My Library





My library, as of 29 February 2016, of primarily of books and articles, but also of letters, pamphlets, reviews, patents, theses, puzzles, exhibition catalogues, conferences, videos, notes, reports and miscellany.





Abas, Syed Jan; Salman, Amer Shaker. Symmetries of Islamic Geometrical Patterns. World Scientific 1995. (12 December 2009)

No Cairo


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


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, P; Kerridge, C. E. 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


Aczel, Amir D. Fermat’s Last Theorem. Penguin Books 1997 (16 July 2007)

Historical quest


Adler, Irving. Mathematics. The Story of Numbers, Symbols and Space. 1958. Juvenile. (3 September 1995)


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

Of limited interest


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

Minor Escher text pages 80-81, Waterfall, Sky and Water I pictures. Bizarrely, the Sky and Water print is asymmetrically cropped!


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

Escher’s Ascending and Descending, page 34, Mobius Strip II, page 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)


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 creditability. Minor aspects of tessellation, within ‘perception’, pp,40, 43 and Islamic design, pp. 66-67.


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, pages 161-180, on Penrose, with a small tiling interest;  Penrose chickens page 177, and Escher's print Waterfall, page 179, along with a popular discussion of Penrose tiles.


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.


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


Anon. Technical Drawing. Letts (22 March 1987)

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


Anon. Letts O Level Geometry (23 March 1987)

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


Anon. The New Mathematics

From  a reference of early maths studies, of 1986-1987 (17 December 1986)


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

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


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


Anon. Practical Plane and Solid Geometry


Anon NGM New General Mathematics?


Abbott.  Algebra (20 January 1987)


Paling, D. Elementary Mathematics. Details Pending

From a reference in my own early studies, of 26 January 1986 1, 2, 8  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.


Hanby, G. A. Geometry I 19-20 March 1986


————. 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 274-84, Puzzles and Curious Paradoxes 286-300. Answers to Puzzles and Paradoxes 301-318. These contain loose geometric dissections, but nothing of particular note.


————. Tricks Played on Hand and Eye’ The UNESCO Courier, Vol., 19, no, 5 (1964), p. 14. (Note the year here, 1964, and not 1966, as given by all authors where this is quoted; all copying from one another)

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 pages 25-28: 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 World of Shape & Number. 1970 Marshall Cavendish Learning System. (6 February 1994.

Advanced Juvenile


————. 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 page 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 amongst 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 20-31, Day and Night. Broadly a retelling of existing illusions. Also see a later companion booklet, of 1975


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 page 29 here, and page 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)


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

Not strictly mathematical, but has the occasional crossover.


? Harpe P. 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?


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


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’, 266-290.


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

Tessellations page 130, barely worth the mention. Textbook


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

Not strictly mathematical, but related



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.

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)


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

Chapter 4, pages 111-141 has much on Quasicrystals and Penrose tiling. Escher’s page and minor text 128-129


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, 105-107 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)


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

Of limited interest; mostly philosophical musings


————. 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.130-131, 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, 23-42, 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 Mataheice, of the same title, which is reprinted in the book. much of the forimalae are too complicated for me, but nonetheless the diagrams are largely accessible.  Howrver, the book largely flatters to decivee.


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


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 80-81. ()


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’, ‘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. 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 as Here’s Looking at Euclid in the US. (27 July 2014).

A personal wander around mathematical aspects of interest to the author, of an overwhelmingly popular level. Occasional references to Escher, 244 and 392 hyperbolic geometry, with Circle Limit IV. Phi, 299-301(and colour plates), with Gary Meisner interview. Martin Gardner 250-253


Belur, Ashwin; Whitaker, Blair. A Practical Solution to Rubik’s Magic. Corgi Books 1986 (two copies, 27 September 1992 and 5 February 1994)


Bergamini, David et al. Mathematics. Time Life 1969 (21 March 1998).

This is a paperback, also see hardback in possession. (False) references are made to the golden ratio appearing in paintings, of which Mario Livio in The Golden Ratio p. 164 rebuts.


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

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, page 228, Metamorphosis I, pp 236-237

Sketch: wall mosaic in the Alhambra, p. 202

Cairo tiling, but not attributed, p. 144


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

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


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


Billings, 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 outside of tiling matters. Tracery seems to be Billings’ main interest, he has another 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)

Self-published booklet of 32 pages. Occasional digressions from polyhedra, with stereo and tessellations


Block, J. Richard, Yuker, Harold E. 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


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).

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. Despite the title, it’s not a book 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, pages 130-169, and other tiling instances scattered throughout the book.


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

Also see Murray T. M Rust, for another in this series


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



————. 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, page 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 147-148 beginners, any quadrilateral will tessellate rule. Also see Activity 76 The Pentominoes, pp.56-77. Other aspects are of interest.


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


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. Page 123 Cairo-like diagram, dual. Has a good bibliography


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

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


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


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


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 (13 or 20 April 2013)

An infant’s book, found by pure chance upon a visit to Cleethorpes library, where in the sale section this was placed, my attention drawn to a symmetrical drawing of cows on the front cover. 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 G. Transformational Geometry. Dale Seymour Publications 1973. (24 October 1998)

Escher’s periodic drawings on cover, swans, and pages 36, Beetles and Flatfish 45, swans, and 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 155-188


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 Schattsneider. 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, 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?


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

From an Escher reference in Schattsneider’s Visions…. Somewhat disappointing in this regards, with a most lightweight treatment indeed of Escher, with two small discussions, as ‘Eschervescence’ Part 1 81-85 (Fish and Frog Optimist/pessimist Birds and Fish) and Part 2 118-121 (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. The book itself is very much in the spirit of Gardner’s The Ambidextrous Universe, of which in the preface Bunch defers to.


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

As sent by Bob Burn


————. 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)


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)

Mostly about time per se. Tilings page 48-49




Cadwell, J. H. Topics in Recreational Mathematics. Cambridge University Press 1966 (13 October 2006)

Occasional aspects of interest; Chapter 1 Regular Polyhedra, Chapter 9 Dissection Problems in Two and Three Dimensions, but mostly too advanced.


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

Chapter 7 Geometry, no tessellations


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, 1958-1972. 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. Nuffield Mathematics Project. Newgate Press Ltd 1968). Books 2 and 3.

Juvenile. *Tessellations 27-28; 32-41(Book 2 9 June 1996) (*Book 3 2 June 1995)


————. Environmental Geometry. Nuffield Mathematics Project. Newgate Press Ltd 1969. Teachers Guide).



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, page 282-298.  ‘Pólya’s ‘Do3’ tiling is shown, page 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


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, 312-313. 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


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. 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, pages 134-135, 152, 153, 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 that it would be suitable for me.


Corbalán, Fernando. The Golden Ratio. The Beautiful Language of Mathematics. Published by RBA Coleccionables, S. A, 2012. Appears to be a English translation of a Spanish work (7 June 2014)

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 ration overlays, such as p. 12-13, 107


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

Juvenile, for classroom work. Very basic indeed, pages 1-6 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, include as it has Escher aspects. Page 63 E97 Bulldogs, E85, Lizard Fish Bat; 81 Print Gallery


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


————. 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. 241-263, 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. 58-73 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. 1-13 and Chapter 2 Regular and Quasi-Regular solids, p. 15-30 and Chapter 6, Star-Polyhedra p. 93-114 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)

Escher pages 57 (Horseman E67) - 59 (Beetles E91), 63. Very brief text.


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 p. 147-150


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.213-217, 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. 2000 (22 September 2007)

Has Cairo diagram page 49. This also has an interesting series of diagrams page 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. Interestingly, in the bibliography, he quotes D.G. Wood, of indirect Cairo tile fame, perhaps he borrowed from him.


