same tiling as of Pólya’s Do3 diagram, and so predates this. Also, pp. 279-280.


Michell, George. The Majesty of Mughal Decoration. The Art and Architecture of Islamic IndiaThames & Hudson, 2007. (7 September 2018)

Oversize coffee table book. Purchased for a specific reason. Upon a  (2018) reading of a 17th century Cairo tile Mughal jali reference in Simon Ray’s Islamic catalogue of 2016, in which this book, being the sole quoted reference, appears to be the source. However, I find that this is not so; it is not in the book! A major disappointment in this regard, to put it mildly. However, there is at least an interesting chapter on Geometry, pp. 68-107, that discusses tilings. Also see Jan Pieper and George Michell for more on jalis.

Michell was a new name to me, although I see that upon a coincidental contemporary chance revisit to Craig Kaplan’s thesis he gets a sideways mention, p. 206, reference 98.


Midonick, Henrietta. The Treasury of Mathematics: 2. A Collection of Source Material in Mathematics Edited and Presented with Introductory Biographical and Historical Sketches. First published in the USA 1965. Penguin Books 1965 (29 October 2005)

Small format paperback, 416 pp. In four sections: Further Development, Algebraic Geometry and Calculus, Logic, Modern Algebra.

Select text from existing works. 24 biographies on ‘the greats’. Most of this is way beyond my understanding. Largely accessible is Albrecht Durer, pp. 104-122. Useful for reference purposes, but nothing more. I have no plans to re-read.


Miller, Charles D., Vern E Heeren, John E. Hornsby, Jr. Mathematical Ideas. Sixth Edition. HarperCollins publishers 1990. (22 July 199? Last number missed; 1998?)

Generally advanced maths, occasional recreational aspects, such as mathematics on stamps liberally throughout the book. Potted biographies of mathematicians liberally sprinkled throughout. No tiling as such. Chapter 9 on geometry.


Millington, Roger. The Strange World of the Crossword Puzzle. M & J Hobbs in association with Michael Joseph. 1974 (5 October 1997)

‘Cairo crossword’ puzzle, by ‘Croton’, from The Listener pp. 100 and 175 (solution), but without further detail. April 2012 research dates this as of 1951 and (not shown) 1954, and so of considerable historical significance. Also see this repeated in Investigation in Mathematics by L. Mottershead, but only indirectly credited.


Mirrow, Gregory. A Treasury of Design for Artists and Craftsmen. 725 paisleys, florals, geometrics, folk & primitive motifs. Dover Publications Inc, 1969 (4 November 2017)

Free, charity shop. Dover pictorial series, and as in the title, of a pictorial nature, without any explanatory text save for the back cover. In five sections, as according to the categories above. Of most interest is geometrics, and in particular a joined/seamless houndstooth, p. 53, that will be studied. Otherwise, there is nothing particularly new or innovative.


Mitchell, James (general editor). Science and the Universe. Mitchell Beazley 1977.

Minor reference to Escher’s prints Angels and Devils, p. 51 and Mobius band, p. 53, with general comment. So lightweight as be barely worth comment.


Mold, Josephine. Circles. Topics From Mathematics. Cambridge University Press 1967. (20 August 2000?)

This book is one of a six-part series from  ‘Topics of Mathematics’, three of which are by Mold (Solid Models, Circles and Tessellations) and three by David S. Fielker (Cubes, Computers and Statisitics). All are of a like presentation and page range of 31-32 pages, from 1967 onwards. Circles and Tessellations are the only ones in possession.

Small, 32-page booklet, for children. Very accessible, with much of interest.


————. Tessellations. Topics From Mathematics. Cambridge University Press 1969 (20 February 1991) photocopied book

School age level, but still of interest. Shows dual Archimedean tiling, p. 25, which can be interpreted as Cairo. Also interesting fish tiling that has dual properties, possibly as a by-product of drawing, rather than purposefully so.

Also of note, as regards Robert Ferréol’s interest in examples of Pavage de Diane, is p. 17, where there is a report of this as an in situ tiling ‘… on the floor of an old shop in Windsor’, with a side reference to Windsor Castle. Upon an initial look, this was not, unsurprisingly, found.


Montrose, Clifford. Games To Play By Yourself. Suitable for young and old a boon to the convalescent. London: Universal Publications Ltd. No date of publication, but given online as 1935, 1936, 1937 (not date stamped, c. 1997 + - 5 years)

Small format paperback, of 90 pages. Stated of a variety of indoor games that you can play alone. Nothing of an overt geometrical nature. Includes: Solitaire pp. 16-19, The Wonderful Puzzle Fifteen pp. 41-43. One of many of its type, with no plans for a re-study.


Moon, Brian. Literary Terms: A Practical Glossary. The English & Media Centre. First published in Australia 1992. (English publication date not stated) (5 November 2011)

Note that although this book is not mathematical, I have decided to include it here in this listing, as it uses Escher's print ‘Drawing Hands’ on the cover, and so is of interest in that regard.


Moore, Alison (ed.) Reader’s Digest Compendium of Puzzles & Brain Teasers. The Reader’s Digest Association Limited 2000 (14 July 2007)

Escher’s Relativity p. 55, with minor text barely worth the mention.


Morgan, Bryan. Men and Discoveries in Mathematics. John Murray, 1972 (24 October 1998)

Poplar account, from over 5,000 years to today. The discussion is in general terms, rather than focusing on specific individuals as an in-depth detail.


Morgan, W; Pickering, J. R. Mathematics I and II Sir Isaac Pitman & Sons, Ltd. 1946 and 1948. 19 July 1992

Textbook, typical of the day, with many problems in calculation, of little interest.


Morris, I. H. and Joseph Husband. Practical Plane and Solid Geometry. Longman, Green and Co. 1944 (26 March 1994, Scunthorpe)

Typical generic geometry text book of the day, one of many that I have; simply, one would have sufficed. The reason for obtaining the book was to be able to look up any geometric construction as and if required, but I do not believe that I have used this in any way. Pattern p.116, tracery p.117.


Moscovich, Ivan. Mind Benders. Games of Chance. Penguin Books 1986. (13 June 1999)


————. Mind Benders. Games of Shape. Penguin Books 1986. (5 July 1998)


————. Ivan Moscovich’s Super-Games. Hutchinson & Co. 1984 (28 November 2004)

consultant editor Ian Stewart.

Various Dudeneyesque puzzles of a one-two page per entry nature, 59 distinct entries, some original, although which is which is not made clear. Lavishly illustrated. No tessellation as such, although plenty of off-shoots.


————. Leonardo’s Mirror & Other Puzzles. BCA. 2005 (2 May 2009)

The book is of a series of 12 (four of which I have) published under the generic theme of ‘Ivan Moscovich’s Mastermind Collection’. c. 100 aspects of ‘simple’ recreational mathematics, pitched at a juvenile level, mostly seen before but nonetheless remain of interest. given such a large number fully documenting the books is problematical, and so I thus outline aspects of immediate interest only. The title given is apparently chosen arbitrarily by Moscovich, given that each puzzle is discussed over one or two pages.

No tessellation. Dissection p. 27, packing discs or circles, pp. 36-45.


————. The Hinged Square & Other Puzzles BCA. 2005 (2 May 2009).

Somewhat disarmingly here, p. 76, on the golden ratio propagates (or at least seems to imply) the ‘belly button’ myth.


————. The Shoelace Problem & Other Puzzles BCA. 2005 (2 May 2009)


————. The Monty Hall Problem. & Other Puzzles BCA. 2005 (2 May 2009)


————. Loopy Logic Problems & Other Puzzles (31 July 2013)

Sterling Publishing Co, Inc, New York, 2006


Moser, Koloman. Turn of the Century Viennese Patterns and Designs. Dover Publications Inc. Mineola, New York 1998 (6 August 2010). New introduction by Leonard Fox and Mark Weinbaum.

Dover states that this: ‘is a republication of all the plates included in the portfolio Flächenschmuck, from the Moser Issue of Die Quelle, published in Vienna and Liepzig by Verlag M. Gerlach, 1901-1902’.

Historically significant, as here are the first true life-like Escher tessellations (I do not consider the ‘Peru’ types bona fide examples). However, separating ‘design’ from tessellation here is fraught with difficulty; there is a definite blurring. The tessellations are not always of a ‘no gaps’ premise. Clearly identifiable as life-like tilings: p.12 (human figure), p. 25 (birds), p. 37 (goldfish), p. 48 (birds), p. 53 (woman), p.57 (woman, with gaps). Birds of p. 25 is noticeably favoured for true premise and inherent quality.


Mottershead, Lorraine. Sources of Mathematical Discovery. Basil Blackwell 1977. (8 March 1997)

A delightful book, albeit with much plagiarism, with much of interest, with a recreational promise, and in particular a unit (chapter) on tessellations. Escher pages: 39, 110, 112-114, 163-166. Horseman, 113; Sky and Water I, 113; Reptiles, 114; Relativity, 163; Waterfall, 164; Belvedere 165; Ascending and Descending 166.

This also features the Cairo tile pp. 106-107 in a section on irregular pentagons. This is also shown as cells in a crossword puzzle. Curiously, Mottershead mentions ‘Croton’ (i. e. the compiler in The Listener!) in association with ‘her’ page of Cairo puzzles! Previously (prior to 2 April 2012), I thought these were original with her, but apparently not! However, to give credit to her, she does indeed mention ‘Croton’ on the page.

The cover has an op art design apparently attributed to one Chris Belson. However, he is not the designer! Carraher and Thurston in Optical Illusions and the Visual Arts (1966), page 59, reproduce this design, with credit given to Franco Grignani.

Of note is that Mottershead shamefully appropriates (1963) Gettings’ diagrams in The Meaning and Magic of Art p. 64 on see p. 128 of Sources…’ without any mention of Gettings!

The first of two books of a like nature by Mottershead, although wide apart in chronology, namely of 1977 and 1985.


————. Investigations in Mathematics. Basil Blackwell 1985. (8 March 1997)

No Escher references or pictures. The book consists of 6 units, or chapters. As with Mottershead’s earlier book, this is very much in the same vein, of a recreational nature. however, here, as an observation, more on numbers, rather than symmetry matters of the other book. That said, there is indeed tilings here, and indeed, this was studied in 1987 (at Grimsby reference library).


Mott-Smith, Geoffrey. Mathematical Puzzles, for Beginners and Enthusiasts. Volume 106 of New home library. Blakiston Company, 1946. Reprinted by Dover Publications, New York,  1954 WANTED

Has geometric dissections p.


————. The Handy Book of Indoor Games. Garden City Publishing  Co, Inc. Permabooks. 1949. (Not date stamped, c. 1997, + - five years)

Small format hardbook, 245 pages. In three parts: 1. Card Games,  2. Board and Piece Games, and 3. Word Games and Pencil-and-Paper Games. The main essence is on card games.  Only of minor passing interest, with nothing really in my field. Note that Mott-Smith was a geometric dissection enthusiast, and is discussed in Frederickson, Plane and Fancy, with a biography p. 111, but there is nothing in this line in the book.


Munari, Bruno. Design as Art. Penguin Books Ltd 1971 (Not date stamped, c 2006, at a guess)

On design, rather than maths. Occasional mathematics. Note that Munari is an associate of Mari.

Murchie, Guy. The Seven Mysteries of Life: An Exploration in Science & Philosophy. Houghton Mifflin Company. Originally published 1978. Part seen on Google Books. (21 April 2020)

P. 467 Cairo tiling in the context of pentagonal chicken wire. Unsurprisingly, the connection is not made. Found by chance upon Twitter responses with T. Sundra Row.

Murphy, Lawrence R. The American University in Cairo: 1918-1987. The American University in Cairo Press, 1987 (9 August 2012)

Although not a mathematics book per se, as it contains incidental instances of the Cairo tile, pp. 64 and 254 (the best picture), I thus include here. A picture of uncertainty is p. 175, possibly of the square format type.


Murphy, Patrick. Modern Mathematics Made Simple. Heinemann London 1982 (7 November 1993)

Among a generally rigorous book on ‘modern mathematics’, with chapters on Relations, Linear Programming, Vectors and more way beyond me, surprisingly tessellation and also Escher-like aspect finds an outlet. Tessellations, Chapter 10, pp. 194-205, and cover design. Cairo tiling, unattributed, p. 200. This book has played a notable role in my early studies, in which in 1987 I studied it extensively. However, the ‘Escher-like’ tessellations by Murphy show a complete lack of understanding of the issues and are a veritable disaster!


Murphy, Patrick and Albert F. Kempf. The New Mathematics. W. H. Allen London 1982 (18 October 1997)




Nath, R.  History of Mughal Architecture  (8 October 2018) PDF

The subject of recent October 2018 interest due to a stated 17th century Cairo tiling jali, in a Simon Ray Indian and Islamic Arts catalogue. Nath appears to be the leading authority, hence my interest in this four-volume set. This being so, I emailed him, asking specifically about the Cairo tiling aspect, but he responded in a non-specific manner.


Nasr, Seyyed Hossein. Islamic Science: An Illustrated Study. World of Islam Festival Publishing Co 1976. (First saw Grimsby library 16 November 1987, or at least the first recorded study, and studied again later, c. 9-10, 12 August 1988, when photocopied pp. 88-90)

Studied as the book has a few geometric aspects, although little on tiling, of which of most interest is pp. 76, 89-90, 147. Without the book to hand, downloaded as a PDF for the sake of convenience (although the book is economically available). Nasr is a prolific author, with 29 publications to his name (on the internet archive site), easily confused. However, my original book title reference is indeed as stated.

Much of the book had been forgotten pending the download.


Necipoğlu, Gülru and Mohammad Al-Asad. The Topkapi Scroll: Geometry and Ornament in Islamic Architecture. Getty Research Institute, U.S. 1995. 384 pp (£621 cheapest copy!) Available as a free PDF on Getty site

From a Peter Cromwell reference, and likely others.


Since few architectural drawings and no theoretical treatises on architecture remain from the pre-modern Islamic world, the Timurid pattern scroll in the collection of the Topkapi Palace Museum Library is a valuable source of information. This text provides an analysis of the scroll dating from the late 15th or early 16th century, and aims to throw light on the conceptualization, recording, and transmission of architectural design in the Islamic world between the 10th and 16th centuries. It compares the Islamic understanding of geometry with that found in medieval Western art. The scroll, with its 114 individual geometric patterns for wall surfaces and vaulting, is reproduced in this volume. A catalogue includes illustrations showing the underlying geometries, in the form of incised "dead" drawings, from which the individual patterns are generated. An essay by Mohammad al-Asad discusses the geometry of the "muqarnas" and demonstrates by means of CAD drawings how one of the scroll's patterns could be used to design a three-dimensional vault.

Gülru Necipoğlu (born 1956 in Istanbul) is a Turkish-born American professor of Islamic Art at Harvard University. She been Aga Khan Professor of Islamic Art and Director of the Aga Khan Program of Islamic Architecture at the Department of History of Art and Architecture in Harvard University since 1993, where she earned her PhD in 1986. She specializes in the medieval and early modern periods, with a particular focus on the Mediterranean basin and the Eastern Islamic lands. She is the editor of Muqarnas: An Annual on the Visual Cultures of the Islamic World and Supplements to Muqarnas. Her books include Architecture, Ceremonial, and Power: The Topkapi Palace (1991); The Topkapi Scroll, Geometry and Ornament in Islamic Architecture (1995); and The Age of Sinan: Architectural Culture in the Ottoman Empire (2005). Her critical interests encompass many subjects, including methodological and historiographical issues in modern constructions of the field of Islamic art.

Nelson, David et al. Multicultural Mathematics. Teaching mathematics from a global perspective. Oxford University Press 1993. (Newark Buttermarket, 11 June 1994)

Chapter 6, Geometry and Art by Julian Williams pp. 142-174 has a small feature on tessellation, but aside from that chapter there is next to nothing here of direct interest.

Of note in the context of Escher cover art is a snippet of Escher’s plane tiling of Swans on the cover (shared with another, unrelated picture). Especially see Chapter 6, ‘Geometry and Art’, with a focus on tessellation, my field of interest. p. 158, of two (semiregular) tessellations, 6, 4, 3, 4 (Pavage de Diane) and 6, 12, 4 as line drawings, said to be from Shibam-Kawkaban, of a minaret in Yemen.


Nelson, David (editor). Dictionary of Mathematics. Penguin Books, Second edition, 1998. First published 1989 (25 August 2007)

Serious reference guide. Tessellation gets a brief mention, with two illustrations.


Newell, Peter S. Topsy & Turvys. (2016)

Of note is the ambigram, p. 31


Newman, James A. The World of Mathematics – Volumes 1-4 Simon & Schuster
2480pp. (4 Volumes) (September 2016, pdf)

(Newman also wrote a foreword to The Universal Encyclopedia of Mathematics)


Nicolas, Alain. Parcelles d’infini Promenade au jardin d’Escher. (in French) Belin Pour La Science. 2006 (2010?)

Delightful! Nicolas is a master of his craft. A must have for anyone interested in Escher-like tessellation.


Nichols, T. B. and Norman Keep. Geometry of Construction. Cleaver-Hume Press Ltd 1959. First published 1947. First saw 1987 (27 August 2000)

Of minor interest. Although of a geometric construction premise, of first principles, at least to begin with, there is indeed some patterns of interest. Fret patterns pp. 88-90, patterns based on squares pp. 90-91, patterns based on circles, pp. 92-93 patterns in circles pp. 94-95 and tracery, pp.196-199. I believe I first saw this book in 1987 (at the college library?) and loosely studied with some geometric constructions of the day. There is no tiling as such.

Nisbet, Harry. Grammar of Textile Design. Scott, Greenwood & Son. First Edition 1906 276 pp. (seen). Second edition 1919, Third Edition 1927 (seen). (26 February, 8 March  2019) PDF

Of an undoubted expert on the subject. Has a Cairo tile unit block, ‘linear zigzag weave’, p. 101 (first edition), but I don’t at all understand the correlation between supposed related diagrams. That said, although doubts remain, surely a Cairo tiling is intended. This is of obvious note as one of the earliest examples, namely that of 1906.