————. 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 pages 2, 171-172 (sketch of a cutaway view of small stellated dodecahedron), 239, 251, 258. Mostly minor text, in conjunction with polyhedra.


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


Cundy, H. Martyn; A. P. Rollett,. Mathematical Models. Oxford University Press 1977 (?) First published 1951

Plane Tessellations Chapter 2.9, 59-65. Cairo diagram but without the attribution page 63.




Daintith, John and R. D. Nelson. (editors). The Penguin Dictionary of Mathematics. 1989 (4 November 2000)


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 pages, 49, 57, 60, 88-89, 103, 137,  252-253

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

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


Day, Lewis, F. Pattern Design. London, B. T. Batsford 1979. First saw 1988? (18 February 2011)

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.


————. The Application of Ornament. B. T. Batsford 1898
Has houndstooth-like basket weave p.25


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


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

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


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


Dearling, Alan; Armstrong, Howard. The Youth Games Book. Resource Centre, Glasgow. 1985 (12 July 1998)

Intermediate Treatment Juvenile.


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, 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 page 53


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



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


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 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


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.




Eastaway, R. Enigmas. The World’s Most Puzzling Book. Arlington Books 1982. (31 October 1993)



Eastaway, Rob and Haigh, John. 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 books 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


Elffers, Joost. Tangram. The Ancient Chinese Shape Game. Penguin books 1984. (19 May 1995)


Elffers, Joost; Schuyt, Michael.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.


Engel, Peter. Origami: from Angelfish to Zen. Dover Publications Inc. 1989 (26 May 2008). Occasional reference to Escher’s tessellations and prints, pages 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



————. 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. Escher print Concave and Convex p. 27. Small section on Escher per se, pp 74-80. Escher Belvedere model by Shigeo Fukada pp. 92-93


Escher, M. C. The Graphic Work of M. C. Escher. Oldbourne, London 1970. (8 August 2004) and Taschen (10 August 1993)


————. M. C. Escher 29 Master Prints. Harry N. Abrams, Inc. Publishers New York 1983 (9 April 1993)

Large format book. In addition to the 29 prints, both tessellation and others, the book includes an essay by Escher, 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.


M. C. Escher’s Universe of Mind Play. edited by Fuji Television Gallery, 1983. (25 August 2010)

Exhibits of travelling exhibition, in Osaka, Fukai, Ishikawi, Tokyo, September-October 1983.

Various Essays: Symmetry in Escher’s World: Its meaning to our times Itsuo Sakane, 13-16; Escher and Endless movement Yuseke Nakahara, 19-25, The Magician of Impossible Space: Psychology and M.C. Escher, Soichi Hakozaki, 27; Escher, Symmetry, Four Dimensional Space, Koji Miyazaki 177-192


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. Turnstone Press 1985.

The 1983 edition has a front cover picture (among others) of Escher’s Drawing Hands


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


Fathauer, Robert. Designing and Drawing Tessellations. No publisher given. 2008 (18 July 2009)

I consider the title a little misleading, given that the premise is one of creating Escher-like tessellation, rather than non-lifelike tessellation per se as the title would otherwise suggest. One of the few books to approach the topic in depth, and so is warmly welcomed.


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, Robert. Mazes Ancient & Modern. Tarquin Publications 2001. (date not stated)


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

No Cairo pentagon. Small booklet, 64 pages.


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 p. 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 page 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). Wilson, Janet. 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

A  series of translated essays of Escher's previously unpublished speeches:

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

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

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

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

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

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

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

Approaches to Infinity (no context or date given). 123-127

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

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

I’m Walking All Round  All By Myself Here, by J. W. Vermeulen  139-153


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

Oversize. The premise is of a ‘grand picture book’ per se, with no text save for the introduction. Useful for seeing Escher’s prints at a larger size than in most books


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, p. 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

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

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, p 24-25 and the three tags, p 50-51. Much of this is Gardneresque nature, albeit pitched at youth. Geometric dissections p. 52-53, but no tiling as such.


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

An absolute delight!  Highlight after highlight, too many to list here. Speculations as to who ‘A. E. Hill’ was, page 157.


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

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


————. Piano-Hinged Tessellations, A. K. Peters, Ltd. 2006 (?)

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’)


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 (24 October 1998).

Juvenile, Junior.


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


Fuller, Buckminster R. Utopia or Oblivion. Penguin Books 1972 (20 September 1992).

Chapter 3 only of direct interest



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


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)

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


————. 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


————. 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

On numerology; a major disappointment! 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, pages 43-51 and Rep-tiles Replicating Figures on the Plane, pages 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


————. 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 referencesp.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 lie 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


————. 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)


————. 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 page 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)



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). however, scattered throughout are various ‘spotlights/biographies, such as Angels and Devils. p. 642-643.  Stanford teapot 647. Reference to par hexagon, 701, 716. Of most interest are Chapter 21, on Symmetry and patterns, and Chapter 22, Tilings 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 (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 Wold, and 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 36-43, but of the usual nonsensical type. snowflakes, spirals and curves 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 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, p. 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 a 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 hare as it has a  small chapter on recreational mathematics: ‘Geometrix (sic) ‘Tricks Involving Geometrical Principles’, 107-114. Included are Mobius strips and lost line, and Hooper’s cut. (Hooper is mentioned in the inteodcution)


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, 111-115. Minor tilings p. 111


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

Uses my bird tessellation from the Leeuwarden 2008 Bridges art exhibit, page 2.


Glenn, Robert. Foundation Maths. For GCSE and Standard Grade. Heinemann Educational Books Ltd 1988 (15 January 2001)

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


Glenn, William H; Johnson, Donovan A. The Theorem of Pythagoras. Exploring Mathematics on Your Own 4. John Murray 1965 (22 October 2005)


————. 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?)

Drawing geometrical figures, with occasional tiling pp.150, 152. Early days of computing, and so all rather dated. Nothing of any interest now


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

Of very limited interest.  Chapter 5, Networks and maps (topology) 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 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, 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 (unknown) Michio Kubo, 1968 on page 91. Frequent occurrence is the tem ‘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.


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

Illusion and Visual Deadlock, 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.


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 page 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 page 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. Illusion in Nature and Art. New York, Scribners, 1973


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

Indispensable! Largely, indeed overwhelmingly, academic, but still accessible on occasion. Cairo-esque 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 pages 297-300, very basic, barely worth mentioning.

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 tilling. 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, 19880

Fun with tessellations, John Rigby

Escheresch, Athelstan Spilhaus

Henry Ernest Dudeney:  Britain’s Greatest Puzzlist, Angela Newing

The Utility of Recreational Mathematics, David Singmaster

Puzzles Old & New: Some historical Notes

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




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.


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.


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

Obtained in relation to possible interest regarding my cluster puzzles, this being a commonly quoted book in jigsaw puzzle circles. As such, for my purposes, somewhat of a let down; it consists mostly of text, with relatively few pictures, and furthermore there is nothing cluster related.

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 pictures. 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)

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. Relatively lightweight, of just 91 pages. Nothing at all in the way of cluster puzzles. However, of slight interest is page 91, where a puzzle has been cut into a tessellation premise of a broad single tile.


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

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


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.


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 , has loose elements, but nothing is entirely satisfactory. Probably the best chapter is  pp.149-165, of Lewis Day-esue, 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. 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 page 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 Schattsneider.


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.


Hemmings, Ray; Tahta, Dick. Images of Infinity. Tarquin Publications 1992 (3 June 1993)

Escher’s Circle Limit 1, page 14


Van der Heyden, 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 see from afar, diagram 39 (book is unpaginated!), although the sighting is strictly not  discernable, 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.


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)

Hopelessly outdated, only obtained due to a known Cairo tiling reference, page 145. Escher tilings: p. 143 Horseman, Birds and fish page 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).


Holiday, Ensor. Altair Creative Colouring Books. Book 3. (9 March 1996 (year semi legible))



Holt, Michael; Ridout, Ronald. The Second big book of puzzles. Puffin Books 1976 (12 September 1993).

Usual fare.


Hoffa, Alan; Koss, Roberta. Focus on Geometry. Addison Wesley Secondary Math. 1998 (15 October 2005)

Tessellations 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. 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?).

Of limited interest. Parquet deformation discussion, not diagram page 477


————. 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, obtained 3 December 2006)

Many uses of Escher’s prints, too numerous to mention here. Book is a bit quirky, if not downright odd.


Hogben, Lancelot. Mathematics for the Million. Pan Books Limited 1940, 1967 (7 March 1993 hardback; 16 April 1995 paperback)


————. 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 pages 315-316, simple, barely worth mentioning


Hollands, Roy. A Dictionary of Mathematics. Longman 1980 (not date stamped, c.10+ years)

Tessellation page 151, inconsequential.


Holt, Michael. What is the New Maths? Anthony Blond Ltd. 1968 (23 September 2000)


Holt, Michael; Ridout Ronald. The Big Book of Puzzles. Puffin Books 1976 (c September 1993) The title is misleading; it’s a standard paperback!


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)


Huff, Darrell. How to Lie with Statistics. Penguin Books 1988. (11 July 1998)

Of limited interest




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. Manufacturers brochure, not dated. (13 July 1995)




Jacobs, Harold R. Mathematics A Human Endeavour. W.H. Freeman and Company 1970 (18 June 2015)

Uses of three of Escher's artworks: Horsemen, on cover, Waterfall p. 19, Horseman again p. 207 and Möbius Band p. 478.  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, 202-208 and Chapter 4, The Regular Polyhedra, 209-244. Certainly, there is nothing ‘new’ here for me.


————. Geometry. W. H. Freeman and Company 1974 (25 August 2007)

Many instances of Escher use throughout the book (although not indexed), on the cover, Ascending and Descending,  pp 128, 227, Beetles, Birds, Flatfish, Bulldogs, 300  birds and Fish. Full of interesting bits of geometry, at a largely accessible level

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, V. The Complete book on Patchwork and Quilting. 1985 (11 June 2013)

Although not a maths book per se, included as this was one of the earliest books I studied; it having many geometric tilings


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.


Jaworski, John & Stewart, Ian. Nut-crackers: Puzzles and Games to boggle the mind. Pan Books Ltd 1976 (14 October 2000)


Jeger, Max. Transformation Geometry. George Allen and Unwin Ltd. 1970 (date not stated)

Of limited interest.


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.


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, pitched at a juvenile audience. This is mostly of ‘simple’ 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. (10 August 1993)


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, pages 71, 134, 191, 248, 265. Many chapters on polyhedra. Cairo tiling featured as the dual of  32 .4. 3. 4, page 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.


Kasnar, Edward, Newman, James. Mathematics and the Imagination. G. Bell and Sons, Ltd.  1970 (25 April 1999)

Space filling curves 343-355


Kay, Keith. Take A A Closer Look. Bright Intervals Books 1991 (3 June 1993)

on optical illusions. Escher tessellations Ascending and Descending, 36 and Belvedere, 42. No text worthy of the name,


Keefe, John O’; Rush, Phillip. Weights and Measures. Methuen and Co Ltd. 1966 (12 October 2002)

Advanced juvenile.


Kelsey, Kenneth; King, David. 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.

As a broad statement, a series on perspective. Much of interest. 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. p. 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 Bezuska. Tessellations Using Logo


Kepler, Johannes. Harmony of the World.

Available on-line:



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 113; Escher inversion  p. 45. Parquet deformation 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


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 pages 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, pages 140-166 re Marjorie Rice and pentagons; Some problems on Plane Tilings, 167-196, Branko Grünbaum; Angels and Devils. H.S. M Coxeter. 197-209. Escher references Colour plate IV, Coxeter article page 198 Angels and Devils, with typical Coxeteresque obscure text. Escher Sphere with Fish page 201. Polyhedron with Flowers, page 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.


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.


Kraitchik, Maurice. Mathematical Recreations. George Allen & Unwin, Ltd. 1943 (18 March 2000, Lincoln) First saw in 1987

Chapter on Geometric Recreations, 2. Mosaics, 199-207. Also see 3. Mosaic on the Sphere, 208-209. Simple tiling diagrams, and ways of tiling with various regular polygons in combination.




Laithwaite, Eric. Engineer Through The Looking-Glass. British Broadcasting Corporation 1980. (11 October 1997, Lincoln)

Brief discussions on Mobius band, flexagons and polyominoes. 28-31; 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 pages 23-25, Solid geometry pages 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. 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, page 13, but without detail or credit!


Langdon, Snape. A Way With Maths


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.


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, page 5. Many computer drawn examples, badly dated. Tiling pages 49-51, 54, the latter of Escher-like ‘gnomes’, by Richard Ladds.


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, pages 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 for internet 10 June 2014)

From a reference on 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, 3540, 46, 51, 55, 63, 67, 69, 77, 89. No tessellation or polyhedra.


Lewis, Donald J. Introduction to Algebra. Harper and Row. 1965 (29 May 1994)

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 pages 232, Reptiles


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, 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.

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 20-21; magic squares 39-43, geometric fallacies 54-55, map colouring 61-62, bees speculations 91-99, 155 pages


Liebeck, Pamela. How Children Learn Mathematics. Penguin Books 1988 (16 February 1995) Tessellation, pages 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 page 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 ‘ From the Tiles to the Heavens’

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)



————. The World of M. C. Escher. Abradale Press 1988 (9 April 1993)


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 page 88. Escher page 4, Shells and Starfish, E42, Fish E41, page 66 Lizards, E56.


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: 65-66, 79, 102, 119-120, 162-169. Pictures include 66 Horseman, 120 Running man, 163 Fish, 164 Lizards, 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.


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 with foreknowledge, as the picture is from afar that without cognisance 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.


Lukas, Edouard. L’Arithmetique 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)

Cairo diagram (but not attributed) page 101, the first (1921) recorded instance? 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, with corrections

Originally saw this in College library (and ‘studied’, broadly stated) in 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. Upon an initial glance through the book, there is nothing original here; the material appears to have been taken from existing sources.


Mallinson, Phillip R. Geometry & its Applications Tessellations. Comap, 1996. (?)

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, 125 Circle Limit IV, 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,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, 164-178 (16 October 2010)


Marjoram, D. T. E. Exercises in Modern Mathematics. Pergamon Press 1975 (18 September 1988?)

The only interest is in Chapter 10, Topology


Marks, Robert W. The New Mathematics Dictionary and Handbook. Bantam Books 1967 (9 April 2007)

Tessellation not mentioned.


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, p. 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.


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.


McGregor, Jim and Alan Watt. The Art of Microcomputer Graphics for the BBC Micro/Electron (First saw 1987) Addison Wesley 1994

See Chapter 5 Night and day – a journey through the world of tesselations (tesselations as spelt as in original)

Cairo pentagon references: text, 196, and picture, 197 Illustrated with a line drawing.

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’.


McLeish, John. Number. From cave people to computers, a revolutionary view of ourselves. Bloomsbury publishing Limited. 1991 (17 December 2005)


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, pages 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 pages 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, 279-280.