The book was mentioned in Grünbaum and Shephard’s article ‘Satins and Twills’, but found independently!

Nixon, J. T. World of Shapes. Oliver and Boyd Ltd. 1968 (5 October 1998)


Norgate, Martin. ‘Cutting Borders: Dissected Maps and the Origins of the Jigsaw Puzzle’. The Cartographic Journal. The World of Mapping Volume 44, 2007 - Issue 4 (May 2020)

Jigsaw/John Splisbury reference. A valuable contribution to the debate on the originator of the jigsaw puzzle. Norgate categorically asserts this to Madame de Beaumont rather than Spilsbury. As a general statement, it is not always clear as to what is original research on Norgate’s part. However, although it repeats established knowledge in parts, I like this piece very much.

Northrop, Eugene P. Riddles in Mathematics. A Book of Paradoxes.The English Universities Press Ltd. 1945 and also Penguin Books 1975. (16 November 1996 and 17 October 1998)

Largely popular account, of  ten chapters with the last of a more advanced nature. Largely of paradoxes and fallacies, derived from stated sources, as detailed in the preface. Not tessellation as such, but of much related and other material of interest; of minor optical illusions, dissections, space-filling curves, Mobius band, four-colour theorem to name but few. The overall tenure is largely of an advanced nature, although the above is indeed of a popular level.


Nunn, G. Modern Mathematics. Macdonald And Evans. 1978 First saw c. 31 July 1987 (22 December 2016)

Small format paperback. As such, this book, first seen at the library, was studied very early, of 31 July 1987, of which my memory has unfortunately considerably dimmed; indeed, I cannot now picture this, or indeed recall the study to any great extent. To aid in reviewing the study, of which for the year of 1987 I am in the midst, I thus ordered (it being reasonably priced), and not least given that it includes a Cairo tiling. The book is typically of ‘modern mathematics’, of fifteen chapters with favoured topics, such as Sets and Algebra, although of course much of this is out of my realm of interest. Nonetheless, it contains dedicated chapters on tessellations, pp. 155-163 and topology pp. 224-268, all of which I had completely forgotten! As such, the chapter on tessellations is somewhat of a let down. All very basic, although couched in technical terms. there is no Escher-like element whatsoever. However, some most rudimentary Escher-like tessellations can be found on in the Appendix, p. 327, which is ‘typical teacher’ i.e. not idea!




O’Beirne, T. H. Puzzles & Paradoxes. Oxford University Press, 1965 (17 February 2015)

A collection of articles which appeared in New Scientist from January 1961 to February 1962. No tiling or Escher. Somewhat of a let down, in that I was expecting some of his tilings columns to be shown.


Obermair, Gilbert. Matchstick Puzzles, Tricks & Games. Sterling Publishing Co., Inc. 1978 Originally published in Germany under the title Streichholz-Spielereien, Wilhem Heyne Verlag, Munich 1975 (14 November 1998)

Small format hardback, 144 pp., 14 Chapters. Popular tricks with matchsticks, with answers. Gives a history of the match p. 7. P. 8 is interesting in that it shows how best  to strike a match, in a counter intuitive way, not the more obvious lengthwise strike, but breadth! I had never even considered this! To what extent the puzzles are original is not readily detectable; certainly, there are no references or bibliography. Likely, they are a re-hash of existing material, given the historical account below. Makes for a mildly amusing coffee-time reading, but there is nothing of significance here.

A brief history: From: Aims Education Foundation on 12/3/2005 Source:

In the 19th century matches were first manufactured. Invented in 1827 by the British chemist John Walker, matches soon replaced the tinderboxes that people had formerly used to light fires. As matches grew in popularity and became ubiquitous later in the 19th century, they spawned a new form of entertainment matchstick puzzles that became quite popular when several match companies printed these puzzles on their boxes. Capitalizing on this interest, publishers began to print books of match-stick puzzles. By the turn of the 20th century, many people had developed a personal repertoire of these puzzles and used them to challenge friends and acquaintances. The toothpick puzzle presented here is modeled after these classical matchstick puzzles. 
Martin Gardner wrote about such puzzles in Scientific American November 1967 with the now famous "cherry in the cocktail glass" matchstick puzzle.


O’Daffer, Phares G and Stanley R. Clemens. Geometry. An Investigative Approach 2nd edition Addison-Wesley Publishing Company 1992. (23 October 2010).

Chapter 4, p. 86-117 Patterns of Polygons: tessellations, albeit very basic in scope. Has Cairo tiling p. 95. Occasional usage of Escher’s prints: Day and Night p. 86-88, Horseman p. 114, Magic Mirror, p. 215.

Oelsner, G. H. A Handbook of Weaves. The Macmillan Company 1915. Translated and Revised by Samuel S. Dale. 1875 illustrations 131 pp. PDF (2019) 

Of fabric interest. As to background matters, Oelsner was the director of the weaving school at Werdau, Germany. The translator has added a supplement. Referenced in Grünbaum and Shepherd’s 1980 Satins and Twills article.

As such, the numerous illustrations are of an abstract nature. As such, no houndstooth or related material. Consequently, the book is of limited interest. Deemed not worthwhile to purchase. has both the 1915 edition (the first?) and a dedicated extract of the diagrams, of which a houndstooth is to be found. WIF (weaving information file) number 44248.


Ogawa, Tohru, ‎Koryo Miura, ‎and Takashi Masunari. Katachi U Symmetry. Tokyo: Springer-Verlag 1996 (23 November 2016, Google book reference). Part 2, pp. 121-287 is available as a PDF. WANTED

Especially see: William Huff. ‘The Landscape Handscroll and the Parquet Deformation’, 307-314. This has four new parquet deformations by ‘new people’, namely:

Alexander Gelenscer; Swizzle Stick Twirl, 1986

Pamela McCracken; Cloisonné, 1990

Loretta Fontaine; Seven of One Make Three, 1991

Bryce Bixby; They Come, They Go, 1991

Of interest: Analysis of Marcia P Sward Lobby Tiling by Teruhisa Sugimoto (Marjorie Rice tiling)

Search of Convex Pentagonal Tiling with 5-valent Nodes by Teruhisa Sugimoto

Oguro, Sabu. Making Marvelous Wooden Puzzles 70 Animal Families : 70 Animal Families

Fox Chapel Publishing, 2012 2014, 160pp. WANTED

Note that this is one of those books that although ostensibly in print, are seemingly simply not available.

O’Keefe, Michael and Bruce G. Hyde. Crystal Structures 1. Patterns & Symmetry. Mineralogical Society of America, 1996, 453 pp NOT SEEN IN FULL

Of Cairo tiling interest, p. 207: The pattern is known as Cairo tiling, or MacMahon’s net and In Cairo (Egypt) the tiling is common for paved sidewalks…

The second use  of the term ‘MacMahon’s Net’ for the Cairo tiling, having previously been used by them in their 1980 paper, ‘Plane Nets in Crystal Chemistry’, but this time in addition with the Cairo association. However, this is very much an ‘unofficial’ description. Upon correspondence with him (2012):

I suspect I got ‘Cairo tiling’ from Martin Gardner who wrote several articles on pentagon tilings. He is very reliable. As to ‘MacMahon's net’, I got the MacMahon reference from Cundy & Rollet….We are mainly interested in tilings on account of the nets (graphs) they carry.

Possibly, and plausibly, this by MacMahon, of 1921, was the earliest known representation, and so in a sense it was indeed broadly justified, even though by 1980 the ‘Cairo tiling’ term was coming into popular use, although if so, it is now been left behind by my subsequent researches. Curiously, the term is used on the Cairo pentagonal tiling Wikipedia page. However, the page leaves much to be desired, including this designation. Toshikazu Sunada has also used this term. However, I do not like this at all; it seems a somewhat artificial, additional naming, and seems unnecessary. Better would simply to have credited MacMahon as the first known instance (at the time) but without naming it after him.


Oliver, June. Polysymmetrics: The Art of Making Geometrical Patterns. Tarquin Publications 1990. (6 April 1993)

Making very simple geometrical patterns, of no real consequence, lightweight in the extreme, some with an Islamic leaning due to her background in these designs.


Opie, Iona and Robert and Brian Alderson. The Treasure of Childhood. Books, Toys, and Games from the Opie Collection. Pavilion Books Limited, 1989 (26 June 2016)

Oversize. Of minor interest as regards puzzles and games, but full of interest in a general sense. Jigsaws, with Spilsbury and others, pp. 152-153.


Opie, James (consultant author) with Duncan Chilcott and Julia Harris. The Collector’s Guide to 20th Century Toys. Bracken Books 1996 (first printed 1995) (26 June 2016)


O’Shea, Donal. The Poincaré Conjecture. In Search of the Shape of the Universe. Allen Lane, 2007. (21 April 2012)

Semi-popular account.


Osler, Dorothy. Machine Patchwork Technique and Design. B T Batsford Ltd London First published 1980. First saw 16 June 1987

Although obviously not a book on mathematics per se, it is nonetheless included in this listing as it has interesting pattern aspects, and indeed, led to minor studies of the day (1987), as indeed with other patchwork books. However, now, and for some considerable time, the nature of the material is considered hardly worthy the time originally devoted to it.

No colour in book.


Ōuchi, Hajime. Japanese Optical and Geometrical Art. 746 Copyright-Free Designs for Artists and Craftsmen. Dover Publications Inc, New York 1977 (9 April 1993)

Flatters to deceive. Essentially of ‘geometrical motifs’, as with Hornung. No tessellation as such, save for one instance. No captions, index or discussions of the graphics renders finding anything is much frustrating. Seemingly op art influenced. A republication of the Japanese edition, titled Leading Part I. Of no real consequence.




Padamsee, Hasan S. Unifying the Universe: The Physics of Heaven and Earth. IOP publishing Ltd, 2003 (Google books, 16 June 2015)

P.132 Eight Heads.


Paling, D. Teaching Mathematics in Primary Schools. Oxford University Press 1982. (15 October 2011)

Only of interest in a historical sense, as it was one of the earliest* books on tessellation (and maths per se) I studied, c. 1986. Tessellation pp. 272-272, with the ‘any triangle, quadrilateral will tile’ rule. * Have I mixed this up with the book immediately below?


Paling, D. and J. L. Fox. Elementary Mathematics. Oxford University Press, 1965

From a reference in my own early studies, of 26 January 1986, 1, 2 December 1986. Not sure if in possession or not. The date here is not of the greatest clarity in terms of certainty; 26 January may be misleading; in that with a 1 December listing, I may simply have forgotten to change the year on the other.


Palmer, Kelvin. The Collector’s Guide To Cluster Puzzles Of The 1960s and 1970s. Self Published. 2003. (5 November 2013)

Essential reading on the subject. History of cluster puzzles (of the type as evinced by Escher’s Plane Tiling I and Plane Tiling II as devised by Palmer’s father, Alex, of the 1960s, with occasional reference to precursors of 1934 and 1943.


Pappas, Theoni. Mathematics Appreciation. Wide World Publishing/Tetra Revised edition 1987. (3 June 1993).



————. The Joy of Mathematics. Discovering Mathematics All Around You. Wide World Publishing/Tetra 1992. (3 June 1993)

Collection of popular mathematics, typically over a two-page spread, sometimes three pages. Tessellation pp. 120-122.


————. More Joy of Mathematics. Exploring Mathematics All Around You. World Publishing/Tetra 1992 (3 June 1993)


————. Mathematical Scandals. Wide World Publishing/Tetra. First published 1997, 3rd printing 1999 (31 August 2002)

A few good (brief) yarns, although apparently no original research per se, with the material seemingly taken from existing sources in the bibliography. In short, essentially cuts to the chase from other, more in-deth treatments.


Paraquin, Charles H. Eye Teasers. Optical Illusion Puzzles. Granada Publishing Limited 1979 (19 July 1992).

Juvenile. Usual repeats of established illusions.


Parsons, Richard. GCSE Mathematics Intermediate Level. Coordination Group Publications 1998. (21 September 2004)


Faryna-Paszkiewicz, Hanna and Zuzanna Fruba. Warszawskie gorseciki zanikajace (Translated: Warsaw’s Vanishing Corsets). Nisza Publishing House (in Polish) 2013, 336 pp. (13 August 2019)

On 'gorseciki' (corset) tiles, of recent (July 2019) interest. Fantastic!


Pasztory, Esther. Pre-Columbian Art. Everyman Art Library. Weidenfeld and Nicolson, 1998 (22 October 2016)

A opportunistic purchase, from a charity shop. Although not a maths book, included here as it discusses Escher-like tessellations in broad terms, of which I recall Branko Grünbaum discussing like aspects here. To be studied and assessed.


Paulos, John Allen. Beyond Numeracy. An Uncommon Dictionary Of Mathematics. Penguin Books 1992 (30 April 1994)


————. Innumeracy. Mathematical Illiteracy and its Consequences. Penguin Books 2000. First published 1988. (no idea as to when obtained, at a guess, 2000-2005)

Popular account. Small format paperback, of mild general interest, but nothing of note as regards my specific interests.


Peak, David; Frame, Michael. Chaos under Control. The Art and Science of Complexity. W. H Freeman and Company 1994 (3 August 2002)


Pearce, Peter and Pearce, Susan. Polyhedra Primer. Dale Seymour Publications 1978 (24 October 1998)

Non attributed Cairo tilings on page 35, and in the context of the Laves tilings, page 39.


Pearcy, J. F. F; Lewis, K. Experiments in Mathematics. Stage 1, 2 and 3 (3 books). Longmans, green Co Ltd 1967. (17 August 1997)

Juvenile. A bit like Mottershead, but for a younger age. Tessellations 14-15 (B1) reptiles 8-9 (B2).


Pedoe, D. The Gentle Art of Mathematics. Penguin Books 1963. First published 1958 (18 April 1993)

Small format largely popular paperback, 160 pp. As such, although largely on popular subjects, there is very little of direct interest to me here. Chapter 1 ‘Mathematical Games’. Chapter 5, ‘Two-Way Stretch’, on topology, has elements of interest. Chapter 6  ‘Rules of Play’, is on symmetry, with a Islamic  tiling p. 120. No other tiling.

From Wikipedia. Dan Pedoe (29 October 1910, London – 27 October 1998, St Paul, Minnesota, USA 1]) was an English-born mathematician and geometer with a career spanning more than sixty years. In the course of his life he wrote approximately fifty research and expository papers in geometry. He is also the author of various core books on mathematics and geometry some of which have remained in print for decades and been translated into several languages. These books include the three-volume Methods of Algebraic Geometry (which he wrote in collaboration with W. V. D. Hodge), The Gentle Art of Mathematics, Circles: A Mathematical View, Geometry and the Visual Arts and most recently Japanese Temple Geometry Problems: San Gaku (with Hidetoshi Fukagawa).


————. Geometry and The Liberal Arts. Penguin Books 1976 (14 July 2001)


Peitgen, Heinz-Otto et al. Fractals For The Classroom: Strategic Activities Volume 1  Springer-Verlag 1991 (6 August 1997)

Volume 1 of a two-volume series. As such, given their content I am at a loss to explain why I purposefully pursued these books (ordered from Claire Publications). Although I have a mild interest in fractals, this is really only in passing, as the subject quickly becomes highly technical, way beyond my understanding. I have a dim memory of them being on offer, at half price, and so it seems that this was enough to induce me. Whatever, the premise of the books are for the classroom, with the six authors (Peitgen, Jürgens, Saupe, Maletsky, Perciante, Yunker) leading lights in their field. Volume I (128 pp) is notably easier than volume 2 (187 pp). Even so, still much is obscure in the first. A graphics calculator seems a prerequisite. Unsurprisingly, no tessellation or Escher. Simply stated, an ‘unwise’ purchase. I have no plans to re-read.


Peitgen, Heinz-Otto et al. Fractals For The Classroom: Strategic Activities Volume 2  Springer-Verlag 1992 (6 August 1997)

As detailed above.


Penkith, F. E. Confidence Mathematics. Macmillan Education Ltd. Reprinted 1990 (first edition 1985) (27 October 2001, Louth).

No tessellation. Basic mathematics, utilitarian, for 12-year-old. Of significance in that this was one of the earliest maths book of all that I studied, c. 1986 or 1987.


Penrose, Roger. The Emperor’s New Mind. Concerning Computers, Minds, and the Laws of Physics. Vintage Books 1989 (25 July 1994)

Mostly too advanced for me. Occasional tessellation, of non-periodic tilings, and their background, pp. 172-178. Occasional Escher references, Circle Limit I p. 203. Quasicrystals pp. 562-563.


————. Shadows of the Mind. A Search for the Missing Science of Consciousness. First published in Great Britain by Oxford University Press, 1994. Vintage 1995. (5 November 2011)

Weighty tome, of  457 pp. As to be expected by the tenure of the book, this is almost entirely beyond me, the only aspect  that is vaguely understandable is a short discussion on tiling in the context of the ‘tiling problem’, in Chapter 1, ‘Consciousness and computations’, with two pages of polyomino diagrams, pp.  29-33, Robert Amman influenced. No Escher.


————. The Road to Reality. A Complete Guide to the Laws of the Universe. Vintage Books London, 2004. Grimsby Library

Minor Escher references and pictures, in conjunction with hyperbolic geometry, 33-35, 39 (all Circle Limit I), 47 (Angels Devils, sphere, plane tiling). Advanced, to say the least!


Perelman, Yakov. Mathematics Can Be Fun. Mir Publishers 1985

————. Recreativa Physics for Entertainment Figures for Fun, Fun with Maths & Physics Arithmetic for entertainment, Mechanics for entertainment, Geometry for Entertainment, Astronomy for entertainment, Lively Mathematics, Physics Everywhere, Tricks and Amusements (1913) 

Figures for Fun: Stories, Puzzles and Conundrums. Gardneresque (10 May 2017, Internet book archive download) Skim read. Minor geometric dissections 133, 140 and end.