Midonick, Henrietta. The Treasury of Mathematics: 2. Penguin Books 1965 (29 October 2005) 24 Biographies


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 pages 100 and 175 (solution), but without further detail. April 2012 research dates this as of 1951 and (not shown) 1954. Also see this repeated in Investigation in Mathematics by L. Mottershead.


Mitchell, James (general editor). Science and the Universe. Mitchell Beazley 1977.

Minor reference to Escher’s prints Angels and Devils, page 51 and Mobius band, page 53, with general comment. So lightweight as be barely worth comment.


Mold, Josephine. Circles. Topics From Mathematics. Cambridge University Press 1967. (20 August 2000?)

Small, 32 page booklet. Very accessible, with much of interest


————. Topics From Mathematics. Tessellations. Cambridge University Press 1969 (20 February 1991) photocopied book

School age level, but still of interest. Shows dual Archimedean tiling, page 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


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 page 55, with minor text barely worth the mention


Morgan, Bryan. Men and Discoveries in Mathematics. John Murray 1972 (24 October 1998)


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 published under the generic theme of ‘Ivan Moscovich’s Mastermind Collection’. c100 aspects of ‘simple’ recreational mathematics,  pitched at a juvenile level, mostly seen before but nonetheless remain of interest. No Tesselation. Dissection p. 27, packing discks or circles, , p. 36-45


————. Loopy Logic Problems & Other Puzzles (31 July 2013)

Sterling Publishing Co, Inc, New York, 2006


————. The Hinged Square


————. The Shoelace Problem


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 state: ‘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. 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 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 a 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 page 64 on  see p. 128 of Sources…’ without any mention of Gettings!

The first of two book of a like nature by Mottershead, although wide apart in chronology, 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 in1987 (at Grimsby reference library)


Mott-Smith, Geoffrey. The Handy Book of Indoor Games. Permabooks. 1949. Mostly card games. (c. 1997)


Munari, Bruno. Design as Art. Penguin Books Ltd 1971 (Not date stamped, c 10+ years)

Design, rather than maths. Occasional mathematics


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)

Tessellations, Chapter 10, pages 194-205. Cairo tiling, unattributed, page 200


Murphy, Patrick; Kempf, Albert F. The New Mathematics. W. H. Allen London 1982 (18 October 1997)




Nasr, S. Islamic Science. (c. 9 August 1988)

First seen in Grimsby library, of which I photocopied a few sheets (recorded on a label)


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)


Nelson, David (ed.). Dictionary of Mathematics. Penguin Books 1998 (25 August 2007) Tessellation gets a brief mention


Newman, James A. The Universal Encyclopedia of Mathematics. Pan Books 1976 (1 April 1993)


Nicolas, Alain. Parcelles d’infini Promenade au jardin d’Escher. (in French) Belin Pour La Science. 2006 (2010?)

Delightful! 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 (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 p.88-90, patterns based on squares 90-91, patterns based on circles, p. 92-93 patterns in circles p.94-95 and tracery, p.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.




Nixon, J. T. World of Shapes. Oliver and Boyd Ltd. 1968 (5 October 1998)



Northrop, Eugene P. Riddles in Mathematics. Penguin Books 1975. (17 October 1998)

Largely of paradoxes and fallacies. Not tessellation as such, but of much related material; of minor optical illusions, dissections, and space filling curves, to name but few. The overall tenure is largely of an academic nature.




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 (14 November 1998)


O’Daffer, Phares.G; Clemens, Stanley R. Geometry. An Investigative Approach 2nd edition Addison-Wesley Publishing Company 1992. (23 October 2010).

Chapter 4, 86-117 Patterns of Polygons: tessellations, albeit very basic in scope. Has Cairo tiling page 95. Occasional usage of Escher’s prints: Day and Night 86-88, Horseman 114, Magic Mirror, 215


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 a Islamic leaning due to her background in these designs.


O’Shea, Donal. The Poincaré Conjecture. In Search of the Shape of the Universe. Allen Lane, 2007. (21 April 2012)

Semi-popular account.


Ōuchi, Hajime. Japanese Optical and Geometrical Art. 746 Copyright-Free Designs for Artists and Craftsmen. Dover Publications Inc, New York 1977 (9 April 1993)




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 pages 272-272, with the ‘any triangle, quadrilateral will tile’ rule.


Paling, D. Elementary Maths. Details Pending

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 forgotoot 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)

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. 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)


————. Mathematics Appreciation. Wide World Publishing/Tetra Revised edition 1987. (3 June 1993).



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 Publications1998. (21 September 2004)



Paulos, John Allen. Beyond Numeracy. An Uncommon Dictionary Of Mathematics.  Penguin Books 1992 (30 April 1994)


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)

As such, although largely on popular subjects, there is very little of direct interest to me here. Chapter 5, Two-Way Stretch, on topology, has elements of interest.


————. Geometry and The Liberal Arts. Penguin Books 1976 (14 July 2001)


Peitgen, Heinz-Otto et al. Fractals For The Classroom. Strategic Activities Volumes 1 and 2. Springer-Verlag 1991 and 1992 (6 August 1997)


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. 172-178. Occasional Escher references, Circle Limit I page 203. Quasicrystals 562-563.


————. Shadows of the Mind. Vintage 1995. (5 November 2011)

As to be expected by the tenure of the book, this is almost entirely beyond me, the only aspect of understandably is a short discussion on tiling 29-33, Robert Amman influenced.


————. 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!


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’, pages 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


Pipes, Alan. Foundations of Art and Design.  Laurence King 2008. Grimsby Library.

Has occasional Escher, with Day and Night


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. 1991 (30 April 1994)


Polster, Burkard (with foreword by John Langdon). Eye Twisters. Ambigrams & Other Visual Puzzles to Amaze and Entertain. Constable, London. 2007 (2010)

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)

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)




Raba, Raoul. Zoo Mathématique ACL Les Éditions Kangourou 1998 ed André Deledicq In French (14 March 2015)


Racinet, A. The Encyclopedia of Ornament. Studio Editions 1989. (Originally published as Polychromatic Ornament by Henry Sotheran and Co. London 1873) (10 August 1993)


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, E. R. Tessellation and Dissection. J. Weston Walsh. 1970 (?  2008?)

Has a variation of the Cairo tiling, with two pentagons, page 36


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 page 90


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**


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’ pages 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. Tranlated 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 57


Reyes, Encarnación and Inmaculada Fernández. Pentágonos. Construcciones. Mosaicos, Geometría sagrada. (in Spanish) 2015. Universidad de Valladolid (21 January 2016)

Our Cairo tile photos on the front cover and pp. 74, 156. Has much of interest in a generalised sense. P. 166 has an interesting ‘mixed’ Cairo tiling, with kites.


Reichmann, W.J. The Spell of Mathematics. Methuen & Co Ltd 1967 (14 July 2001).

Of limited interest. Too advanced.


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, page 37. An interesting technique for drawing patterns is given, page 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.


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 (12 December 1998).

Rangoli and Islamic tilings page 21. Use of Escher’s Relatively print Frontispiece and page 25. Juvenile.

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)

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)


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).

Pattern in the broader sense, rather than confined to tessellation. Also see A. E. Bolt for another book of this series




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. (27 August 1997)



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 mathematics book per se, included as it has an Escher print, 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. Dover Publications Inc 1954 (7 December 1994)


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 pages 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. The Search for Pattern. Penguin Books 1970. (17 September 1994).

Not recreational maths


Prelude to Mathematics. Penguin Books 1961. (9 July 1994).