Petersen, Ivars. The Mathematical Tourist. snapshots of modern mathematics. W. H. Freeman and Company New York. 1988 (23 August 1994)

Chapter 7, pages 200-212, ‘The Fivefold Way’, with Penrose tiles.


————. Islands of Truth: A Mathematical Mystery Cruise. W. H. Freeman and Company New York. 1990 (30 April 1994).

See ‘Paving the Plane’, pp. 83-86.


Petrie, Flinders W. M. Egyptian Decorative Art. Arno Press. 1978. First published 1895 (19 November 1994, York)

Checked for Cairo pentagon – no reference.


————. Decorative Patterns of the Ancient World. Bracken Books. First Published 1930. Studio Editions Ltd 1995 (26 August 1995)

Checked for Cairo pentagon – no reference.


Phillips, Peter, and Gillian Bunce. Repeat Patterns: A Manual for Designers, Artists and Architects. Thames & Hudson 1992.

First saw in art school library, and duly studied, disproportionately so, the exact memories of which have long since faded. The premise is of using a computer for drawing tessellations.


Pick, J. B. (Compiler). Dictionary of Games. Aldine Paperbacks, J. M. Dent  & Sons Ltd., 1952

Subtitled as ‘Outdoor, Covered Court, Gynnsuim and Indoor’. How to play 501 games

Of game and puzzle interest. Five chapters, of (I) Full Dress, Outdoor Games, (II) Informal Outdoor Games, (III) Covered Court Games, (IV) Gymnasium Games,  (V) Indoor Game, with many subchapters. Potted details, rules and history of games, not illustrated. Of  most interest is (V) Indoor Games, with a subchapter on Board and Table Games, pp. 229-263. However, there is nothing overtly mathematical here. The background of Pick went unresolved.


Pickover, Clifford. (24 July 2016)


Pickover, Clifford. The Pattern Book: Fractals, Art, and Nature 1995. WANTED

Dewdney’s ‘informal tesselation (sic) of Cats’ cluster puzzle picture. Not that he told me about this when we corresponded!


Pieper, Jan and George Michell, editors. The Impuse to Adorn. Marg Publications 1982

From a reference in Craig Kaplan’s thesis, p. 206. A reference to Haresh Lalvani and Pattern regeneration, on jalis.


Pinto, Edward & Eva R. Tunbridge and Scottish Woodware. G. Bell & Sons, 1970. c. summer 2016

Non-mathematical interest, primarily of jigsaw history. Especially see plate 8, of a ‘treadle operated jigsaw, by W. Fenner, about 1760’.


Pipes, Alan. Foundations of Art and Design. Laurence King 2008. Grimsby Library.

Has occasional Escher, with Day and Night.

Pizzuto, J. J. and P. L. D'Alessandro. 101 Fabrics. Analyses and Textile Dictionary. New York: Textile Press, 1952 (8 April 2019). Viewed online at Hathitrust, seemingly not available as a PDF

From a reference in Grünbaum (Satins and Twills article). The premise is of a illustration of a term (cashmere etc.) with a swatch,  technical details and then dictionary entry at the end of the book. Popular account. Houndstooth p. 43 Glen Plaid p. 39. Note that no other terms I have studied are included, such as Shepherd’s Check, Border Tartan etc. Useful, convenient reference to the fabric terms in a generalised sense.

Plichta, Peter. God’s Secret Formula. Discovering the riddle of the universe and the prime number code. Element 1997 (11 September 2000)


Pohl, Victoria. How to Enrich Geometry Using String Designs. National Council of Teachers of Mathematics. 1986. Third Printing 1991 (30 April 1994)

Geometric string designs, in two and three dimensions. Ostensibly for children, 68 pp. No tessellation. As such, now, and for a considerable while, only of limited interest. For a while, amid my geometric studies of 1987, I was interested in such designs, and so likely with that in mind obtained the book by chance, from my visit to John Bibby in York, where I had to make a choice of ‘buy or lose’ at the the time, this of course predating the internet. Although of a popular geometric nature, if seen for the first time I would now not be so enthused, and certainly not at cost price! Further, unlike other books from Bibby, this was not studied in any way, and so this, even of 1994, reflects a realisation that this was not of direct interest. Be that as it may, it just may have been, and so in that sense the purchase was justifiable.


Polster, Burkard (with foreword by John Langdon). Eye Twisters. Ambigrams & Other Visual Puzzles to Amaze and Entertain. Constable, London. 2007 (2010)

A prize in an Australian tessellation contest that I won run by Polster. Very nice indeed, in the same spirit as with John Langdon’s Wordplay. Escher section: ‘Escher & Co, with Drawing Hands, Magic Mirror, Day and Night, Relativity and other tessellations by Hop David, Ken Landry, Jos Leys, Peter Raedschelders, Henry Segerman, William E. Wenger, and Alain Nicolas.


Pólya, G. How to Solve It. Doubleday Anchor Books. 1957 (Date not stated, 10+ years).

Of limited interest.


Price, Jeffrey. M. C. Escher Amazing Images. (privately published book/catalogue). (28 March 2011)

Gift of Jeffrey Price. Much of interest, with many previously unpublished materials and Price’s own insights concerning Escher.


Priestly, J. B. Man and Time. Aldus Books London 1964.


Pye, David. The Nature & Aesthetics of Design. Barry and Jenkins Ltd 1978 (18 October 2008)


Pythagoras (the entire archive). Mathematics journal in Dutch. 

Of interest:

No. 4, April 1998 (1 April 2016). A whole issue devoted to Escher. Also of note is an article by Rinus Roelufs on the Cairo tiling ‘Tegels kleuren’, 22-23





Raba, Raoul. Zoo Mathématique. ACL Les Éditions Kangourou 1998 editor André Deledicq (In French) (14 March 2015)


Racinet, A. The Encyclopedia of Ornament. Studio Editions 1989. Originally published as L’Ornement Polychrome, 1856 and in English translation, Polychromatic Ornament by Henry Sotheran and Co. London 1873). First saw Grimsby Central reference library, 8 or 9 January 1989 (10 August 1993)

Essentially of ornament rather than tessellations. Although of a tome of major undertaking, it is of little direct interest as regards tessellation. Pages of interest, with tessellations, include 77, 123, 129, 135 and 149, albeit there is nothing in the way of innovation. Time constraints forbid an considered examination of the text. The two red and blue diagrams of Egyptian tilings, p. 77 are repeated in**.  p. 129 and 135 are Arabic patterns.

Note that this book was studied briefly, almost derisorily, on 10 January 1989, albeit seemingly of just a single sheet (amid non-related studies) and is of no consequence.

Fom Dover: Presents one hundred plates in color, comprising upwards of two thousand specimens of the various styles of ancient, oriental, and medieval art; including the Renaissance and the seventeenth and eigtheenth centuries. Though he himself was a distinguished painter and illustrator, Albert-Charles-August Racinet (1825–1893) is best remembered for two monumental color-plate publications he edited: Le Costume historique (Historic Costume) and L'Ornement polychrome (Color Ornament).
L'Ornement polychrome, a visual record in color of ornament and decorative arts from all over the world and throughout history to the end of the 18th century, eventually included 220 plates. The first 100 plates (Series I) appeared in ten installments between 1869 and 1873. A first edition of 5000 copies in volume form was published shortly after the completion of the installments, a second edition appearing as early as 1875.
This edition contains all the plates from Series I, with brief new English captions that summarize the French text. The copious material, ranging from Europe to Oceania and from ancient Egypt to just before 1800, is derived from architecture, painting, woodwork, metalwork, leatherwork, textiles, and many other art forms. Racinet's often-repeated purpose in publishing these decorative masterpieces was the encouragement and improvement of the arts of his own day, not only so-called fine arts but also the commercial arts involved in the designing and selling of manufactured goods. Dover's reissue of the plates, recognizing their perennial value and appeal, naturally is meant to serve the same purpose. Racinet's breadth of insight and catholicity of taste, truly enlightening for his day, give his selection a welcome variety and a consistently high standard of excellence; while the consummate skill of his artistic fellow workers and of his printer/publisher, the celebrated Firmin-Didot company, make these plates true works of art in their own right.


Raeburn, Michael. An Outline of World Architecture. First published 1973 Octopus Books Limited. (23 October 2015)

Although not a maths book per se, included here nonetheless as it includes occasional tiling, and more specifically a fused pentagon of a Cairo-like tiling at Amber Place, India, p. 55, having not seen before.


Ranucci, E. R. and Teeters, J. L. Creating Escher-Type Drawing. Creative Publications 1977. (15 October 1994)

Of its type, a good account of the general procedures of creating Escher-like tessellations, although as neither Ranucci and Teeters (and Ranucci in particular) can make any great claims as to talent in the field, the book is held back somewhat. The all-important issues underlying life-like tessellation are not discussed. Broadly, the book appears to be aiming at a juvenile audience.


Ranucci, Ernest R. Tessellation and Dissection. J. Weston Walsh. 1970 (The date stamp is only semi legible, apparently 2008 or 2009)

Somewhat of a lightweight production, of just 79 pages. The mathematics is of a popular level, seemingly of a school age nature, of about 12-14 years. Has a variation of the Cairo tiling, with two pentagons, p. 36. As such, it has not influenced my studies directly.


Ravenna, Daniele. Australia Puzzle: Contemporary Silverware & Jewellery. Photographs Mario Tedeschi, text Daniele Ravenna. Sydney: Puzzle Pty, 1994. (November 2016)

Gift of Lorenzo Logi. Of note is that it contains example of Angiolo Logi’s cluster puzzle work.


Rawson, Phillip. Creative Design A New Look at Design Principles. Macdonald and Co (Publishers) Ltd 1987. (29 August 2005)

First came across the term ‘simulacrum’ page 150 from this book. Islamic pattern p. 90.

Ray, Simon. Indian & Islamic Works of Art. Self Published, 2017, pp. 177-178 (2019)
Of Cairo tiling interest, due to a jali, said to be of the 17th century and so possibly of the utmost significance historically of the first known extant pentagonal tiling. As such, the 17th century dating here is thus largely taken on trust; the provenance is lacking in its caption (in contrast to others). To try and clarify matters naturally I attempted to contact Simon Ray (a dealer in Indian and Islamic Works of Art, in London, UK) for more details. However, despite two emails from myself, stressing the historical importance, without reply, and then at my behest two others from interested parties, namely professors Gregg De Young and Chaim Goodman-Strauss, Ray once more chose not to reply! Hence the stated date given is thus on trust. In the (most unlikely) event of you reading this Simon, do by all means redeem yourself here!

Rayner, D. Higher GCSE Mathematics: Revision and Practice. Oxford University Press, 1994 (4 August 2001)

Textbook, and as such, of limited interest; the book has no tessellation aspects per se, save for some ‘regular pentagon loops’, albeit strictly of ‘patches’, p. 51.


Razzell, Arthur G. and K. G. O. Watts. Symmetry. Mathematical Topics 3. Rupert Hart-Davis 1967 (22 January 1994)



Read, Ronald C. Tangrams - 330 Puzzles. Dover Publications, Inc (18 March 2000)


Reader’s Digest Books and Articles – see Moore, Alison, Keeton, Greg.


Rees, Martin. Just Six Numbers

Has Escher’s * and * pp**.


Reichelt, Gotz-Peter. Tier welten (in German) c. 2003 (6 June 2016)

On his interlocking wood carved animal puzzles, namely cluster puzzles. Most pleasing indeed, with quality examples throughout.


Renko, Hal; Edwards, Sam. Tantalizing Games for your TI99/4A. Addison-Wesley Publishers Limited. 1983 (10 October 1993).

‘Early’ computer book, badly dated. Purportedly ‘Escher’ pp. 50-54, with computer instructions, although none of Escher’s tilings/prints are illustrated. So lightweight as regards Escher to be barely worth the mention.


Rey, Marc Lachieze- and Jean-Pierr Luminet. Translated by Joe Laredo. Celestial treasury. Form the music of the spheres to the conquest of space. Cambridge University press, 2001 (15 August 2015)

Although on astronomy, has sideways references to mathematics, namely with polyhedra, pp. 48-51, Jamnitzer and Kepler p. 57.


Reyes, Encarnación and Inmaculada Fernández. Pentágonos. Construcciones. Mosaicos, Geometría sagrada. (in Spanish) 2015. Universidad de Valladolid (21 January 2016)

Has much of interest in a generalised sense, although hindered in understanding in that it is in Spanish. P. 166 has an interesting ‘mixed’ Cairo tiling, with kites. A mention of  myself and collaborator Helen Donnelly on pp. 74 and 156, and  photos on the front cover.


Reichmann, W. J. The Spell of Mathematics. Methuen & Co Ltd First published 1967 (14 July 2001).

Small format hardback, 15 Chapters,  272 pp. Some popular philosophical musings as to the attraction (or ‘spell’) of mathematics. Of limited interest. Too advanced in parts, but still largely accessible. No tessellation or Escher. Perhaps of most interest is on the cycloid, pp. 160-161. Also occasional geometry throughout the book.


Richardson, Margaret H. The Sign of the Motor Car. Dennis, Massachusetts, 1926, privately printed. (PDF 2020)

By a pioneer of cluster puzzles. ‘A biographical sketch’, as quoted by Anne Williams.


Riley, Noel. A History of Decorative Tiles. Grange Books 1997 (11 June 2015, Grimsby library)

Examined on the likely possibility of tessellation, but not so, at least of any substance.


Robertson, Bruce. Learn to Draw Step-by-Step. Macdonald & Co (Publishers) Ltd 1987 (undated c. 1997?)

Although not a mathematics book by any stretch of the imagination, as it is primarily of art procedures, as it contains Escher and pattern aspects, albeit briefly, I thus include. A pastiche on Escher's Day and Night, p. 37. An interesting technique for drawing patterns is given, pp.178-179. This influenced my studies of the day when first seen, in December 1987.


Rogers, James T. The Story of Mathematics. Hodder and Stoughton 1979 (8 August 2004) History, 16-year-age range.


Rogers, Nigel. Consultant editor Dr Ian Gordan. Incredible Optical Illusions. A Spectacular Journey Through The World Of The Impossible.  Quarto, 1998. First published by in Great Britain by Simon & Shuster Ltd. (28 April 2018)

Rooney, Anne. The Story of Mathematics. From creating the pyramids to exploring infinity. Arcturus Publishing Limited, 2008. (30 November 2019)
Some good stories, some of which I was unaware of. Nothing on tiling as such, albeit Escher is mentioned on pp. 115 and 118.


Roojen, Pepin van. Islamic Designs From Egypt. Pepin Press, 2007 (7 August 2014)

Obtained on the off chance of a Cairo tiling appearing, of whatever form. However, there is no Cairo tiling in the book. Indeed, the whole book is one of relative disappointment, it consisting solely of pictures, with each page of a tiling or pattern, but without any text to put the pictures into context. Without such information, this thus loses any overall value it may have had. On occasion, I recognise the picture source (such as the ‘fused Cairo’), but this is indeed rarely.

The accompanying CD-Rom is of a like nature.


Ross, Alistair. The story of Mathematics (as in original). A & C Black (Publishers) Limited 1984 Fist saw in Cleethorpes library c. 29 August 1987 (12 December 1998).

Rangoli and Islamic tilings p. 21. Use of Escher’s Relatively print Frontispiece and p. 25. Juvenile. This led to studies of p. 21 (not Escher-like), of three different periods, of August/October 1987, July 1988 and January 1991. As such, the studies were of relative depth of the day, albeit now, and for some time, somewhat overstated as to their inherent importance.

Row, Sundara T. Geometrical Exercises in Paper Folding. Madras, 1893. Edited and Revised by W. W. Beman and D. E. Smith, Chicago 1917 (Downloaded from Internet 13 May 2015)
From a reference in MacMahon, although noted before elsewhere. Begins with a few simply polygon folds, before moving on to more advanced work. A book full of interest, although whether I will be able to find the time to study this is any degree of depth (or indeed in passing) is doubtful. Has a small section (five pages) on pentagon folding, but not relevant to tiling matters.

Rowland, Kurt. Looking and Seeing. Notes for teachers. Book 1 Pattern and Shapes. Book 2 The Development of Shape. Book 3 The Shapes We Need. Ginn and Company Ltd. 1965 (2 July 1995)

All books are text only.


Roza, Greg. An Optical Artist: Exploring Patterns and Symmetry. The Rosen Publishing Group, Inc. 2005 (28 March 2011)

Gift of Jeffrey Price. Has Escher cover of Hand with Reflecting Sphere (Juvenile).


Rubin, Don. What’s the Big Idea? And 35 other unusual puzzles. J.B. Lippincott Company 1979. (9 July 1995)


Rucker, Rudy. The Fourth Dimension: A Guided Tour of the Higher Universes. Houghton Mifflin, 1984. A later edition is of 2012, with a change of subtitle (30 December 1989) NOT IN POSSESSION

A popular account of advanced concepts. A minor study worked on on 2 January 1990, first seen at Grimsby central library, long since deleted from stock. Nothing on tessellation per se in my study.

The book is available for free on his website (and with other publications of his, notably Infinity of the Mind), but without the page numbers, which means finding references is a chore. Minor Escher reference with pseudsosphere p. 108.


Russell, Betrand. Wisdom of the West. First published by MacDonald & Co (Publishes) Ltd, 1959. (28 May 2005)

On philosophy, with occasional mathematical references. However, finding and sorting ‘useful’ maths here for my purposes is few and far between.


Rust, Murray- T. M. Mathematical Pattern. Mathematics for the Majority. Chatto & Windus 1971 (22 August 2004).

One book of the seven-part ‘Mathematics for the Majority’, series, of which I have two. The book seems to have been compiled by a ‘project team’, with one primary author stated. The books are stated as ‘Chatto & Windus for the Schools Council’, which thus gives the intended audience. The back cover states ‘This Schools Council project was set up to further the teaching of mathematics in secondary schools to children of average and less-than-average ability’. Also see Machines, Mechanism and  Mathematics by A. E. Bolt and J. E. Hiscocks for another I have in this series.