Of limited interest. Better than his other book The Search for Pattern


————. Vision in Elementary Mathematics. Penguin Books 1964 (2 April 1994)

Of limited interest


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.


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)



Schattschneider, Doris; Walker, W. M. C. Escher Kaleidocycles. Tarquin Publications 1982 (19 August 1988)

Cairo like tiling, page * . 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 Probelms112-126 (129-138), but without tilings. Volume III appears of a more technical nature, mostly text, of few diagrams. Nothing on tilings and polyhedra.

Seckel, Al. Incredible Visual Illusions. Arcturus Publishing Ltd, 2005 (not stated) (guess 2008?)


————. 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 290.


Scripture, Nicholas E. Puzzles and Teasers. Faber and Faber. 1970 (24 October 1998) Dudeneyesque.


Sealey, L.G. W. The Shape of Things. Basil Blackwell Oxford 1967 (12 October 2002)

Juvenile 10-years-old audience


Seiter, Charles. Everyday Math for Dummies. Hungary Minds Inc. 1995 (17 April 2005)


Seymour, D; Britton, J. Introduction to Tessellations. Dale Seymour Publications 1989 (8 March 1995)

Cairo tiling, but not attributed, page 39


Seymour, Dale. Introduction to Line Designs. Dale Seymour Publications 1992 (10 August 2006)

Advanced Juvenile


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.


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 page 180, albeit by default of quadrilateral tilings 176-179. Escher lizards, unicorns figures 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.


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


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


Slade, Richard. Geometrical Patterns. Faber and Faber Limited 1970. (24 October 1998) Contains interesting historical French curve source reference, page 16, different from others. 16-year-old audience


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 it. Indeed, I might even have 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.


Sly, A. J. GCE O-level Passbook Modern Mathematics. 1976 (20 September 1992) Textbook


Smeltzer, Victoria; Smeltzer, Patricia. Mathematics Encyclopedia. Burke Books 1980 (18 February 2000?)

Juvenile, 10-year-old. Tessellation page 75. Hexagons, not worth mentioning


SMP Book 1. Cambridge University Press 1965 (6 August 1994) Patterns (tessellations) 159-163 basics (Hardback)


Smith, Thyra. The Story of Measurement. Basil Blackwell Oxford. 1968. (12 October 2002). Juvenile


————. 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)

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


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)


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)

Substantial tessellations, albeit of a basic nature, pages 1-5, 13-22


SMP Book F. Cambridge University Press 1970 No tessellation (10 February 1994) (A series of eight books, A-H, for CSE. SMP 1-5, is for O-level)


SMP Book H. Cambridge University Press 1972. (16 October 1993) Doncaster

Minor tessellation, pages 83-84, with a P. Murphy chicken-like motif


SMP Book T. a 9 December 1986 refernece.


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?)

Logic Puzzles


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 pages 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 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). Third printing, 1987 (c. 2008). 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


————. 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.


Stewart, Ian. 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.


————. Concepts of Modern Mathematics. Penguin Books 1982 (16 May 1999) (17 November 1994 and 24 August 2004)

Of limited interest, somewhat technical.


————. 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.


————. 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. (17 June 2012)

Popular account of hard to understand concepts.


Stewart, Ian and Martin Golubitsky. Fearful Symmetry: Is God a Geometer? University Press 1993

Occasional Escher pictures, Circle Limit IV, page 45, Lizards 237; Penrose tiling page 95, Kepler Aa to Z patch, page 96; Pólya diagram page 239, with Pólya’s annotations, but generally all these references are in passing only


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; Star, Phyllis. 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 & Leanne Rylands. Cube Games. 92 Classic Games, Puzzles & Solutions. Penguin Books 1981. (20 June 1993)


Taylor, Don. Mastering Rubik’s Cube. Penguin Books 1981. (29 August 1993)

Very small book


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)

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 , 179181, 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. .


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


Thompson, D’Arcy. On Growth and Form. Abridged edition by J. T. Bonner. Cambridge University Press. 1975 (13 July 2009)


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.


Todd, Audrey. The Maths Club. Hamish Hamilton London. 1968. (26 September 1991)


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.

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.


Tyler, Tom. British Jigsaw Puzzles of the 20th Century. Richard Dennis 1997 (22 March 2014)

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.




Valette, G. Les Partages d’un Polygone Convexe en 4 Polygones Semblables au premier (in French)


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




Wade, David. Crystal & Dragon. The Cosmic Two-Step. Green Books 1991 (27 November 1993)


————. Geometric Patterns & Borders. Wildwood House Ltd. 1982 (16 September 1995)

Simply a pattern book, with line drawings from various cultures around the world. Text is purposefully kept to a minimum at the beginning of the book. Has many interesting designs. No Cairo tiling


————. 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.


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.


Walter, Marion. The Mirror Puzzle Book. Tarquin Publications, 1985 (30 January 1999)

Juvenile, of no interest


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.

Page 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. 158 paving tiles from fourteenth century, France.  p. 172, stylised swans from Escher, p. 206 arrowhead tiling variation, p. 219 Escher's lizards, 232 Chinese traditional design.


Wells, A. F. The Third Dimension in Chemistry. University Press, Oxford, 1956

Cairo tilings 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 page 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 N. 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)

N.B. Rubik’s puzzle.


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-. Mathematics. The Science History of the Universe. The Waverley Book Company 1911 (Date not stated)


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 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


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. Pieceful solution plate. Richardson pp. 55-57, 59 Escher, Savage p. 107 and endnotes p. 217. No mention is made  of ‘A Bad Dream’ or Palmer, or indeed the concept of cluster puzzle beyond Pieceful Solution.


————. 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. Page 37 is a dedicated section. Sam Savage’s Schmuzzle puzzle p.325, which mentions a 16-page instruction book of tesselated [sic] figures not seen.

Just for general interest, I list Richardson’s puzzles known: The Gleaners, Kentucky Belle, Romeo and Juliet at the Window.


Williams, Robert. The Geometrical Foundation of Natural Structure. A Source Book of Design. Dover Publications, Inc. 1979 (3 June 1993)

Cairo tiling page


Willson, John. Mosaic and Tessellated Patterns. How to Create Them. Dover Publications, Inc. 1983. (30 April 1994)

Cairo tiling plate 3. (Neglected until 7 May 2013!)

Very pleasing indeed, with many simple, but interesting tilings, and ideas thereof.


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.


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, pages 28-29 (one with potential as a human figure), and 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, 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)

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


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, page 6, with Relativity print, and book page 17.




Zechlin, Katharina. Games you can build yourself. Sterling Publishing Co., Inc. 1975 (23 August 1994)

Mostly board games






Abas, Jan. ‘Islamic Patterns: The Spark in Escher’s Genius’. In Coxeter, et al, Eds. M.C. Escher: Art and Science. Amsterdam: North-Holland 1986. 100-112 (30 April 1994).

Acevedo, Victor. ‘Space Time with M.C. Escher and R. Buckminster Fuller’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration (31 August 2005)

Adams, Colin C. ‘Tilings of Space by Knotted Tiles’. The Mathematical Intelligencer Vol. 17, No. 2, 1995, 41-51 (16 December 2011)



Akiyama, Shigeki. ‘A Note on Aperiodic Ammann Tiles’. Discrete & Computational Geometry (2012) 48: 702-710. (2 November 2012)

Largely of an academic nature throughout, of which although there are plenty of largely accessible diagrams, the tone of the paper is way beyond me. Of no practical use.


Akelman, Ergun. ‘Twirling sculptures’. Journal of Mathematics and the Arts. Vol. 3, No. 1, March 2009, 1-10


Alexanderson, G. L. and Leonard F. Klosinki. ‘Mathematicians’ Visiting Cards’. The Mathematical Intelligencer (Mathematical Communities) Vol. 25, No. 4, 2003 (22 December 2011)

Includes MacMahon’s visiting card.


Alexanderson, G. L. and K. Seydal. ‘Kürschak’s Tile’. Mathematical Gazette 62 1978 192-196

From a reference in Tilings and Patterns. Semi-popular


Albers, Don. ‘Mathematical Games and Beyond’.  Part II of an interview with Martin Gardner

College Mathematics Journal Vol. 36, No. 4 September 2005


Albright, Thomas. ‘Visuals’. Rolling Stone. 52. p. 40, February 21, 1970 (28 March 2011)

An in-depth essay on Escher, although of a single page of the magazine, in broadsheet format. Various prints, too numerous to list, are discussed, although these is nothing here in the way of originality. Illustrated with three prints LiberationSelf Portrait and Waterfall.


Amiot, B. ‘Mémoire sur les Polygones Réguliers’. Nouvelles Annales de Mathématiques, 264-278. 1844 (23 April 2015)

From a reference in Bradley. Disappointing, not a single diagram! Academic throughout.


Alvarez, Josefina and Christina Varsavsky. ‘Tilings’. Function Volume 28 Part 4 94-102 August 2004. (15 January 2015)

On convex pentagons, among other tilings. As an aside, this (Australian) journal only came to my attention to as late as 15 January 2015, it having begun as far back in 1977!  


Ammann, Robert, Branko Grünbaum, G. C. Shephard. ‘Aperiodic Tiles’. Discrete & Computational Geometry 8: 1-25 1992 (25 May 2012)



Anonymous. ‘Life’s rich patterns’ (28-29) ‘Art by numbers’ (30-31) and ‘Get into shape (32-33). Junior Education. January 1994 (30 December 1993)

N.B. The chronology discrepancy is explained by the journal appearing before its stated date.

Escher’s prints in Life’s rich patterns’,  Circle Limit IV and a fragment of Metamorphosis  shown p 28. No mention of Escher beyond the caption! A lifelike Tesselation tutorial in ‘Art by numbers’p.30. Nothing of any great significance here, pitched at a child level.


Anonymous. ‘Islamic Art and Geometric Design’. Metropolitan Museum of Art 2004. 1-46

Pseudo Cairo tiling from India, picture 14 (2010)


Anonymous. ‘The Yellow Book’. An Illustrated Quarterly. Vol. XI October, 1896. John Lane, The Bodley Head Ballantyne Press. London and New York. London & Edinburgh

From Andrew Crompton’s reference in his article Lifelike Tessellations; only a single page is of interest this is the only tessellation, and even this is not  clearly of the premise. The artist is apparently Nellie Syrett. Various authors, of a collection of writings. note that 


Anonymous. ‘The Geometer’s Sketchpad Workshop Guide’. 2002 Key Curriculum Press.


Anonymous. ‘Geometric Investigations on the VoyageTM with Cabri’. Teacher’s guide: Tessellations and Tile Patterns 25-31. 2003 Texas Instruments Incorporated (2010)

Brief discussion of the Cairo tiling, page 30


Anonymous. ‘Drawing Tessellating Guide in Illustrator: Pen Tool Basics: Tantalizing Tessellations (Drawing a Mosaic) Design and Print’, Illustrator Module 5 of 15 (2010)


Anon. Life 7 May 1951. ‘Speaking of Pictures’ 8-10 (28 March 2011)


Anon. Time. 25 October 1954. ‘The Gamesman’, p. 68 (28 March 2011)


Anon. Time. 2 April 1951. ‘Prying Dutchman’, p. 50 (28 March 2011)


Anonymous. H. S. M. Coxeter. Biography


————.  Scholastic Art. ‘Art Meets Math’. February 2010. 2-15 (16 October 2014)

A Canadian pedagogy magazine pitched at school age children. Varied content as regards Escher, such as analysis of how Escher created his tessellations and  a series of tutorials of how to create Escher-like tessellations, not always good advice. Not of any significance.


Anon. ‘Pavages du plan avec des polygones’. No bibliographic detail. Cairo tiling aspects (18 December 2012)


Anon. Ad by the AtlanticRichField Company, p. 97 (Sphere Spirals). In Prodhoretz. ‘The Literary Light as Eternal Flame’. The Saturday Review. August 24 1974  90-119 (29 July 2015)


Anon. Ad by the AtlanticRichField Company, p. 97 (Relativity). In Prodhoretz. ‘The Literary Light as Eternal Flame’. The Saturday Review. August 24 1974  90-119 (29 July 2015)


Appel, Kenneth and Wolfgang Haken. ‘Every Planar Map is Four Colorable’. Illinois Journal of Mathematics 21 429-567, 1977 (22 January 2015)

From a reference in Hinged Dissections: Swinging and Twisting. Academic throughout, albeit with numerous diagrams, but, of no practical use.


Appel, K. and W. Haken. ‘The Four Color Proof Suffices’. The Mathematical Intelligencer  Vol.

8, No. 1, 1986, 10-20 (9 December 2011)

Academic nature throughout


ApSimon H. G. ‘Almost Regular Polyhedra’. The Mathematical Gazette, Vol XL, No. 332 May 1956, p. 81-85 (7 March 2013)

Academic nature throughout


————. ‘Periodic forests whose largest clearings are of size 3’. Philosophical Transactions Royal Society, Series A, 266. 1970 113-121 (7 March 2013)

From a reference in Tilings and Patterns. Academic, of no practical use.


————. ‘Periodic forests whose largest clearings are of size n 4’. Proceedings of the Royal Society, London A 319. 1970 399-404 (7 March 2013)

Academic, of no practical use.


Aljamali, Ahmad, M. and Ebad Banissi. ‘Grid Method Classification of Islamic Geometric Patterns’. Publication is unclear, appears to be Proceedings of February 2003 (20 October 2010)

Of general interest.


Austin, David. ‘Penrose Tiles Talk Across Miles’. Feature column in AMS, web, unknown date

Largely popular account




Baboud, Roland. ‘Pavages De Pentagones’. Bulletin ATMEP No. 423 September to October 1999 (in French)

Cairo-like diagram, David Wells mentioned, Rice, James, Stein, new types of pentagon


Baeyer, Hans C. Von. ‘Impossible Crystals’. Discover, February 1990 69-78 (10 February 2012)

Article on Quasicrystals, Penrose tiles.


Badoureau, M. A. ‘Mèmoire sur les figures isoscèles’. Journal de l’ École Polytechique 30, 1881, 47-172 (2 January 2015)

From a reference in Tilings and Patterns. Of an academic nature throughout, numerous diagrams, but mostly of polyhedra; one page of plane tilings. Of no practical use.


Bagina, O. ‘Tiling the Plane with Congruent Equilateral Convex Pentagons’. Journal of Combinatorial Theory Series A., 221-232 (February 2012)

Academic. Something of a let down, in that the text is of a technical nature; I though this might have been illustrated with Cairo-like tiles, or at least of pentagons, but there’s not a single tiling diagram per se at all!


————. ‘Convex pentagons which tile the plane’. (4 June 2012)

Another largely theoretical paper, no diagrams!


————. ‘Tilings of the plane with convex pentagons’ (in Russian). Vestnik KemGU 4(48): 63-73, 2011.


Bain, George Grantham. ‘The Prince of Puzzle-Makers’. An Interview with Sam Loyd.