Of ‘pattern’ in the broader sense, of number and geometry. The topics of this book are broadly out of my mainstream interest, but it still has isolated aspects of interest. There is no tessellation. Symmetry pp. 25-28, Polyhedra pp. 36-37, Golden Section pp. 61-63. Has interesting book list, pp. 66, with unknown E J. James reference and series of  ‘Topics of Mathematics’, Cambridge.




Sabin, Francene and Louis. ‘The One, The Only, The Original Jigsaw Puzzle Book’. Chicago Henry Regnery Company, 1977 (11 April 2017)

From a reference in Williams, although found first by ‘favoured chance’ on the web, of the first chapter, An Irreverent History of the Jigsaw Puzzle which ‘showed promise’, hence a speculative purchase. A somewhat quirky book, seemingly primarily of a humorous premise rather than any attempt at scholarly insight. Overall, rather silly, with alternate chapters of a single page…. Although there is indeed a history, in which Spilsbury’s place is detailed, the impression given is that this is the authors own research, as no references are given. However, this is not so; without doubt, the Sabins are borrowing from Hannas. You win some and you lose some…


Sackett, Dudley. The Discipline of Number. Foundations of Mathematics. Sampson Low, Marston and Co: London 1966. (Junior) (24 October 1996 or 1998)


Sackson, Sid. A Gamut of Games. Hutchinson & Co. Ltd. 1983. First published 1969 (27 August 1997)

Gardneresque. Stated on the back cover as ‘diversified collection of 38 remarkable, intellectually stimulating indoor games…. many of the book’s best games are the invention of the author’. Martin Gardner praises it. In six sections. Each section begings with a small essay, followed by the games. Has a useful section on ‘short reviews of ganes in print’, pp. 188-221. Sackson is widely recognised as an authority on games, and game history. Of general interest, but much of the material, being non-geometrical, is of little direct interest. It is  ideal for reference purposes, but not for actual study.

Sagan, Carl. The Cosmic Connection. Book Club Edition. First published in 1973. Second edition 2000. Seen on Internet Archive as ‘community texts’ (24 December 2019).

Makes occasional use of Escher prints, albeit without any commentary at all. Another World, Part 1

Wikipedia: … an expanded edition with contributions from Freeman Dyson, David Morrison, and Ann Druyan was published in 2000 under the title Carl Sagan's Cosmic Connection. The book contains artwork by Jon Lomberg and other artists.

Of note is that Sagan had a decided interest in Escher. Also see his Dragons of Eden: Speculations on the Evolution of Human Intelligence for more of Escher’s works. Further, he used Another World as a background to the film of the (famed) series Cosmos (Episode 10: The edge of forever). (As noted in a Facebook Escher group posting by Jose David Avila Arevalo). Note that this was not mentioned in the book of the series.

It would be interesting to know more of Sagan’s interest.

————. Dragons of Eden: Speculations on the Evolution of Human Intelligence. Random House. First published 1977 (and numerous reprintings). Seen on Internet Archive (24 December 2019).

Makes occasional use of Escher prints, albeit without any commentary at all. Plane Filling II on cover, but not on 1977 edition! Reptiles print inside cover. Three Spheres, desaturated, is used on all nine chapter headings. Plane Filling II, p. 78. Stars, p. 231. Escher is not mentioned in the index or seemingly credited elsewhere aside from the captions

Wikipedia: ... the author combines the fields of anthropology, evolutionary biology, psychology, and computer science to give a perspective on how human intelligence may have evolved.

… The book is an expansion of the Jacob Bronowski Memorial Lecture in Natural Philosophy which Sagan gave at the University of Toronto.


Salvadori, Mario. The Art of Construction Projects and Principles for Beginning Engineers & Architects (25 October 2014) Chicago Review Press 1990, third edition

Occasional crossover to mathematics.


Sanchez, Miguel. The Alhambra and the Generalife. Publisher unclear. 1976. (5 December 1992, small and 30 August 1998, large)

No Cairo pentagon.


Sarcone, Gianni A. and Marie-Jo Waeber. Amazing Visual Illusions. Arcturus Publishing Limited, 2011 (5 January 2013)

Although not a maths book it is included here as it has crossovers. Popular account. Yoshifugi Utagawa (not credited) cover and p. 27, with elephants and children; Duck/Rabbit p. 58 in Fliegende Blatter, 1892, Escher’s Relativity, p. 74. Convex and Concave, p. 74. Occasional new illusions.


Sardar, Ziauddin, and Iwona Abrams. Ed. Richard Appignanesi. Introducing Chaos. Icon Books UK 2002. (date unclear, 2002?)

Popular account of chaos, as a part of a series of like books.


Sarhangi, Reza (Ed). Bridges. Mathematical Connections in Art, Music, and Science. Conference Proceedings 1998 (7 March 2006) South Western College, Kansas. (The first Proceedings)


Sarhangi, Reza (Ed). Bridges. Mathematical Connections in Art, Music, and Science. Conference Proceedings 1999 (7 March 2006) South Western College, Kansas. (Second)


Sarhangi, Reza (Ed). Bridges. Mathematical Connections in Art, Music, and Science. Conference Proceedings 2000 (7 March 2006) South Western College, Kansas (Third)


Sarhangi, Reza (Ed). Bridges. Mathematical Connections in Art, Music, and Science. Conference Proceedings 2001 (7 March 2006) South Western College, Kansas (Fourth)


Sarhangi, Reza (Ed). Bridges. Mathematical Connections in Art, Music, and Science. Conference Proceedings 2002 (7 March 2006) Towson University (Fifth)


Sarhangi, Reza; Carlo Séquin (Ed). Bridges. Mathematical Connections in Art, Music, and Science. Conference Proceedings 2004 (7 March 2006). South Western College, Kansas (Seventh)


Sarhangi, Reza; Moody, Robert V. (Ed). Renaissance Banff. Bridges. Mathematical Connections in Art, Music, and Science. Conference Proceedings 2005 (7 March 2006). Canada (Eighth)


Sarhangi, Reza; John Sharp (Eds). Bridges. Mathematical Connections in Art, Music and Science. Conference Proceedings 2006 Tarquin (10 August 2006) London, England (Ninth)


Sarhangi, Reza (Ed). Bridges. Mathematical Connections in Art, Music, and Science. Conference Proceedings 2008 (7 March 2006) Towson University Leeuwarden, Netherlands (Tenth)

Cairo reference and diagram page 102. B. G. Thomas and M. A. Hann in ‘Patterning by Projection: Tiling the Dodecahedron and other Solids’ gives an equilateral pentagon


There are, however, equilateral convex pentagons that do tessellate the plane, such as the well known Cairo tessellation shown in Figure 1. Also, other minor references essentially in passing.


Bridges & Passages. Outdoor Exhibitions. Bridges 2008 Leeuwarden Catalogue

Collection of essays of featured artists in churches: Istvan Orosz, Yvonne Kracht, Ulrich Mikloweit, Koos Verhoeff, Rinus Roelufs, Oscar Reutersvärd, Gerard Caris, Elvira Wersche.

Occasional use of Cairo tiling by Roelufs, but not credited.


Sarton, George. The Study of the History of Mathematics & The Study of the History of Science. Two volmes bound as one. Dover Publications Inc. New York 1954 (7 December 1994)

Free, college library. A serious, though still readable  discourse. I find the book a little odd. It is essentially a study of a study!  I can’t see how I could gain from re-reading this.


Sattin, Anthony and Sylvie Franquet. Explorer Egypt. AA Publishing. Reprinted 2000. First published 1996 (not seen). (18 May 2013)

Although not a maths book in any way, included as it has incidental instances of the Cairo tiling. Typical tourist guidebook, picture heavy. Two sightings, page 47, of the Old Cataract hotel, and page 222, of the relics in the Al-Alamein war museum grounds. Both pictures are not ideal, with as usual the subject matter being not the pavings themselves. Of the two, the Cataract instance is by far the best, but even so, one requires foreknowledge to discern individual pentagons, albeit it is not too far from being identifiable as distinct pentagons. The Al-Alamein sighting is much the poorer, taken at a raking angle, and only with foreknowledge is the tiling known, the picture is essentially of square tiles in a chequerboard formation.


Sautoy, Marcos du. Finding Moonshine. A Mathematician’s Journey Through Symmetry. Fourth Estate, London. 2008. Library.

Many occasional references to Escher, mostly in passing. Those of note include pp. 24-26, 76-79.


————. Symmetry. A journey into the Patterns of Nature. Harper Perennial 2009. First published in Great Britain as Finding Moonshine. (3 January 2015)

Much of general interest, of 12 chapters built around the year, but especially of October: The Palace of Symmetry 62-87, with Escher heavily featured, Alhambra tiling discussion. An old fashioned ‘good yarn’, with complex mathematics discussed in simplified terms for the layman. Nice discussion on Simon Norton among others.


Sawyer, W. W. Prelude to Mathematics. Penguin Books 1961. (9 July 1994).

Small format paperback, albeit of 214 pages. Of limited interest, due to the nature of Sawyer’s writings, which largely focus on calculation, not my forte. although there are chapters ostensibly of interest, such as Chapter 6, ‘Geometries other than Euclid’s’, and Chapter 12, ‘On Transformations’, these are still advanced for me. As such, I find Sawyer’s writings (like Coxeter’s), not conducive to my understanding (my fault, not theirs!). I am not inclined to re-read this.

From Wikipedia: Walter Warwick Sawyer was born in St. Ives, Hunts, England on April 5, 1911. He attended Highgate School in London. He was an undergraduate at St. John's College, Cambridge, obtaining a BA in 1933 and specializing in quantum theory and relativity. He was an assistant lecturer in mathematics from 1933 to 1937 at University College, Dundee and from 1937 to 1944 at University of Manchester. From 1945 to 1947, he was the head of mathematics at Leicester College of Technology.

In 1948 Sawyer became the first head of the mathematics department of what is now the University of Ghana. From 1951 to 1956, he was at Canterbury College (now the University of Canterbury in New Zealand). He left Canterbury College to become an associate professor at the University of Illinois, where he worked from winter 1957 through June 1958. While there, he criticized the New Math movement, which included the people who had hired him. From 1958 to 1965, he was a professor of mathematics at Wesleyan University, where he edited Mathematics Student Journal. In the fall of 1965 he became a professor at the University of Toronto, appointed to both the College of Education and the Department of Mathematics. He retired in 1976.

Sawyer was the author of some 11 books. He is probably best known for his semi-popular works Mathematician's Delight and Prelude to Mathematics. Both of these have been translated into many languages. Mathematician's Delight was still in print 65 years after it was written. Some mathematicians have credited these books with helping to inspire their choice of a career.

Sawyer died on February 15, 2008, at the age of 96. He is survived by a daughter.


————. Vision in Elementary Mathematics. Introducing Mathematics: 1. Penguin Books 1964 (2 April 1994)

Small format paperback, albeit of 346 pages.Of limited interest, due to the nature of Sawyer’s writings, which largely focus on calculation, not my forte. The first in the series of four books (of which I have two) on the pemise of ‘Introducing Mathematics’. Better than his other book The Search for Pattern. However, I find Sawyer’s writings (like Coxeter’s), not conducive to my understanding (my fault, not his) I am not inclined to re-read this.


————. The Search for Pattern. Introducing Mathematics: 3 Penguin Books 1970. (17 September 1994).

Written in the same vein as with the first book in the series, as detailed above, and of which the same comments apply.

Not recreational maths.


Sawyer, W. W (ed.) Mathematics in Theory and Practice. Odhams Press Ltd. 1948 (29 November 1992)

Very much of its day, with much calculation, although that said, much is readable.


Scharf, Aaron and Stephen Bayley. Introduction to Art. The Open University. An Arts Foundation Course, Units 16, 17 and 18 (30 March 1994, Boston)

Escher’s Circle Limit IV, pp. 172-173, discussed in the context of dual function of shape.


Schattschneider, Doris. Visions of Symmetry. Notebooks, Periodic Drawings, and Related Work of M. C. Escher. New York. W. H. Freeman and Company 1990. (20 February 1991)

Revised edition 2004 (23 March 2010)

Indispensable! Highligh after highlight.


Schattschneider, Doris and Wallace Walker. M.C. Escher Kaleidocycles. Tarquin Publications 1982 (19 August 1988)

Cairo like tiling, p. 26, and a short discussion as to Escher’s. I also have a German edition, M. C. Escher Kaleidozyklen. Taschen 1992 (10 August 1993).


Schattschneider, D. and M. Emmer (editors). M. C. Escher’s Legacy. A Centennial Celebration. Springer. First edition 2003, paperback 2005. Springer (31 August 2005)

41 papers from the conference, full of interest. Highlights include Rice’s, ‘Escher-like patterns from Pentagonal Tiles’, pp. 244-251. Brief von Hippel reference p. 60.


Schlossberg, Edwin; Brockman, John. The Pocket Calculator Game Book 2. Corgi Books 1978 (18 October 1997)

Schubert, Hermann. Mathematische Mussestunden. Volumes I, II and III. Leipzig 1898. (Downloaded from internet 14 May 2015)
From a reference in MacMahon. In three volumes. Volume I has nothing in the way of tilings or polyhedra. Chapter on 1-15 puzzle, 133 (142). Volume II. Again no tilings or polyhedra. Has a chapter on Geometrical Problems pp. 112-126 (129-138), but without tilings. Volume III appears of a more technical nature, mostly text, of few diagrams. Nothing on tilings and polyhedra.

————. The Fantastic World of Optical Illusions. Carlton Books 2002 (date has faded, 2007)

Although not strictly a mathematics book it is included here nonetheless, as it has a loose crossover. Delightful. Mattheau Haemakers dressed as man holding an impossible cube. p. 14, Escher portrait tiling by Ken Landry on frontispiece and p. 272, a physical model of Escher’s Belvedere, p. 273. Penrose stairs p. 290.


Scripture, Nicholas E. 50 Mathematical Puzzles and Oddities. Van Nostrand 1963 Viewed (not downloaded) at Internet Archive, 1 March 2018


Small format, 83 pages. Stated as for teachers to enliven lessons. Covers a broad mathematical puzzle spectrum: Oddments in Artithmetic, Oddments in Algebra, Oddments in Geometry, Miscellaneous Oddments, Answers, Book List (although the book list is not seen!). By far the most interesting is on Oddments in Geometry, although there does not appear to be anything too original here. Discusses Geometric Dissections, pp. 52-54, including Perigal’s dissection pp. 52-53. Although most is popular, it has all been seen before. No tessellation.

The author, largely unknown,  appears to be of a puzzler/mathematician, as he has (at least) two other like books to his name.


————. Puzzles and Teasers. Faber and Faber. First published 1970 (24 October 1998)

Small format hardbook, 96 pages. Dudeneyesque. Covers a broad puzzle spectrum: Mathematical Puzzles,  Logic Puzzles, Crossword Puzzles, Word Puzzles, Vocabulary, Literature and General Knowledge, Oddity-Box, with solutions. Nothing geometrical. To what, if any extent these are original is not made clear.


Sealey, L.G. W. The Shape of Things. Basil Blackwell Oxford 1967 (12 October 2002)

Juvenile 10-years-old audience.


Seckel, Al. Incredible Visual Illusions. Arcturus Publishing Ltd, 2005 (not stated) (guess 2008?)

Wide ranging. Yohifugi Utagawa Ten Bodies and Five Heads p. 158, titled as a ‘change in meaning illusion’ (not credited) C. 2005 Escher ‘section’ pp. 117-119, with Belvedere, Waterfall and Ascending and Descending. Fish Tesselation p.50 (unaccredited (stated as), Original face/vase illusion p. 48 ‘American Puzzle cards’ by E. K Dunbar and Co. Boston 1880.


Seiter, Charles. Everyday Math for Dummies. Hungary Minds Inc. 1995 (17 April 2005)


Seitz, William C. The Responsive Eye. The Museum of Modern Art, New York, in collaboration with the City Art Museum of St. Louis, and others, 1965. (23 December 2016) PDF

The Responsive Eye catalog commemorates the show of the same name at the MoMA in 1965. A show several years in the making, it was the first to introduce the public to Optical (or ‘Op) art.

Artists featured in the show and catalog include the well-known Victor Vasarely and Joseph Albers as well as the sensational and underappreciated Paul Feeley and collective work by Equipo 57, a group of Spanish artists, among others.

Of note is painted tessellation by Equipo 57 (a Spanish collective), p. 23; Schröeder’s staircase p. 31; and Mavignier p. 33, of whom has a loose parquet deformations of sorts and of which upon subsequent searching has other works a like nature.


Serra, Michael. Discovering Geometry. Key Curriculum Press, 2008 (30 August 2016, select part seen previously)

Note that I have only seen a small part of the book, namely Chapter 7, made available on the web, namely transformations and Tessellations. Of most interest is chapters 7.4-7.7. The book is aimed at a school age audience, of 11-16. Of perhaps most note is that of P*, where I discovered Rice’s connection of the Type 13 pentagon, derived from a Cairo tiling. the conjunction of the tiling, and the Cairo tile, put the seed in my mind, although this is not made clear in the book. Also of interest are some children’s tessellations. Although these are mostly typical, of poor understanding, a few are markedly better than others, such as ‘Perian Warriors’ by Robert Bell and ‘Sightings’ (Elvis Presley) by Peter Chua and Monica Grant respectively. Use is made of Escher's prints. As such, there is nothing new here, aside from the original artwork, but nonetheless a welcome basic introduction to tessellation and Escher-like aspects.


Seton-Williams, Veronica and Peter Stocks. Egypt. Blue Guide. A & C Black Publishers Ltd Second Edition 1988

Obtained, by a chance finding, of possible Cairo tiling interest. Described as a description and travel guide, and furthermore of an scholarly (although still readable), extensive nature, of a narrow fomat paperback, of 743 pages! Simply stated, this was seen at a car boot sale, and of which it was thus  impractical to view for Cairo tiling aspects. On the off chance of usefuness, duly obtained. Upon a more leisurely read, there was no Cairo tile references in any capapacity. But there might have been! The matter is at least settled, rather than regretting having left the possibility open-ended. Previously, the term Blue Guide (from the colour of the cover) was unknown to me. From Wikipedia:

The Blue Guides are a series of highly detailed and authoritative travel guidebooks focused on art, architecture, and (where relevant) archaeology along with the history and context necessary to understand them. A modicum of practical travel information, with recommended restaurants and hotels, is also generally included.