The Strand, 1907, 771-777. (18 May 2015)


Bakos, T. ‘2801 On Note 2530’ (Correspondence on C. Dudley Langford Cairo tile reference)’. The Mathematical Gazette, Vol 42, No. 342 December 1958, p. 294 (1 March 2013)

Of importance, due to Cairo tiling reference, referring to Rollett’s and Langford’s pieces in the Gazette (Note 2530 and correspondence). Gives an interesting discussion in terms of minimum values of hexagon and pentagon.


Bandt, C. and P. Gummelt. ‘Fractal Penrose tilings I. Construction and matching rules’. Aequationes Mathematicae 53 (1997) 295-307 (30 May 2012)



Boots, Barry and Narushige Shiode. ‘Recursive Voronoi Diagrams’. Environment and Planning B Planning and Design 2003, Vol. 30 pp. 113-124.

Academic. Found upon a search for tessellation in Environment and Planning archive. On voronoi diagrams, of no practical use.


Barcellos, Anthony. ‘A Conservation with Martin Gardner’. The Two-Year College Mathematics Journal. Volume 10 No. 4 Sep 1979 233-244.

Gardner interview, illuminating. I had no inkling of this journal to as late as February 2013!


Baracs, Janos, ‘Juxtapositions’. Structural Topology 1, 1979 59-71. (2016)

Has a Cairo tiling p. 65

On polyhedra, with Cairo-like simmarities


Baracs, Janos et al. ‘Habitat polyhedrique’. Structural Topology 2, 1979 7-38. (2016)



Barnette, David W. ‘The Graphs of Polytopes With Involutory Automorphisms’. Israel Journal of Mathematics. Vol. 9, 1971, 290-298 (25 October 2012)

From a reference in Tilings and Patterns. Of an academic nature throughout, no diagrams! Of no practical use.


Bar-On, Ehud. ‘A Programming Approach to Mathematics’. Comput. Educ. Vol 10, No.4, pp 393-410, 1986 (18 November 2011)

The subject per se is too obscure for me; the only aspect of interest is a minor reference to the Cairo tiling, page 339 that is barely worth mentioning.


Barron, Roderick. ‘Bringing the map to life: European satirical maps, 1845-1945’. 6th  international BIMCC conference 16 November 2007, 25-27 (2 September 2014)

Although not a mathematics reference per se, included as it is of Cluster puzzle-esque nature


Barnes, F. W. ‘Algebraic Theory of Brick Packing I’. Discrete Mathematics 42 1982 7-26 (23 September 2013)

From a reference in Tilings and Patterns. Of an academic nature throughout, no diagrams! Of no practical use.


————. ‘Algebraic Theory of Brick Packing II’. Discrete Mathematics 42 1982 129-144 (23 September 2013)

From a reference in Tilings and Patterns. Of an academic nature throughout, no diagrams! Of no practical use.


Basin, S. L. ‘The Fibonacci Sequence as it Appears in Nature’. The Fibonacci Quarterly, 1 1963 53-64 (9 September 2014)

Re golden section. From a reference in Livio


Baylis. John. 73.44 ‘Fault lines and the pigeon-hole principle’. The Mathematical Gazette 318-319 (11 April 2013)



Bays, Carter. Cellular Automata in the Triangular Tessellation’. Complex Systems Publications, Volume 8, Issue 2, 127-150, 1994 (14 December 2012)

Cairo aspect p. 148 … the Cairo tessellation (a tiling of identical equilateral pentagons)…

Cursory Cairo mention in passing


————. ‘Further Notes on the Game of Three Dimensional Life’. Complex Systems Publications, Volume 8, Issue *, 67-73, 1994 (14 December 2012)
No Cairo aspect..

————. ‘A Note on the Game of Life in Hexagonal and Pentagonal Tessellations’. Complex Systems Publications, Volume 15, Issue 3, 245-252, 2005. (28 January 2011)
Cairo aspect pp. 249-250. Also see two 1994 references by Bays.

Beard, R. S. ‘The Fibonacci Drawing Board Design of the Great Pyramid of Gizeh’

The Fibonacci Quarterly, 6 (1968) 85-87 (9 September 2014)

Re golden section. From a reference in Livio


Beard, Robert. S. (Colonel) Tessellated Polygons. Scripta Mathematicae Vol. XVII, Nos. 1-2 March-June 1951. (31 December 2008)

Article as reprinted in his 1973 book ‘Patterns in Space’


Beech, Martin. ‘Escher’s Stars’. The Journal of the Royal Astronomical Society of Canada. Vol. 86, No.4, 1992 (1 June 2011)

Discussion of polyhedra used in Escher’s prints


Beevers, Brian. ‘Filling the Gap’. Mathematics in School, March 1999. 40-41 (18 February 2013)

Forming tilings by taking polygons and in effect tiling these, leaving gaps. Begins at a popular level then becomes academic.


Beineke, Lowell and Robin Wilson. ‘The Early History of the Brick Factory Problem’. The Mathematical Intelligencer. Vol. 32, No. 2, 2010 41-48. ‘Years Ago’ (28 December 2011)

On Paul Thurán and Anthony Hill


Bellos, Alex. ‘Magic numbers: A meeting of mathemagical tricksters’. New Scientist 2010

Gathering for Gardner


————. ‘Gardner’s Question Time’. May 2010

Full transcript of Bellos’s interview with Martin Gardner in 2008


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

Of note is my contribution to this, of the back cover and two works inside.


Belov, N. V. ‘On One-dimensional Infinite Crystallographic Groups’. In ‘Coloured Symmetry’ by A. V. Shubnikov, N. V. Belov and others, 222-227 (13 October 2006)

Published in Kristallografiya 1, 474-476, 1956


Belov, N. V. ‘Three-dimensional Mosaics with Colored Symmetry’. In ‘Coloured Symmetry’ by A. V. Shubnikov, N. V. Belov and others 238-247 (13 October 2006)

published in Kristallografia 1, 621-625 1956


Belov, N. V., N. N. Neronova, T. S. Smirnova. ‘The 1651 Shubnikov Groups’ (Dichromatic Space Groups. Trudy kad. Nauk SSSR, Inst Kristall 11, 33-67 1955. In ‘Coloured Symmetry’ by A. V. Shubnikov, N. V. Belov and others 175-219 (13 October 2006)


Belov, N. V. and E. N Belova. ‘Mosaics for the Dichromatic Plane Groups’. In ‘Coloured Symmetry’ by A. V. Shubnikov, N. V. Belov and others 220-221 (13 October 2006)

A part of the paper Mosaics for 46 plane (Shubnikov) antisymmetry groups and for 15 (Fedorov) color groups published in Kristalloga 2, 21-22 1957, and reproduced from the translation in Sov. Phys Crystall 2, 16-18.


Belov, N. V., E. N. Belova and T. N. Tarkhova. Polychromatic Plane Groups. In ‘Coloured Symmetry’ by A. V. Shubnikov, N. V. Belov and others 220-221 (13 October 2006)

A part of the paper Groups of colored symmetry, published in Kristallographie, 1 10-13 (1956), with subsequent corrections and emendations in Kristallographie 1, 615, 619-621 (1956), 2 21-22 (1957), and 3. 618-620 1958


Bennett, Curtis D. ‘A Paradoxical View of Escher’s Angels and Devils’. The Mathematical Intelligencer  (24 November 2011)

The title indicates a likely popular article, but in actuality it’s of an advanced nature; studying the premise of hyperbolic geometry; decidedly obtuse, far too difficult for me.


Berend, Daniel and Charles Radin. ‘Are There Chaotic Tilings?’ Communications in Mathematical Physics 152, 215-219 (1993) (20 September 2012)

Academic throughout. All theory, with not a single diagram!