Seymour, D; Britton, J. Introduction to Tessellations. Dale Seymour Publications 1989 (8 March 1995)

Cairo tiling, but not attributed, p. 39.


Seymour, Dale. Introduction to Line Designs. Dale Seymour Publications 1992 (10 August 2006)

Advanced Juvenile.


————. Geometric Design. Dale Seymour Publications 1988 (24 October 1998)

Various geometric designs, based upon circles, as in the style of Hornung. The book is pitched at a late junior school age level, and is picture-led, with simple geometric constructions given, and then latterly, in the appendix, such as a bisecting a line; the only text is the appendices. There is no tiling as such. As such, the book has not influenced my studies in any way.


Sharp, Richard; John Piggott (ed.) The Book of Games. Artus Publishing Company. Date faded 2000?

Card and board games.


Shaw, Sheilah. Kaleidometrics: The art of making beautiful patterns from circles. Tarquin Publications 1981 (3 June 1993)

Broadly, a ‘geometric design’ book per se. This concerns making symmetrical designs of a ‘Kaleidoscope’ theme using circles as the underlying framework, with 22 examples, and with text, likely purposefully, at a minimum. It is not clear as to the target audience. No mathematics at all really. The book lacks structure; it has no formal contents and introduction. As such, there is very little of direct interest for me here, save for page 23, which has a ‘whirling squares’ tessellation. The designs are somewhat repetitive and trite; a multitude of such examples are possible. No tessellation as such. The book is lightweight, of just 40 pages.


Shefrin, Jill. Neatly Dissected, for the Instruction of Young Ladies & Gentlemen in the Knowledge of Geography: John Spilsbury & Early Dissected Puzzles Cotsen Press, 1999 (6 December 2016)

Of jigsaw puzzle interest. Speculative purchase on account of the book being frequently quoted in serous jigsaw bibliographies. Some outstanding research of the highest order on Spilsbury by Shefrin. In particular, each of the five puzzles in the cabinet are examined and described in depth. Although a slim volume, of just 40 pages, the content is most interesting. One shortcoming is that it lacks an index. Darton mention on p. 17.


————. Such Constant Affectionate Care: Lady Charlotte Finch, Royal Governess & the Children of George II. Cotsen Occasional Press, 2003 (3 June 2017)

Of jigsaw puzzle interest. Some outstanding scholarship by Shefrin. Has much new insight on Mme Beaumont pp. 69-76 (and elsewhere) and Spilsbury, and with a inventory of his known dissected maps. Also the much discussed cabinet, with attached note as to provenance and claim. And of course on Finch herself. Will stand numerous re-readings. Also of note is a possible precursor to the four-colour problem, p. 8


————. The Dartons - A Bibliographic Checklist. Hes & de Graaf 2009 WANTED £125

Of jigsaw puzzle interest.

Shubnikov A. V. and N. V Belov. Coloured Symmetry. Pergamon Press 1964 (13 October 2006)

Largely academic, and so mostly beyond me; mostly concerning group theory and crystallography elements. Very occasional tessellation – see ‘Mosaics for the Dichromatic Plane Groups’, p. 220, with a pull-out. However, even this is theoretical. One aspect of interest here is diagram 10, which resembles the famous Café wall illusion, but with parallelograms, rather than rectangles. Also see Plate 1, on p. 229 for further tiling diagrams, but of such simplicity of no real interest.

This also contains obscure crystallography articles by Russian authors, such as Belov, as an English translation.


Shubnikov, A. and V. Koptsik. Symmetry in Science and Art. Plenum Press 1974 (12 December 2006)

Symmetry in all aspects. Somewhat difficult to assess. Largely of an academic nature, but with occasional aspects of a recreational level. Cairo tiling p. 180, albeit by default of quadrilateral tilings p. 176-179. Escher lizards, unicorns figures pp. 228-229 (colour plate), birds p. 364, winged lions p. 365.

Interestingly, as regards to the winged lions’, Schattschneider [1990] also refers to this as a ‘winged lion’, despite these creatures bearing little resemblance to a lion, wings or not. Was her description taken/influenced by Shubnikov? She knew of this book.


Sibbald, Tim M. and Miranda Wheatstone.. ‘Advancing Escher art through generalization’. Ontario Association for Mathematics Education Gazette, 54(4), 23–26, 2016. (9 February 2018)

The Escher aspect is somewhat overstated.


Singh, Simon. Fermat’s Last Theorem. Fourth Estate, London 1998 (19 February 2007)


————. The Code Book. Fourth Estate, London 1999 (30 June 2013)

General interest.

The Simpsons and Their Mathematical Secrets. Bloomsbury, 2013 (28 December 2019).

Escher 91-92, in relation to Relativity.


Singmaster, David. Notes on Rubik’s ‘Magic Cube’. Self Published, Fifth Edition, 1980.  (20 September 1992)

Small format paperback 68 pp, of condensed text. One of the earliest books on the Rubik cube, at the height of the craze. Written from a group theory viewpoint, with much of the text way beyond my understanding. Likely, I quickly gave up on this! Mentions a few high profile names who I didn’t know were interested in this, such as Roger Penrose and PeterMcMullen.


Silverman, David L. Your Move. Kaye & Ward. 1971 (24 October 1998)

100 various puzzles and games under various descriptions, all at a popular level, such as ‘Potpurri I’, ‘Bridge’, ‘Chess and Variations’, ‘Checkers and Variations’ etc., with each puzzle on a single page followed by the answer. No tiling or polyhedra.


Sirett, Natalie. Drawing Visual Illusions. How to Have Fun Creating Masterpices of Deception. Arcturus Publishing,  First saw I believe 2010, Grimsby Library. (20 May 2017)
Popular account, 128 pp., of visual illusions, with Escher’s Relativity on the cover. A veritable disaster, from start to finish; the author has no grasp of the subject whatsoever! Where to begin? Shortcomings and incorrectness abound. The book is loosly themed upon six sections titled as ‘games’. Escher and tessellation is to the fore, with many references. Section 3, Games with Pattern, pp. 56-83 contain tessellations, of both ‘Escher-like’, as devised by the author, and ‘pure’ tilings, such as the semi regular. The premise is one of a tutorial nature, with guidance of how to create Escher-like motifs. Quite frankly, this section has to be seen to be believed! The tessellations shown are typical of people with no understanding of the subject. Just four tessellations are shown, supposedly of an octopus, two different birds (one in conjunction with a star), and a starfish. I cringe at these. All of these are particularly bad. No, that’s too generous, instead read quite appalling, of no artistic merit whatsoever, all undeserving of being in a book. Not a single tessellation here is of any worth. The standard is quite appalling. What can one say of this quite sorry mess? This is unforgivable. Aside from Escher and tessellation, a series of other illusions are supposedly discussed, but the treatment is so slack. Even aside from the content there are errors in the text and incorrect statements:

P. 18. Cube illusion of ‘three  ways’
P. 40. Narcisus starving to death
P. * absolulutely unique’.
P. 69. It is  stating that Escher’s periodic drawing 110 is undated. Really? This is news to me. This can be seen on the drawing itself as 1961 - see
 Visions of Symmetry. A case of simple lack of checking, of which Sirett couldn’t even be bothered to spend a few seconds finding out. And I suspect many others, of which I lack the will to confirm.
Further, her artwork is no nore than that an average school child. Amazingly she was ‘trained in Fine Art at the University of Newcastle-upon-Tyne. Studied painting and drawing at the Royal Academy Schools, London’.

The book lacks a bibliography.
Finally, Sirett is completely unknown in tessellation circles. How she can have the audacity to pronounce herself as an authority on the subject with these truly sorry examples is unbelievable.


Slade, Richard. Geometrical Patterns. Faber and Faber Limited, 1970. (24 October 1998)

The jacket describes as ‘This is a fascinating book for children….’, which gives the intended audience. Slade describes himself as a teacher of handicrafts. He is also the author of nine other books on handicrafts, with possible interest Patterns in Space although from the jacket description inclined towards the handicraft aspect. Many photos are from the British Museum and Victoria & Albert Museum. The term patterns is used in a broad sense, with lines, polyhedra, divisions of squares, circles etc.  The book is not just on patterns per se, but includes basic  drawing tools and hints and definitions. Of most interest as regards tilings is pp. 66, 69, describes as ‘networks’, and ‘some types of pattern’ containing a variety of  simple tilings. However, there is nothing here that has not been seen before, as to be expected, given the intended audence. Reference to counterchange p. 68.

Has an interesting historical French curve source reference, page 16, crediting this to a ‘Professor French, a mathematician’ different from others. Islamic design on front cover, and repeated p. 37. Eight chapters, with Chapter 6, the most indepth, and  of most interest.


Slocum, Jerry, and Jack Botermans. Puzzles Old & New: How to Make and Solve Them. 1999 third edition. (10 August 2006)

Mostly of manipulative puzzles, with historical details, all of a popular level. Delightful. Upon a re-reading of 6 June 2014, I happened to notice a cluster-type premise puzzle, p. 40, of animals based on the set of 12 Pentominoes in a rectangle, as designed by the Japanese teacher Sabu Oguro, and produced commercially by U-Plan, Japan! Somehow, in previous re-readings, I must have seen this and overlooked its significance. Indeed, I do recall that I was entirely been dismissive of it! Only with the foreknowledge of the cluster puzzle can it now be appreciated. As such, I have seen this puzzle elsewhere in recent times, but without background detail as given by the authors; this I can now follow-up.

Sam Loyd Trick mules and true source p. 34. No jigsaw puzzles as such. Good bibliography.


————. The Book of Ingenious and Diabolical Puzzles. Three Rivers Press; 1st edition November 1999 (7 March 2017)

Popular puzzle book. Primarily purchased in regards of my cluster puzzle investigations, in that it recently came to my attention that one of the puzzles mentioned ‘The Jayne Fishing Puzzle’ p. 15, has possible relations, and so thus purchased, although of course with the bonus that the book per se would be of likely interest. However, in this instance, this was merely of a general packing nature, rather than of a higher standard of double contours. But it could have been…. Has much of a general interest without being of an overarching concern. Has eight chapters of various puzzle classification. Of note is Mayblox of MacMahon, pp. 34-35. Much is indeed new, and can be read again with profit.


Sly, A. J. GCE O-level Passbook Modern Mathematics. 1976 (20 September 1992) Textbook


Smeltzer, Victoria and Patricia Smeltzer. Mathematics Encyclopedia. Burke Books 1980 (18 February 2000?)

Juvenile, 10-year-old.Tessellation p. 75. Hexagons, not worth mentioning.


Smith, Cyril Stanley. A Search for Structure. The MIT Press, 1983. WANTED

Has Cairo tiling diagram, without attribution, found indirectly in a  excerpt page in the Indian journal ‘Resonance’, of June 2006.


Smith, Thyra. The Story of Measurement. Basil Blackwell Oxford. 1968. (12 October 2002).



————. The Story of Numbers. Basil Blackwell Oxford. 1969. (12 October 2002).



Smith, Charles N. Student Handbook of Color. Reinhold Publishing Corporation New York, 1965. (24 January 2015) First saw at least 5 January 1987, College library

Although not a maths book per se, it is included nonetheless as it was studied with my early maths studies of 1987, it containing a few geometric tilings, such as p. 57, as well as optical illusions.


Smith, David T. Miscellaneous Musings. Published by J. W. Northend Ltd, West Street, 1929, and 1936. Illustrations by Elspeth Eagle-Clarke (24 November, 6 December 2016).

Upon my interest in Elspeth Eagle-Clarke’s work in cluster puzzles, I investigated her further, of which I found a book reference, with illustrations by her, of Miscellaneous Musings. Therefore, I thus decided to investigate this reference, albeit with the likelihood of nothing cluster puzzle related, albeit with the possibility of background details of her. As I presumed Elspeth Eagle-Clarke illustrations/references would be in the 1929 first edition but not there! Continuing, I then obtained the 1936 edition where this reference was indeed stated. as such, in relative terms it was a disappointment. save for the credit, no other mention is made of Eagle-Clarke. There are 14 illustrations, in black and white, on pp. 5, 9, 16, 27, 45, 46, 52, 65, 73, 79, 85, 111, 117 and 119. The artwork is nothing special per se, a mixture of straightforward scenes from life and occasional fantasy aspects. Many references in the book allude to Eagle-Clarke’s cluster puzzle work, with Pterodactyl and her Yorkshire background. An intriguing possibility is that she was married to Smith. This idea was put in my head by a reference to the book, where she was so titled, of which initially she was so called. Initially, I though this was a mistake, and indeed likely so, as it would be most unlikely a modern-day bookseller with no interest in Eagle-Clarke would be privy to such detail. However, perhaps it is indeed true! Who knows?


SMP Book 1. Cambridge University Press 1965 (6 August 1994)  Hardback

First , I place all my SMP books under a single grouping, as ‘SMP’, for the sake of convenience of reference. There seem to be many different editions and contributors to various books, the machinations of which I lack the time and desire to unravel, hence the en masse recording here. Of note is that this was the maths books series I studied during my school days (how I would love to see this now!), and of which I vaguely recall a mild interest in tessellation than in other aspects, but this is indeed so vague as to possibly being a false memory. Of note is that Escher is mentioned, but not illustrated in any book. Generally, tessellations are  neglected.

Within a chapter on area, and sub Patterns (tessellations) 159-163, of basics. P. 163 appears to have been the source for some SMP-inspired studies of November 1987. Of note is a tiling later used by myself for a bird tiling, the tilings of which was used again in Book B.


SMP Book 3. Cambridge University Press 1976 (c November 1995)

No tessellation (Hardback).


SMP Book 4. Cambridge University Press 1979 (6 August 1994)

No tessellation (Hardback). P. 266 has a clown figure of a five-fold nature which I studied in 1989. Parabolas p. 149


SMP Book B. Cambridge University Press 1974 (29 August 1993) (A series of eight books, A-H, for CSE. SMP 1-5, is for O-level)

In contrast to other books in the series, of a substantial reference to tessellations, in relative terms, with a prelude on ‘Tiling Patterns’, albeit of a basic nature, of first premises, pp. 1-5, and a dedicated Chapter 2, Tessellation, pp. 13-22. By far, of the SMP series, this book is the most substantial regarding tessellations, albeit this is in relative terms. Diagrams of particular note include p. 18, Figs. 11a, b. Of note is a tiling later used by myself for a bird tiling, the tilings of which was used again in Book 1,  p.163, and a clamshell, p. 19.


SMP Book F. Cambridge University Press 1970 (10 February 1994) (A series of eight books, A-H, for CSE. SMP 1-5, is for O-level)

No tessellation.


SMP Book H. Cambridge University Press 1972. (16 October 1993) Doncaster

Minor tessellation, pp. 83-84, with a P. Murphy chicken-like motif p. 84, within a chapter on Geometry, pp. 76-90


SMP Book T. a 9 December 1986 reference.


SMP Book X. Cambridge University Press 1973 (25 August 1991)

A follow-on from books A-H, for O-level. No tessellation.


SMP Book Y. Cambridge University Press 1973 (25 August 1991)

No tessellation.


SMP Book Z. Cambridge University Press 1974 (25 August 1991)

No tessellation.


SMP Teacher’s Guide for Book X. Cambridge University Press 1974 (25 August 1991)

No tessellation.


SMP Teacher’s Guide for Book C. Cambridge University Press 1971 (18 April 1993)

No tessellation.


SMP Book X. Cambridge University Press 1975 (not dated)

No tessellation.


SMP 11-16 R3. Cambridge University Press 1989 (14 August 1994)

Impossible objects 116-117 ‘Penrose-like’ stairs page 125. No tessellation.


Smullyan, Raymond M. What is the Name of this Book. The Riddle of Dracula and other Logical Puzzles. Prentice-Hall Inc. 1978 (date semi legible – 2000?)

271 popular logic puzzles in four parts. Of minor recreational interest, nothing more. Has occasional stories on mathematicians. Overall, I have limited patience with the genre.

Snowden, Robert, Peter Thompson and Tom Troscianko. Basic Vision. An introduction to visual perception. Oxford University Press. First published 2006 (22 June 2019)

Of vision interest. Of a broad popular account, with leanings at times to the academic. Escher print ‘Ascending and Descending’ p. 198, albeit only discussed in passing.

P. 96. Cafe wall illusion. I was under the impression that this came about by the cafe wall tiling in Bristol, but the book contradicts this, saying it was known well before. Investigate.

P. 123. Idea for Ponzo illusion, p. 123, fade upper bar as in perspective gradient, and see the effect.

Sommerwell, Edith. A Rhythmic Approach to Mathematics. Classics in Mathematics Education, Volume 5.

This book is a reproduction of a monograph written in 1906 to advocate the use of curve stitching in the early school years. The book was originally accompanied by a set of punched cards depicting geometric shapes; each card could be used in the construction of many varied designs. The book's preface is written by Mary Boole, to whom the technique is attributed by the author. Both the preface and the text itself praise the use of curve stitching as promoting both aesthetic satisfaction and subconscious awareness of pattern, harmony, and relationships among objects. The importance of using pleasing colours and of allowing the child to work out his own rules for stitching is stressed. Methods of developing the curve of pursuit, the parabola, and other curves are described. Many figures illustrating the principles used and plates displaying complex designs completed by children of various ages are included. (SD)


Springett, David. Woodturning Wizardry. Guild of Master Craftsman Publications Ltd 1973. (18 May 2014)

Although strictly not a mathematics book, included nonetheless as it has certain crossovers, albeit most tenuous indeed. I seem to recall John Sharp quoted this author in an article, and so I was ‘primed’ to notice this. This includes a historic polyhedral instance from a book I was unfamiliar with: Manuel du Tourneur, 1816 by Hamelin Bergeron. Also, it reveals how the seemingly impossible ‘arrow through bottle’ was achieved, pp. 54-63. Much of interest in a generalised sense with polyhedra carving.

A later colour edition was subsequently seen in both Cleethorpes and Grimsby libraries.


Staněk, V. J. Beauty in Nature. Artia, Prague 1955 (c. 1995-2000). Oversize.

Not really a maths book, has occasional pattern by default.


Stannard, Dorothy (Editor). Egypt. Insight Guides. Fifth edition 1998, updated 2000, reprinted 2002) first edition 1987, not seen (obtained 13 April 2013, Cleethorpes library sale. First saw 5 May 2011, Grimsby central library)

Although not a maths book per se, has an instance of the Cairo tiling, and so is thus included here. Has Cairo tiling page 171, clearly displayed, outside a mosque in the City of the Dead. This is of note as the first pictured reference seen by myself, although subsequently I have found other, and indeed earlier instances. Somewhat ironically, given extensive searching in maths books this reflects badly on me, it being under my nose at least since when the library obtained, in 2001, but I simply didn’t think of a possibility of it being in travel guide books.


Steadman, Philip. Vermeer’s Camera. Uncovering The Truth Behind the Masterpieces. Oxford University Press 2001 (26 October 2007)

Camera obscura conjectures.


Steinhaus, H. Mathematical Snapshots. (Third American edition, Revised and Enlarged, with a new preface by Morris Kline). Oxford University Press 1983. (30 April 1994)

Many aspects of recreational interest. Chapter 4, tessellations pp. 75-83.


Stephens, Pam. Tessellations: The History and Making of Symmetrical Designs. Crystal Productions (19 March 2010)

Juvenile content, despite the serious title, of only 40 pages. Stephens apparently wrote the entire text, with the artwork (tessellations) by Jim McNeil. Pages 1 and 2 cut out, hence this lacks bibliographical detail.


Stevens, Peter S. Handbook of Regular Patterns. An Introduction to Symmetry in Two Dimensions. The MIT Press, Cambridge, Massachusetts and London, England). First printed 1981, Third printing, 1987 (c. 15 December 2007, through receipt). First saw 4 October 1990

First saw 4 October 1990 (ordered through the library), this sparking a concerted study of the day, throughout the month of October. Illustrated throughout with various Escher periodic drawings. Occasional Cairo tilings arising from my studies, although not directly from the book itself. A most pleasant read, with a crystallographic leaning, best described as a compilation (as the author, an architect, admits) of tiling and patterns throughout the ages. A feature, scattered liberally throughout the book, is of geometric Escher-like tessellations, of which I believe acted as the moving spring of my own. These would appear to be by Stevens, albeit there are doubts here, as this is not made explicit. In the preface Mollie Moran is credited with drawing most of the illustrations, hence my uncertainty.

If there's one criticism, the illustrations need more detail; sometimes nothing at all is given, whereas with others there is only a bare minimum.

Of direct interest:

Houndstooth tiling (not standard model), pp.195-196. Said to be from Sandwich Islands, of which likely this refers to Owen Jones’ Grammar (who is mentioned in the bibliography).

Dogbone tiling, p. 294, Arabian.

————. Patterns in Nature. Penguin Books 1977. First published 1974 by Little, Brown & Co. (16 September 2007)

Although not a maths book in the conventional sense, included nonetheless as it is of interest. Tiling is mentioned only briefly, in Chapter 1, with a small section on polyhedron and mosaics, pp. 11-16. Even so, some innovations here. The semi-regular tilings are presented as according to the number of corners, of which off hand I don’t believe I have seen as in this particular presentation.


Stephenson, C. and F. Suddards. A Text Book Dealing With Ornamental Design For Woven Fabrics. First edition 1897, Methuen & Co. Ltd and Fourth Edition. Methuen & Co. Ltd 1924. N.B. Not seen the first edition but I have the fourth edition as a PDF. (March 2019)

Of houndstooth and weave interest. Very much in the Lewis Day tradition, from a weave perspective. Of most note is p. 16, of Plate III, Fig. 9, of a houndstooth (but not stated as such) in black and white as a counterchange design, and further not as a weave but of the tessellation type. I presume that this is also in the first edition.

Stewart, Ian. Concepts of Modern Mathematics. Penguin Books 1982. First published 1975 (16 May 1999) (17 November 1994 and 24 August 2004)

Of limited interest, with somewhat technical, advanced concepts way beyond me. 339 pp. paperback. Topology, Chapter 10 pp. 144-158. No tessellation, Escher. No plans (2018) to re-read.


————. Game, Set and Math. First published 1989. SEEN

From a reference on an old cardboard ring binder, of 3 May – 21 days library book. The content is now (2018) long forgotten. I do not recall any studies arising from this.


. ————. Nature’s Numbers. Discovering Order and Pattern in the Universe. Weidenfeld & Nicholson London 1995 (7 November 1998) Science masters series.

Popular account, but of general interest only, no tessellation.


————. Does God Play Dice? The New Mathematics of Chaos. Penguin Books 1997. (Date not given).

Of limited interest.


————. Taming the Infinite. First published 2008 by Quercus, 2009 paperback (21 February 2015)

Chance finding. Although the subject matter is mostly beyond my understanding, it contains the occasional snippet of interest. For instance, fuel efficient planetary probe orbits by Edward Belbruno. In all my time in astronomy, I was unfamiliar with the fuel concept as outlined by Belbruno p. 372, and indeed of himself! Escher p. 223, a single line mention in the context of hyperbolic geometry. No tessellation


————. Professor Stewart’s Cabinet of Mathematical Curiosities. Profile Books 2008 (4 June 2011)

Popular maths.


————. From Here to Infinity. A Guide to Today’s Mathematics. Oxford University Press 1996. First published 1987 (17 June 2012)

Largely popular account of hard to understand concepts. Quasicrystal tiling p. 101-103. Chapter 12, Squaring the Unsquarable, Geometric Dissections pp. 168-171. No references to Escher, tessellation or tiling. Shortage of time (2018) forbids a re-read.


Stewart, Ian and Martin Golubitsky. Fearful Symmetry: Is God a Geometer? University Press 1993

Occasional Escher pictures, Circle Limit IV, p.45, Lizards 237; Penrose tiling p. 95, Kepler’s Aa to Z patch, p. 96; Pólya diagram p. 239, with Pólya’s annotations, but generally all these references are in passing only.


Stewart, Desmond. Early Islam. Great Ages of Man. 1967, Time-Inc. (First saw, or at least recorded 13 September 1987, and again studied 9-11 August 1988) recording as seen Central library and Willows library? Not in Possession. Saw at Internet archive 1 February 2018.

Part of a 20-book Time-Life series. Not on tilings as such, but of course with many side references. Only tiling aspects of interest pp. 150-151 and 156-157, with double page spreads.


————. and the editors of the newsweek book division. The Alhambra. TBS The Book Service Ltd; First printing 1974, 1976 (Internet Archive 1 February 2018)

Recorded on a menu card, c. September 1987. A single study of 11 July 1988 study.

General discussion on the Alhambra without a dedicated study of tilings per se. Many pictures have not been seen before. Although tilings do appear, this merely illustrates the discussion. Of most interest pp. 15, 33, 73, 101, 125, 183, where a tiling is shown full on, as a square each time.

Sturgis, Alexander. Magic in Art. Belitha Press Limited, first published 1994 (18 August 2019)

Juvenile, 32 pp. The cover caption states: Perspective, Tricks and Illusions. A series of topics are discussed over a two-page spread, with most of the interest being ‘The Impossible World of Escher’, 16-17, illustrated with Belvedere, Relativity and Ascending and Descending, with brief commentary. Relativity is also on the front cover.

Of standard fare. Somewhat to my astonishment, given a juvenile book, I found not one but two aspects new to me! This concerned excised columns appearing in Leonardo's Mona Lisa and op art by Victor Vasarely’s son, Jean-Pierre Yvaral, also known as Jean-Pierre Vasarely, whose work I was unfamiliar with!

Sutton, O. G. Mathematics in Action. G. Bell and Sons Ltd 1966 (24 October 1996 or 1998)

Semi-popular, although tending towards the advanced.


Sykes, Mabel. Source Book of Problems for Geometry. (subtitled as … Based upon Industrial Design and Architectural Ornament). Dale Seymour Publications. Originally published 1912 by Norwood Press, Norwood, Mass. (1 March 2012)

From a reference in Britton. As such, I consider this book poorly titled in the (obviously modern day, but year not stated) reprinting, as the cover does not give the full title to adequately describe the contents; only with the full title does it make sense.

There is very little tiling here per se; rather, the book is concerned with designs in a variety of given shapes, such as church windows. And what tiling there is, is from other sources, rather than from Sykes herself. Part 2 is on tiled floors, pp.13-22, and parquet floor designs. Even, there are some tilings I have not seen before, such as p. 19, of regular octagons and isosceles right triangles. Throughout the book, exercises are given, most of which are beyond me, not that I have the time to do these in any case….




Tallack, Peter, ed. Science Book. Cassell & Co., 2001 (30 May 2015)

Overweight coffee table book, occasional maths. Escher’s print Möbius Strip II p. 144. Reference to Arnold (Nol) Escher, p. 206 as regard mountain formation. p. 482 Quasicrystals.


Tammadge, Alan and Phyllis Starr. A Parents’ Guide to School Mathematics. Cambridge University Press 1977 (4 October 1997)


Tapson, Frank. Oxford Study Mathematics Dictionary. Oxford University Press, First published 1996, fourth edition 2006 (21 February 2015)

Chance finding. Intended for a 11-16 audience, albeit even here, much of this remains obscure. Gives simple definitions of mathematical terms. Of perhaps most note is a Cairo tiling (not attributed) on p. 139.


Taylor, Don and Leanne Rylands. Cube Games. 92 Classic Games, Puzzles & Solutions. First published in Great Britain by Penguin Books 1981 (my copy). First published in Australia by Greenhouse Publications Pty Ltd 1981. (20 June 1993)

Small format paperback, 50 pp. In short, a how-to on Rubik’s Cube, published with many others at the height (1981) of the craze. Quite where the ‘92 classic games’ is derived is unclear; it’s simply a monologue on Rubik’s Cube. This is perhaps better than most of the day, at least in theory, with colour diagrams, which is surely better than line diagrams with initial letters from colours. Taylor and Rylands are both (mathematical) Australians, hence the Australian publication.

Such books are essentially in my past now (2018). There is simply a lack of time to re-study.


Taylor, Don. Mastering Rubik’s Cube. Penguin Books 1981. (29 August 1993)

Very small book.

The Chronicler (Lord High Keeper). Fred Learns the New Mathematics. Continua Productions Ltd 1978, 160 pp (‘First saw’ c. 8 December 1989 (day of study).

Book is not to hand, lost in storage. I only have a photocopy of relevant pages, and so the discussion below is as best in the circumstances permit, pending a more detailed piece upon finding. This seems to be a children’s book, of which the author’s name is unclear. Of most interest is Chapter 4, ‘Tessellations and Topology’ with tessellations of pp. 146-148. This includes both simple, pure tilings as well as Escher-like, mostly tilted somewhat extravagantly, with bird, ‘flat chested thrusters’ (dinosaur-like), ‘curved back leaper’, ‘phantasmagorial fibbertigibbet’ (dog-like).

Phantasmagorial - having a fantastic or deceptive appearance, as something in a dream or created by the imagination. having the appearance of an optical illusion, especially one produced by a magic lantern. Changing or shifting, as a scene made up of many elements.

Flibbertigibbet - is a Middle English word referring to a flighty or whimsical person, usually a young woman. In modern use, it is used as a slang term, especially in Yorkshire, for a gossipy or overly talkative person.

One of the pure tilings, reminiscent of a fish, led to extensive studies.


Thé, Erik, Designer. The Magic of M. C. Escher. Joost Elffers Books Harry N. Abrams 2000. Foreword by W.F. Veldhuysen. Introduction by J. L. Locher. (2 September 2004)

A major work on Escher, one of the ‘core value’ books. Oversize, with numerous gatefolds. The premise is visual rather than text. Indeed, there is no text save for accompanying quotes from Escher in various letters. the larger format thus enables the prints and drawings of Escher to be more properly shown at their larger sizes. Has occasional sketches that up to this date, I had not seen before, such as pp. 72-73, 96-97, 107, 111, 113, 150-151, 163, 166-167, 177, 179, 181, 184, 187-189. Surprisingly, there is very little tessellation in the book; it’s mostly on prints without the tessellation element, and certainly no concept sketches, at least worthy of the name.

Has a serious bibliography, titled ‘Selected Bibliography’, p. 196, which is a facsimile, reference for reference (checked 6 October 2016) of Locher, Escher The Complete Graphic Work.


The Yellow Book. Some early designs of later 1890s that can be interpreted as of a tessellation nature.

As given by Andrew Crompton. Author unknown.

Thomas, Briony G. and Michael A. Hann. Patterns in the Plane and Beyond: Symmetry in Two and Three Dimensions. 2007. The University of Leeds and the authors. Ars Textrina, No. 37

Cairo tiling pp. 52-53, 70-71, 79. Stated (incorrectly) as equilateral.

Thomas, Frank and Ollie Johnston. The Illusion of Life. Disney Animation. Disney Editions N York 1981 (5 December 2009)

Obtained solely due to Craig Kaplan’s reference to it in his thesis (and reference to it is as the likely anonymous reviewer of my Bridges paper, as regards the ‘staging principle’). As such, as regards tessellation aspects re ‘staging’, I do not find anything of relevance. Undoubtedly, a good book in its field, but not for tessellation


Thompson, D’Arcy Wentworth. On Growth and Form. Abridged edition by J. T. Bonner. Cambridge University Press. 1975 (13 July 2009). First published 1917, Cambridge University Press

A single-sheet study of this is dated 31 March 1988, of which by the page numbers quoted is clearly of an edition by Thompson (1,116 pp.), rather than the abridged (346 pp.) by Bonner. Likely this was by following a reference somewhere (possibly The Art of Microcomputer Graphics for the BBC Micro/Electron which quotes this), but if so is long forgotten. I do not recall the circumstances of whether this required ordering; I suspect yes. However, as regards tiling, it is a relative disappointment, with only p. 505 studied, albeit as it not as such on the subject, one would thus not expect such matters to appear, and if so, only in passing. Simply stated, an inconsequential study.


From Wikipedia: On Growth and Form is a book by the Scottish mathematical biologist D'Arcy Wentworth Thompson (1860–1948). The book is long – 793 pages in the first edition of 1917, 1116 pages in the second edition of 1942. The book covers many topics including the effects of scale on the shape of animals and plants, large ones necessarily being relatively thick in shape; the effects of surface tension in shaping soap films and similar structures such as cells; the logarithmic spiral as seen in mollusc shells and ruminant horns; the arrangement of leaves and other plant parts (phyllotaxis); and Thompson's own method of transformations, showing the changes in shape of animal skulls and other structures on a Cartesian grid.

The work is widely admired by biologists, anthropologists and architects among others, but less often read than cited. Peter Medawar explains this as being because it clearly pioneered the use of mathematics in biology, and helped to defeat mystical ideas of vitalism; but that the book is weakened by Thompson's failure to understand the role of evolution and evolutionary history in shaping living structures. Philip Ball and Michael Ruse, on the other hand, suspect that while Thompson argued for physical mechanisms, his rejection of natural selection bordered on vitalism….


Thorndike, Joseph J. (Editor-in-Chief). ‘Escher's Eerie Games’. Horizon 8, no. 4 (1966): 110-115. (24 May 2014)

First, note that as such, the article, in a ‘general arts’ book published three-monthly, is not credited with an author (other articles in the same book are the same.)

As Thorndike is the main editor, I this file under his name for wont of anything better. Does anyone know who the author is?
A brief essay on Escher, illustrated with eight prints, Hand with Reflecting Globe, Tetrahedral Planetoid, Magic Mirror, Horseman, Tower of Babel, Three Intersecting Planes, Waterfall, Belvedere. The text is most lightweight indeed, with a picture bias; no real insight is offered by whoever wrote this.


Thornburg, David D. Exploring Logo Without a Computer. Addison-Wesley Publishing Company 1984 (27 June 1993)

Being on a popular computer program of the day, Logo, now some thirty years later somewhat dated. Note that the book is not just about tessellation. Of most interest, relatively, is Chapter V, on Tiling and Symmetry pp. 59-100. Escher’s Pegasus p. 73, Shmuzzle pp. 74, 99. Pentagonal tiles pp. 66, 67, but seen previously. Author’s own (poor) dog tessellation p. 77. Of no consequence.


Thyer, Dennis and John Maggs. Teaching Mathematics to Young Children. Holt, Rinehart and Winston, Second edition 1981. First published 1971 (27 July 1997)

On teaching Infants (rather than a textbook). Of limited interest. Tessellations are briefly mentioned and illustrated 84-85, 95, 209, 213, 217, but are not of any significance.


Todd, Audrey. The Maths Club. Hamish Hamilton London. First published 1968. (26 September 1991

For a 9-16 age range school maths club. No tessellation. Chapter 4 of a substantial nature, pp. 43-65, ‘Curve Drawing and Stitching’ may have influenced some c. 1986-1987 studies. Chapter 5, pp. 66-78 ‘Geometrical Solids’ as well.


Tolansky, S. Optical Illusions. Pergamon Press 1964. (26 July 1997)

A scholarly account in a popular manner (in contrast to mostly others, of a lightweight nature). No maths at all.


Tóth, Fejes L. Regular Figures. Pergamon Press 1964 (12 December 2010), partial copy, of Chapter 1 up to p. 43...

Largely theoretical. Mostly concerning group theory, which is out of my remit. Occasional tiling. Escher mention p. 39. Tilings Plates 1-3. As such, of what I have seen (Chapter 1 Plane Ornaments only), of no consequence (likely, the book is even more obscure in succeeding chapters).


Townsend, Charles Barry. The World’s Greatest Puzzles. 1996. Quality Paperback Book Club New York. (3 June 2007?) (The date stamped year has faded in the book). An anthology of four books: The World’s Most Challenging Puzzles; World’s Most Baffling Puzzles; World’s Greatest Puzzles; World’s Most Incredible Puzzles

As a general statement, the puzzles are of a Dudeneyesque nature, in both style and substance (with black line drawing reminiscent of the period, early 1900s). Very little is said of the source of the puzzles. ‘Professor Hoffman’ (primarily) and Sam Loyd gets a credit, and no one else. Looking at the puzzles, albeit admittedly briefly, many of these are well-known, of which it is unlikely that there is too much, if any, in the way of originality by Townsend here.


Travers, James. Puzzles and Problems. The Education Outlook, 145, 1933. WANTED


————. Puzzling Posers.  London George Allen & Unwin Ltd, 1952. WANTED


Tufnell, Richard. Introducing Design and Communication. 1986 Nelson Thornes Ltd (c. 14 April 1987) SEEN, BUT NOT IN POSSESSION.

Some minor tessellation studies, nothing in the way of originality.


Turner, Harry. Triad Optical Illusions and How to Design Them. Dover Publications 1978

72 pp. Drawn to my attention by  Alan Bridges of art college. First saw 22 February 1993, where I seemingly borrowed and photocopied the entire book the next day, upon which I then studied on the sheets themselves. I canot recall if I have subsequently obtained the book. I cannot find it if so; I get mixed up with other similar Dover publications by Locke and Willson.


Tyler, Tom. British Jigsaw Puzzles of the 20th Century. Richard Dennis 1997 (22 March 2014)

Of jigsaw puzzle interest. Although by its nature this is not a maths book, as it includes two aspects of tiling (albeit brief, pictures only) I nonetheless include here for the sake of convenience. These references on p. 110 are Penrose’s ‘Perplexing Poultry’ and a new name to me in regards of cluster puzzles, George Luck, who shows a ‘animal map’ of the British Isles. Upon following this up, I see that he has many other examples of (likely independent discovery) cluster puzzles, of a decided simplified nature, of which they can be described as relatively ‘pleasing’, but certainly not outstanding.

An excellent piece of research, one of the few ‘must have’ books. Among the jigsaw puzzle aspects of note include:

Hamley’s (in regard of a newspaper report of one of the earliest cluster puzzles), in which there is scant detail of this source. P. 8 includes a box, described as ‘THE GREAT Society Picture Puzzle’. A brief, three-line discussion of this is given in Chapter 9, p. 127 Coronation puzzle of HM King George V and Queen Mary. An open question is to whether Hamley’s made this themselves, or outsourced.

Treadle history, p. 16, described as ‘in use by 1900’.

As such, there is no apparent mention of their connection with jigsaws on the company history, going back to 1760 (Wikipedia, and elsewhere)

surprisingly so for such a major company. Wikipedia: Hamleys is the oldest and largest toy  shop in the world and one of the world's best-known retailers of toys. Founded by William Hamley as "Noahs Ark" in High Holborn, London, in 1760, it moved to its current site on Regent Street in 1881…. . Dreweatts gives ‘label of Hamley Bros. on sliding lid, 1909’. £150 - £200.

A company history is given at, albeit this is most lightweight indeed.




Varnedoe, Kirk. Vienna 1990 Art, Architecture & Design. New York: The Museum of Modern Art, 1986. (8 September 2017)

From a reference in Visions of Symmetry, p. 42 re Moser designs, where Schattsneider states:

… Only recently have Escher’s  designs been compared with Moser’s patterns; for instance in a 1986 essay by Marianne Teuber in M.C. Escher: Art and Science and in the 1986 exhibition catalog Vienna 1900 by Kirk Varnedoe.

A major disappointment! As such, I am more than a little under whelmed with such a brief references of no particular insight of just a single  sentence; perhaps influenced by Tuber’s in-depth essay, I was expecting a like treatment, but this piece (if it can be called that) is emphatically not so. Though the book may come in useful in a generalised sense, as to Moser, this is not why I obtained it! I was hoping for more Escher comparisons from Varnedoe, of an essay.

As a bonus, but nothing more, there is extensive discussions on Moser, both focussed and scattered throughout the book, but disappointingly nothing at all on Erwin Puchinger.


Valette, G. Les Partages d’un Polygone Convexe en 4 Polygones Semblables au premier (in French)


Van Delft, Pieter, and Jack Botermans. Creative Puzzles of the World. Harry N. Abrams. Edition? First seen around 15 June 1987, the first recorded date of study.


Vecht, N. J. van de. De grondslag voor het ontwerpen van vlakke versiering (Fundamentals for the Design of Surface Ornament), Rotterdam, 1930. From a refence in The World of M.C. Escher, p. 22 WANTED


Veldhuysen, W. F. (the author is unclear; Veldhuysen wrote the foreword, hence placed accordingly). M. C. Escher International Ex libris Competition. Homage to the Dutch Graphic Artist M. C. Escher. 1998? (Bridges Leeuwarden 2008 free)

On a Escher theme of ex libris, on a competition marking the 100th anniversary of his birth. This collects all of Escher fifteen ex libris works (pp. 6-20), with an pleasing, insightful essay on these by Jos van Waterschoot, along (pp.21-23) with the best of the competition. Only two names are known to me, Kenneth Landry (p. 57), with his enigmatic repeating portrait of Escher, and István Orosz (p.33). Many examples of ex libris prints from artists in tribute to Escher are shown. I do not generally find favour with most entries; however, an honourable exception is Frank-Ivo van Damme (p.47), with an original Escher-like tessellation/composition, of a human figure.


Vermeulen, Jan W. Escher on Escher. Exploring the Infinite (original title, or published as Het Oneindige English Translation by Karin Ford). Harry N. Abrams, New York 1989.) 29 May 1991

Escher's writings collected.


Vorderman, Carol. How Maths Works. Dorling Kindersley, 1998.

Tessellation 130-131, Polyhedron 152.


————. Help Your Kids with Maths. Dorling Kindersley, 2010. (2 July 2016)

Covers the basics, but even here, I’m struggling in more places than I care to  (embarrassingly) list…




Wade, David. Crystal & Dragon. The Cosmic Two-Step. Green Books 1991 (27 November 1993)


————. Geometric Patterns & Borders. Wildwood House Ltd. 1982 (16 September 1995)

The premise is of a geometric pattern book, with line drawings and colouring (in black and white) from various countries around the world. Text is seemingly purposefully kept to a minimum at the beginning of the book. The book is (regrettably) not paginated, but rather is ordered by diagram numbers. No bibliography. Has many interesting designs, worthy of study, of which I return to at intermittent intervals. Countries of origin generally accompany the diagrams, but no other detail, which is frustrating where more specific detail is sought. Nonetheless, even with shortcoming of presentations, the book is a veritable visual feast, to be returned to time and again. No Cairo tiling as such.

Diagrams of interest include:

193, double axe head tiling

194, Cairo tile-related bowtie tiling.

260, houndstooth, stated from Hawaii, likely referring to the Owen Jones reference in Grammar of Ornament, p. 15. However,  Wade’s instance is different in proportion, based on an isometric grid.

333, Cairo tile and regular hexagon in combination.

383, houndstooth in nature, of a ‘pixelated weave’ type, and is briefly discussed in the introductory text, ‘… from African basketwork…’. However, beyond this, no specific detail as to source.

555, houndstooth as a frieze.


————. Pattern in Islamic Art. Studio Vista 1976 (12 March 2010)

Primarily a diagram led book, with little text. The diagrams are not sourced. No Cairo. The first half of the book s of more interest than the second. The second is more concerned with complex patterns, and their construction. Of interest: pp.10-11, square root of 2 and 3 triangles, and rectangle thereof. ‘Fused Cairo’ p. 20, octagon based patterns p. 34 and others.


Wade, Nicholas. Vision, Illusion and Perception Art and Illusionists. Springer 2016 (2 March 2017)

Complimentary copy from Springer. Popular account. Lots of interest. Chapter specifically on tiling, with frequent references to Escher.


Wallis, Denis (Principal writer). Reader’s Digest Why in the World? 1994 First edition, The Reader’s Digest Association Limited (10 August 2015)

One minor reference to Escher, p. 87, with Waterfall and general text.


Walker, Michelle. The Complete Book of Quiltmaking. Guild Publishing London Book Club edition 1989. First published 1985. First saw 1 May 1987 (15 May 2015)

Although obviously not a book on mathematics per se, it is nonetheless included in this listing as it has interesting pattern aspects, and indeed, it is one of the best books, in relative terms, there is on the interrelation between the two, and indeed led to extensive studies of the day (1987, as indeed with other patchwork books). However, now, and for some considerable time, the nature of the material is considered hardly worthy the time originally devoted to it.

Interestingly in the bibliography, Walker quotes David Wade (2) and Johannes Itten, not to mention Jinny Beyer.


Walter, Marion. The Mirror Puzzle Book. Tarquin Publications, 1985 (30 January 1999)

Juvenile, of no interest.


Walter, Marion. Boxes, Squares and Other Things. A Teacher’s Guide for a Unit in Informal Geometry, NCTM. 1970 PDF (February 2018) Internet download.

For children 8-11. Tessellations pp. 57-59. Has a most intesting biblography, with some  tesellation references not seen before!


Wark, Edna. The Craft of Patchwork. Saw 7/8 July 1987

Although obviously not a book on mathematics per se, it is nonetheless included in this listing as it has interesting pattern aspects, and indeed, led to minor studies of the day (1987), of just two (dual-sided) pages as indeed with other patchwork books. However, now, and for some considerable time, the nature of the material is considered hardly worthy the time originally devoted to it.


Washburn, Dorothy K. and, Donald W. Crowe. Symmetries of Culture. Theory and Practise of Plane Pattern Analysis. University of Washington Press. 1988 Second Printing 1992. (30 April 1994)

Simply stated, although widely quoted in tiling literature, in truth there is very little here for the tessellator. It mostly consists of notation systems for pattern, not necessarily tessellation.

P. 7 gives the first likely apparent reference in print to the term ‘counterchange’, by A Text Book Dealing with Ornamental designs for Woven Fabrics, by Stephenson and Suddards, 1897, p. 18.  P. 158 paving tiles from fourteenth century, France; p. 172, stylised swans from Escher; p. 206, arrowhead tiling variation, p. 219 Escher's lizards; p. 232 Chinese traditional design.

Watson, William. Colour in Textile Designing.  Elementary Weaves and Figured Fabrics. First edition 1912. Second edition 1921 (Internet Book Archive and Longmans, Green and Co. There is an at least a 7th edition. 369 pp. (2019) PDF

Of fabric interest. Referenced in Grünbaum and Shepherd’s 1980 Satins and Twills article. Some confusion as to specifics here. Watson has other books to his name. The book is replete of interest. Liberally illustrated, with nearly every page containing diagrams. A must have!

Obvious houndstooth (but not named as such) p. 156, 164. Shepherd’s Check p. 157

Watson, William (ed.) The Great Japan Exhibition: Art of the Edo Period 1600-1868. Royal Academy of Arts London 1981-2. Catalogue published in association with Weidenfeld and Nicolson, London (22 June 2019)

Has minor aspects of pattern and tiling.


Weaire, Denis. The Kelvin Problem. Taylor & Francis, 1997. WANTED

Of note is that in the preface, by Charles Frank, in so many words surely is discussing the Cairo tiling, seen at Glasgow Physics lab!


Wells, A. F. The Third Dimension in Chemistry. University Press, Oxford, 1956 (8 February 2016?).

Wildly quoted in tiling mathematics despite being a chemistry book! Cairo tilings pp. 24-25, in the context of Laves tilings, although not named as such (8 February 2016). This is pleasing as Wells mentions, p. 24, the aesthetics, with ‘… a very elegant arrangement of pentagons…’.


Wells, David. Hidden Connections Double Meanings. Cambridge University Press 1988. (30 April 1994)

A somewhat hard to describe book, loosely on ‘popular geometry’, with a subject followed by answers. Pp. 22-23 dissections of dodecahedron into rhombs; tiling pp. 24-26, 45, 57, 121.


————. The Penguin Dictionary of Curious and Interesting Geometry. Penguin Books 1991. (30 April 1994)

Cairo line drawing, and discussion p. 23.


————. The Penguin Book of Curious and Interesting Puzzles. Penguin Books 1992. (30 April 1994)

This is best described as a compilation of puzzles from a variety of other authors (as noted in the acknowledgements), notably by Dudeney and Loyd. Nothing of originality from Wells himself.


————. The Penguin Book of Curious and Interesting Numbers. Penguin Books 1987 (31 March 1995, first saw 16 June 1990)

Popular account of properties of numbers, of the same premise of Neil Sloane, but much more accessible.


————. You Are A Mathematician. Penguin Books 1995. (23 April 1998)

The book is somewhat mistitled, as it is essentially a (popular) book on geometry. Has only occasional tessellation, on pp. 246 and 319, but of a lightweight treatment.


Wenninger, Magnus J. Polyhedron Models. Cambridge University Press 1989. (3 June 1993)

Foreword by Coxeter. Popular account, of 119 polyhedra. Discusses colouration (although the book is in black and white) and history.


————. Polyhedron Models for the Classroom. National Council of Teachers of Mathematics (NCTM) 1986 (3 June 1993)


Werneck, Tom. Mastering the Magic Pyramid. The Secrets of the Pyraminx (sic) Unlocked. Evans Brothers Limited 1981 (11 June 1994)

Small format paperback, 112 pp, with instructions for solving the Pyraminx, a Rubik’s cube-type puzzle, published at the height of the Rubik cube craze. The puzzle is more properly known as the Pyraminx. Originanally, I thought this was Rubik-inspired of the day until latterly re-reading the book and seeing the Wikipeda entry:

Fom Wikipedia: The Pyraminx was first conceived by [Uwe] Mèffert in 1970. He did nothing with his design until 1981 when he first brought it to Hong Kong for production. Uwe is fond of saying had it not been for Erno Rubik's invention of the cube, his Pyraminx would have never been produced.

On Tom Werneck, of whom I was unfamiliar (Form Boardgamegeek): Tom Werneck (born 1939) is a pioneer of board game journalism in Germany, who wrote more than 5000 articles for newspapers, journals and radio. In addition to that, he is a game designer and author from Haar, Bavaria, Germany. He is a co-founder and former member on the jury for the Spiel des Jahres awards.


Wesley, R. (ed.) Mathematics for All. Odhams Press Ltd 1954 (18 March 1994).

Very much of its day, with much laborious calculation.


Weyl, Hermann. Symmetry. Princeton University Press, Princeton, New Jersey. 1989 (11 June 2007)

Although this little book is much praised in the tiling world, I must admit that for my purposes I was a little disappointed with it. Certainly, it is of interest, but the audience it is intended for is not clear; there are both recreational and academic instances of study. Tiling as such is at a minimum, subsumed under ‘Ornamental Symmetry’. Of note is that an earlier edition, in Russian, of 1953? shows Escher’s Lizards, the first such usage his work as cover art.


Wheeler, Francis Rolt- (Managing Editor). The Science History of the Universe. Mathematics. Vol. VIII. In ten volumes, Volume VIII, Mathematics. The Waverley Book Company Limited Copyright 1909 and 1911 (Date of publication not stated)

Small format hardback. Semi-popular,semi-scholarly account of mathematics. Of most interest is geometry, Chapter 5,  pp 107-154, with Kepler pp. 112-113, Perigal dissection of Pythagoras diagram. p. 114, Designs on Tombs of Bernoulli, Archimedes, p. 137. Liberally illustrated, but a bit too advanced for me. No tessellation,per se, although there is ‘tessellated muliplication’, p. 44,  of which I was unfamilar with. Looking on Google, I cannot find any other instances of this description.

From WorldCat:

I. Astronomy, by W. Kaempffert; introduction by E.E. Barnard.--II. Geology, by H.E. Slade and W.E. Ferguson.--III. Physics, by G. Matthew. Electricity, by W.J. Moore.--IV. Chemistry, by W.A. Hamor; introduction by C. Baskerville.--V. Biology, by Caroline E. Stackpole.--VI. Zoology, by W.D. Matthew. Botany, by M.E. Latham; introduction by W.T. Hornaday.--VII. Anthropology, by F. Rolt-Wheeler. Medicine, by T.H. Allen; introduction by F. Starr.--VIII. Pure mathematics, by L.L. Locke. Foundations of mathematics, by C.J. Keyser. Mathematical applications, by F. Bellinger; introduction by C.J. Keyser.--IX. Art, by B.S. Woolf. Literature, by F. Rolt-Wheeler; introduction by E.J. Wheeler.--X. Schools of Philosophy, by C.G. Shaw. Sociology and political economy, by L.D. Abbott. Ethics, by F. Rolt-Wheeler; introduction by H. Münsterberg.


Whistler, Rex and Laurence Whistler. AHA. First published 1948 again 1978. John Murray (23 September 2017)

Chance finding. Not strictly mathematical. Of topsy turvey heads, an early instance (although not the first) in the field of such double imagery. Relatively lightweight, of 21 images. The images are by Rex, with accompanying verse by Laurence, his brother. Some are better than others.

Note that although not ‘officially’ accompanying the book, inside was a small booklet of a related theme, ‘Turn Me Round’, with 18 images published by Tobar Limited, Norfolk (said to be 1997) from Dreh’ mich um, rund herum’ by Otto Bromberger, published in Germany in the late 1890s. This was without any text whatsoever, not even a caption or paginated.


White, Gwen. A World of Pattern. John Murray 1957. (23 September 1996? The last digit is unclear).

Juvenile, mostly patterns in the real world. Occasional tessellation.


————. Perspective. A Guide for Artists, Architects and Designers. 1982. Batsford Academic and Educational Ltd.

Recommended by art class tutor Peter Bendelow, c. 1983.


White, William F. A Scrap-Book of Elementary Mathematics. Chicago The Open Court Publishing Co, 1908 (Downloaded from Project Gutenberg 8 June 2015)

Lots of recreational aspects, with most of interest to me: geometric dissection pp. 91-99, tiling p.100, four-colour theorem p. 120-121.


Whitelaw, Ian. a measure of things (sic). The story of measurement throughout the ages. David & Charles, 2007. (5 August 2018)

160 pp, small format hardback. Popular account, of 11 chapters, in bite size.


Whitehouse, F. R. B. Table Games of Georgian and Victorian Days. Peter Garnett Ltd, London, 1951 (27 November 2018). (PDF)

Of game interest. In-depth treatment, although of a popular nature. Liberally illustrated.  Twelve chapters, of particular interest Chapter X, ‘Jig-Saw Puzzles’, pp 84-85, on John Wallis, albeit lightweight in depth. Book quoted in Hannas, The English Jigsaw Puzzle. First saw on Giochidelloca Italian site.


Williams, Anne D. The Jigsaw Puzzle. Piecing Together a History. Berkley Books, New York, 2004 Foreword by Will Shortz (17 July 2014).

Although not strictly a maths book, included here as it has certain crossovers to my recent interest in cluster puzzles. All pages and photos are in black and white. Some outstanding scholarship is displayed. One of the few ‘must have’ jigsaw books.

Of perhaps most interest is that of Margaret Richardson’s entry, of pp. 55-57, 59, and one of the unnumbered plates, ‘plate 8’. Pp. 55-56 gives a detailed account, whilst pp. 57, 59 are mentions in passing. Plate 8 shows a picture of ‘Kentucky Belle’, of 908 pieces. No other puzzles are mentioned by name; certainly no mention is made of ‘A Bad Dream’. Pieceful Solution (Shumaker and Power) plate. Escher p.107 (a mention in passing, on Savage), Savage p. 107 and endnotes p. 217. Palmer, or indeed the concept of cluster puzzle beyond Pieceful Solution. Has an excellent bibliography and endnotes. Includes Richardson’s Kentucky Belle. Just for general interest, I have a list of Richardson’s puzzles known from Williams.


————. Jigsaw Puzzles. An Illustrated History and Price Guide. Wallace-Homestead Book Company, 1990. (c. November 2015)

Although not strictly a maths book, included here as it has certain crossovers to my recent interest in cluster puzzles. The book is more properly described as an American history, as stated in the preface. The price guide aspect (pp. 326-330) is most minor, and should arguably have been left out of the title; the book is overwhelming of history puzzles, and not a price guide per se. Minor references are made to Margaret Richardson, pp. 12, 37, 149, 153. P. 37 is of a dedicated section. Sam Savage’s Schmuzzles puzzle p. 325, which mentions a 16-page instruction book of tesselated [sic] figures that I have not seen.


Williams, Robert. The Geometrical Foundation of Natural Structure. A Source Book of Design. Dover Publications, Inc. 1979 (3 June 1993)

Of most note is a Cairo tiling pp. 38, 204, in the context of the dual and transfromation between squares and basketweave tessellations. Quite how best to describe Williiams is unclear. Architect, designer? He deos not appear to be a mathemtaican per se. Further how much of the book is original with him is unclear. I suspect, from the books and articles quoted, that he is borrowing heavily. Of note is that on p. 42 he quotes the most obscure D. G. Wood Cairo tile reference. Of most interest per se is Chaper 2, ‘Natural Structure and the Two Dimensional Plane’, on tilings and circle packings, pp. 31-52.


Willson, John. Mosaic and Tessellated Patterns. How to Create Them. Dover Publications, Inc. 1983. (30 April 1994)

Slim volume, of just 30 pages. Cairo tiling plate 3. (Neglected, or not noticed, until 7 May 2013!)

Very pleasing indeed, with many simple, but interesting tilings, and ideas thereof. Discussion of tesellations, in a simple manner, pp. 1-14, 15-18, these being separated by wirfeame plates. Studied tessellating letters p. 15?


Wilson, Eva. Islamic Designs. British Museum Press. First published 1988, Fourth Impression 1992 (3 June 1993)

The title is a little less than exact, in both scope and content. The introduction states the designs are in effect ‘restricted’, from ‘the illuminated Koran’, ‘metalwork’ and ‘pottery’. These are all hand-drawn, rather than of photographs. The premise is overwhelmingly one of illustration rather than discussion. Much use is made of material from Critchlow and El-Said & Parman. As such, it is more of a general introduction to Islamic designs of the above, rather than of a groundbreaking, definitive work. Given that it essentially repeats other authors, of no consequence.


Wilson, Robin. Four Colours Suffice. How the Map Problem was Solved. Penguin Books, 2003.

(6 July 2017)

Popular account.


————. Stamping Through Mathematics. Springer 2001. PDF (31 December 2015)

Wiltshire, Alan. The Mathematical Patterns File. Tarquin Publications. 1988 (3 June 1993)

Subtitled as ‘mathematical patterns in the classroom’, with a leaning towards pedagogue of 10-12 year old group as far as I can tell. Discusses, or more accurately illustrates, symmetry (rather than pattern as in the title) in the broader sense, with reflection, arcs, hexagons, octagons, tessellations, polar graph, quadrants, spirals, envelopes, overlaps, grids, enlargement, all of no particular merit. Text, aside from the initial page, is non-existent. No Escher-like tessellation. Not at all impressed, even for the level it is pitched at.


————. The Geometrics File. Tarquin Publications. 1983 (3 June 1993)

A Tarquin Mathematics Resources File. Broadly, this is of creating ‘geometrical mathematical designs’, of a relatively substantial nature, of 79 pages, aimed at a 10-12 year group. Text is at a minimum, with a caption for each aspect under discussion. Occasional tessellation, pp. 28-29 (one with potential as a human figure), and pp. 41-42, but it’s not really a book on tessellation as such. No Escher-like tessellation. Of little direct interest now. Also see Wiltshire’s ‘companion’ book The Mathematical Patterns File.


————. Symmetry Patterns: The art of making beautiful patterns from special grids. Tarquin Publications 1989.


Wood, Elizabeth Armstrong. Crystals and Light. An Introduction to Optical Crystallography . Speicial edition for Bell Telephone Laboratories, Inc. (1964). Dover Publications; 2nd edition 1977) (First saw, or at least recorded, 24 September 1987, at college library)

A minor study, in which the crystal studies are shared with other books of a like nature. Note that the book has been through variuous editions, although which edition I saw is long forgotten; however, likely the more substantial Van Nostrand, that than the more slim-line Dover second edition of 1977. Seen on Internet Archive 29 December 2017.


Wollny, Wolfgang. Reguläre Parkettierung der Euklidischen ebene durch Undeschränkte Bereiche. Bibliographisches Institut, Manheim, 1969


From a reference in Tilings and Patterns. Also see four other articles of Wollny in Geometriae Dedicata.

Wood, Donald G. Space Enclosure Systems. Identification and Documentation of Cell Geometries. Bulletin 203. Engineering Experiment Station, The Ohio State University, Columbus Ohio. 1967 or 1968, 52 pp. No publication date is given in the booklet (11 December 2012).

An obscure publication, seemingly little known or quoted in tiling circles, when perhaps it should otherwise be. Although of a polyhedral nature, in the form of prisms, has occasional tiling aspects of note. Much of his work here, and elsewhere in the book, is in regards to modelling prisms, in cardboard.

This bulletin came to my attention (belatedly) from a footnote in The Geometrical Foundation of Natural Structures by Robert Williams, p. 43, as regards my Cairo tiling interest. Has occasional Cairo-tile instances (non-attributed), of an equilateral pentagon of pp. 3-5, 30-31, derived  (and credited) from MacMahon and Cundy and Rollett’s works.

Wood makes a curious observation as regards tilings with equal length sides, with the Cairo tiling being one of five such instances (with the others being an equilateral triangle, square, hexagon, and rhomb); as such, I do not recall seeing this simple and possibly significant (in relative terms) observation elsewhere.

Donald G. Wood (1922-2011), was a professor of industrial design in the School of Art at Ohio State University, Columbus, Ohio, US.

Wood gives 15 references, some popular, and known, and some obscure, never before (as far as I know) quoted in tiling circles. In the hope of finding pentagon tilings, or indeed interesting  tilings in general, I found (historical) references on Lewis, Davey, and Egleston, but no tiling as such was found in their respective works.

Note also that there is one other (later) like publication by Wood, Space Enclosure Systems: The Variables of Packing Cell Design, Bulletin 205, 1968, 52p. Not seen.


Wood, Mary. The Craft of Temari. Search Press 1991 (30 April 1994)

Although strictly a craft book and not a mathematics book per se, I include this here, as it loosely it is of a geometric nature. Note that the only reason I got this was that I had seen a reference to temari balls in M. C. Escher: Art and Science, pp. 237-238 and colour plate on p. 398, and upon an opportunity of a book on the subject (at John Bibby’s) I thus obtained. However, my interest in this per se is decidedly minimal; I have no intention of ‘studying’ the subject.


Woodman, Anne; Eric Albany. Mathematics Through Art & Design: 6-13. Unwin Hyman. (14 August 1995, Hull central library)

Many pages concerning Escher-like tessellations, beginner’s level, very poor standard indeed, even for children.




Yarwood, A. Graphical Communication. Hodder and Stoughton. 1975 (20 August 1995) Tessellations 190-197

This was first studied between 7, 9, 12 October 1987. Within a ‘graphical communication premise’, this has a small chapter on tessellations (not Escher-like), titled ‘Geometrical Patterns’ pp.190-195. The tilings are simple, of no consequence.


Young, Jay. The Art of Science. A Pop-Up Adventure in Art. Walker Books 1999. (16 April 2010)

Devised and paper engineered by Jay Young, written by Martin Jenkins. Oversize. Various illusion/perception effects illustrated by pop-outs. Also see accompanying booklet, which discuses the pictures. Minor reference to Escher p. 6, with Relativity print, and book p. 17.




Zechlin, Katharina. Games you can build yourself. Sterling Publishing Co., Inc. 1975 (23 August 1994)

Mostly board games.



Zusne, Leonard. Visual Perception of Form. New York: Academic Press Inc 1970. (18 August 2016)

From a reference in Schattsneider and Locher. Of an academic nature, not surprisingly given the publisher! Large tracts are simply not of direct interest or understandable. A relative disappointment as regards Escher aspects, with only a few pages devoted to him, and some in passing, too: pp. 17-19, 55, 114-115, 417. Prints include Day and Night 18, Circle Limit IV, p. 115. However, the book itself seems interesting in itself, although academically inclined, but it’s finding the time to study! Aspects of interest include figure ground, notably with pp.116-118, where Zusne discusses aspects of the Rubin vase I had not considered consciously. Pp.300 and 316 are of interest as regards visual form.

Specific Aspects

Aside from the en masse listing above, it is also possible to compile inventories of specific aspects, such as spiral tiling. In short, this is a convenience, to save wading through the above extensive listing.

Spiral articles and books of broad interest, some more direct than others, with a leaning towards tiling. 

A compilation of 1 April 2020+, primarily for purposes of aiding Peichang Ouyang et al (including myself) paper ‘Generation of Advanced Escher-like Spiral Tessellations’.

Books and Articles

Burgiel, H. and M. Salomone. ‘Logarithmic spirals and projective geometry in M.C. Escher’s Path of Life III’. Journal of Humanistic Mathematics, Volume 2 Number 1, January 2012, pp. 22-35.

Some advanced mathematics of a substantial article. Oddly, Escher’s Path of Life III is not shown, which seems strange given the premise, and the frequent mentions throughout.

Gailunas, Paul. ‘Spiral Tilings’. In Bridges 2000, 133-140

Nice treatment indeed. Comments on Grunbaum and Shephard comment on little literature on the subject. Building on their work, Gailunas shows a ‘Zig-zag spiral tiling’. It is not entirely clear the extent of originality here. I suspect  it may be based on others, with prominent use made of the versatile. Profusely illustrated. 

No Escher-like tilings.

Gardner, Martin. ‘Extraordinary nonperiodic tiling that enriches the theory of tiles’. Scientific American. January 1977 110-121.

On Penrose tiling. Minor mention and illustration of Voderberg spiral, p. 111. Reprinted in Penrose Tiles to Trapdoor Ciphers, pp. 2-4. 

Goldberg, M. ‘Central Tessellations’, Scripta Math. 21, 1955,  253-260.


Grunbaum B. and Shephard, G. C. ‘Spiral Tilings and Versatiles’, Mathematics Teaching, no.88, Sept. 1979, pp. 50‑51.


Grunbaum, Branko and G. C. Shepherd. ‘Some Problems on Plane Tilings’, in The Mathematical Gardner by David A. Klarner. Pringle, Weber & Schmidt Boston, 1981 pp. 167-196

Amid a discussion on various problems on plane tiling, a spiral discussion, with ‘Problems 10-12’, pp. 191-194. Of particular note is that Problem 12 asks: ‘Give a precise definition of a spiral tiling’, in relation to saying what exactly a spiral tiling is. Is this the first explicit instance? Voderburg tile as a plane tiling and spiral tiling pp. 189-192, 196, Versatile 192-194 as a spiral tiling.

Grunbaum, B. and Shephard, G. C. Tilings and Patterns.  W. H. Freeman, 1987. 512-518  related p. 123

Chapter 9.5 on spiral tilings, pp. 512-516, 518. Begins with mention of Voderburg. Note and references  pp. 517-518, where they lament the lack of literature, but not here the definition [CHECK]

Grunbaum, B. ‘Patch determined tilings’. The Mathematical Gazette. 31-38.

In the context of ‘patch determined tilings’, a five-arm spiral tiling is shown, Fig. 10. 

No Escher-like tilings.

Hatch G. ‘Tessellations with Equilateral Reflex Polygons’, Mathematics Teaching, no.84, September 1978, p.32.


Kanon, Joseph. ‘The Saturday Review December’ 16 1972 ** 

Sphere Spirals

Klaassen, Bernhard. ‘How to Define a Spiral Tiling?’ Mathematics Magazine December 2017, pp. 26-38

On the difficulties of defining a spiral tiling, implicit building on Grunbaum and Shephard’s conjecture.

Shows Voderburg spiral. Largely popular, with occasional advanced maths.

Lalvani H. US Patent 4,620,998, 1986. Cited in Meta Architecture, in Architecture and Science (ed. Di Cristina G.), Wiley Academy, 2001.

Occasional spiral-like tilings, but not stated as such.

Mann, Casey. ‘A Tile with Surround Number 2’. The American Mathematical Monthly. Vol. 109 No. 4 April 2002 pp. 383-388

On coronas, something of which I am not particularly interested in. In the course of the ‘surround’ study, there is a Voderberg tile discussion, but not in the context of spirals.

Marcotte, James and Matthew Salomone. ‘Loxodromic Spirals in M. C. Escher's Sphere Surface’. Journal of Humanistic Mathematics Volume 4 Issue 2 July 2014

Palmer, Chris K. ‘Spiral Tilings with C-curves Using Combinatorics to Augment Tradition’. In Bridges Renaissance Banff 2005, pp. 37-46

The use of the word spiral in the title is somewhat overblown; it is then not mentioned again until the references page! Of no real interest as to spirals as such.

Pickover, Clifford. ‘Mathematics and Beauty: A Sampling of Spirals and Strange Spirals in Science, Nature and Art’. Leonardo Vol. 21, 1988, No. 2, pp.173-181

A good general, popular guide as to all things spiral applications as to the real world, and more, with much interest. No tiling or Escher-like as such.

Rice M. and Schattschneider D. ‘The Incredible Pentagonal Versatile’, Mathematics Teaching, no.93, Dec. 1980, pp. 52‑53.


Sharp, J. ‘Golden Section Spirals’. Mathematics in School. November 1997pp.  8-12

Of general interest in spirals. No tiling. Notable authors such as Keith Devlin and Ian Stewart are taken to task for misattributing the Nautilus shell cross section as a Golden Section Spiral.

Simonds, D. R. ‘Central Tesselations (sic) with an Equilateral Pentagon’. Mathematics Teaching No. 81, December (1977), pp. 36-37

————. Untitled note Mathematics Teaching 84 (1978), p. 33

Stock, Daniel L. and Brian A. Wichmann. ‘Odd Spiral Tilings’ Mathematics Magazine Vol. 73, No. 5 (Dec., 2000), pp. 339-346

Seemingly borrowing from Grunbaum and Shephard, Stock and Wichmann comment on the little literature on the subject. Building on their work, with a regular decagon on odd numbers of arm spirals, they show any number of odd numbered spiral tilings. A versatile is also shown.

No Escher-like tilings.

Tóth, Fejes L. Regular Figures. Pergamon Press 1964 (12 December 2010), partial copy, of Chapter 1 up to p. 43...

Regulare Figuren. Akademi kiado, Budapest , 1965. English translation

Largely theoretical. Mostly concerning group theory, which is out of my remit. Occasional tiling. Escher mention p. 39. Tilings Plates 1-3. As such, of what I have seen (Chapter 1 Plane Ornaments only), of no consequence (likely, the book is even more obscure in succeeding chapters).

Voderburg, H. ‘Zur Zerlegung der Umgebung eines ebenen Bereiches in kongruente’.

Jahresbericht der Deutschen Mathematiker-Vereinigung 46 pp. 229-231, 1936 

The first of two articles by Voderburg, in German. Four figures of Voderberg spiral tile, with one figure of the resultant spiral tiling.


 ————. ‘Zur Zerlegung der Ebene eines in kongruente Bereiche in Form einer Spirale’.

Jahresbericht der Deutschen Mathematiker-Vereinigung 47 pp. 159-160, 1937

In German. Two figures of Voderburg spiral tile, but not shown as an actual tiling.

Waldman, Cye H. ‘Voderberg Deconstructed & Triangle Substitution Tiling’ 2014. No article 

Much of interest; spiral tilings. Both popular and academic.


Crompton, Andrew 

Voderburg bird tiling.

Ribault, Dominique. Polytess website

Escher-like Elephant spiral tessellation. See notes