Bilney, Bruce. ‘Ozzie The Magic Kangaroo’. The Australian mathematics teacher vol. 52 no.4, 24-25 1996. Also see p. 29, of  review by Paul Scott of his ‘Ozzigami’ polyhedra models (2012)

Despite the specific title, a few musings on a variety of tessellation aspects: in effect the Droste effect, Escher, polyhedra, and only latterly is ‘Ozzie’ discussed.


Bookchin, Murray. ‘Toward an Ecological Solution’. Ramparts Magazine, 7-15 May 1970 (29 July 2015)

Use of Escher’s images ‘Metamorphosis’, ‘Liberation’, ‘Fish and Frogs’, ‘Three Worlds’ and ‘Verbum’. No references to Escher are mentioned in the article.


Bollabas, Bela. ‘Filling the Plane with Congruent Convex Hexagons Without Overlapping’. Tudomanyegyetem Annlaes Sectio Mathematica 1963, 117-123 (19 April 2015)

From a reference in Schattsneider’s pentagon ‘Tiling the Plane with Congruent Pentagons’ article, but pentagons are not mentioned by Bollabas! Interesting in its own right, albeit academic, with ‘understandable’ diagrams.


Bolster, L. Carey. ‘Activities: Tessellations’. Mathematics Teacher. April 1973 66 (April 1973 339-342 (20 February 2013)

Child mathematics


Bolster, L. Carey and Evan M. Maletsky. ‘Tangram Mathematics’. December 1975, 143-146

Child-inclined mathematics


Boreland, Gareth. ‘Tessellations, Polyhedra and Euler’s Theorem’. Mathematics in School 8-10, November 2007 (20 February 2013)

Very minor reference to tessellation, and surpassingly, given the journal, of a largely academic nature.


Bosch, Robert. ‘Simple-closed-curve sculptures of knots and links’.  (10 April 2013)


Boule, François. ‘Variations Autour D-’un Pavage Semi Regulier’ (in French). No bibliographic detail is given; could be taken from a book, as it begins at page 15. (18 December 2012)

Paper is titled as ‘François Boule, Dijon, 2001’

Interesting in many ways: 1. It uses the term ‘semi regular’ as used elsewhere. 2. It gives an interesting ‘Saigon paving’. 3. Some fearsome Cairo mathematics!


Bowers, Philip L. and Kenneth Stephenson. ‘A ‘regular’ pentagonal tiling of the Plane. Conformal Geometry and Mathematics’. Electronic Journal of the American Mathematical Society. Volume 1, 58–86 (14 November 1997) (2010)

Somewhat advanced, although there is the occasional diagram of interest


Boyd, David. W. ‘The disk-packing constant’. Aequationes Mathematicae 7 1971, 182-193 (31 December 2014)

From a reference in Tilings and Patterns. Academic throughout, no diagrams. Of no practical use.


————.‘The osculatory packing of a three dimensional sphere’. Canadian Journal of Mathematics 25, No. 2 (1973), 303-322 (1 November 2013)

From a reference in Tilings and Patterns. Academic throughout, no diagrams. Of no practical use.


Brade, Sam. ‘Create interlocking motifs’. Computer Arts May 2010, 66-69 (19 June 2014)

Tutorial on drawing tessellations of his cluster puzzle.


Bradley, H.C. and E. B Escott. Problem 2799.  American Mathematical Monthly 28 186-187 1921


————. Problem 2933 1921, 467 Dudeney’s problem, 1902 Solution 147-148

(Not in Frederickson)


————. Problem 3048. American Mathematical Monthly 37 158-159, 1930 (4 March 2013)

(Reference in Frederickson)


Bravais, A. ‘Mémoire sur les systèmes formés par des points distributés regilièrement sur un plan ou dans l’espace’. Journal de École Polytechique 19 1850, 1-128. (2 January 2015)

From a reference in Tilings and Patterns. Academic, of no practical use. Not a single diagram!


Breen, Marilyn Some tilings of the plane whose singular point form a perfect set. Proceedings of the American Math Society 89, Number 3, 1983, 477-479 (2 January 2015)

From a reference in Tilings and Patterns. Of an academic nature throughout, no diagrams!


————.‘Tilings Whose Members Have Finitely Many Neighbors’. Israel Journal of Mathematics, Vol. 52, Nos. 1-2, 1985, 140-146.

Academic throughout, of no practical purpose. Minimal diagrams. From bibliography in Tilings and Patterns.


Brecque, Mort La. ‘Quasicrystals. Opening the Door to Forbidden Symmetries’. Mosaic Volume 18 Number 4 Winter 1987/8. 3. (24 January 2014)

Heavily slanted to the quasicrystal element (but mostly of a popular account), Penrose ‘section’ pp 14-16


Bricard, R. ‘Sur une question de géométrie relative aux polyhèdres’. Nouvelles Annales de Mathématique 15, 331-334 1896, Series 3 (13 January 2015)

From a reference in Dissections: Plane & Fancy. All text, of no practical use


Brisse, François. ‘La Symétrie Bidimensionelle et le Canada’. The Canadian Mineralogist. Volume 19, May 1981, Part 2, 217-224 (in French Canadian) (18 November 2013)

As such, somewhat disappointing in that tessellation is secondary to symmetry.  Has one life-like tessellation, of a polar bear, semi-respectable, page 222. Of no consequence. I can’t recall the source of this article.


Britton, Jill. ‘Escher in the Classroom’. Mathematics Teaching in the Middle School. 480. (23 February 2013)

Simple ideas for use in the classroom


————. ‘Escher in the Classroom’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration (31 August 2005) 305-317


Broos, C. H. A. ‘Escher: Science and Fiction’. In The World of M. C. Escher. Abradale Press 1988 30-38 (9 April 1993)


Broos, C. ‘M. C. Escher’. In Holland-a century of form and colour. Netherlands National Tourist Office. No date, but I have a 1963 reference for this (17 January 2015)

A minor ‘article’ (apparently not previously referenced) on Escher in the context of Dutch painters of the 20th century in a booklet in association with the Netherlands National Tourist Office. This uses (‘pp.22-23’) three of Escher’s prints: Day and Night, Other World, Waterfall, with a general commentary on Escher, albeit not on the works above, and what text there is most brief and of no new insight.

Presumably ‘C. Broos’ is he same person as C. H. A. Broos as the author in an article in  The World of M. C. Escher, immediately above. The booklet, of 32 pages, but without an introduction, contents or pagination, does give brief details as to Broos’s background not available above; he  was curator of modern art, Municipal Museum, The Hague.


Broug, Eric. ‘Escher and Islamic Design’. 20-27 (19 October 2015 )

From catalogue of 2012 exhibit.


Browne, Cameron. ‘Duotone Truchet-like tilings’. Journal of Mathematics and the Arts. Vol. 2, No. 4, December 2008, 189-196 (24 April 2013)

Of general interest. Can be described as Truchet tiles brought up to date.


Bruckman, P. S. ‘Constantly Mean’. The Fibonacci Quarterly, 15, 1977 236 (9 September 2014)

Re golden section. From a reference in Livio


Bruijn, N. G. de. ‘Updown generation of Penrose patterns’. Indagationes Mathematicae, 1 (2), 201 -220, June 18 1990 (27 September 2013)

Largely academic, of no use.


————. ‘Penrose patterns are almost entirely determined by two points’. Discrete Mathematics 106/107 1992, 97-104. (27 September 2013)

Largely academic, of no use.


Buchman, E. ‘The impossibility of tiling a convex region with unequal equilateral triangles’.  American Mathematical Monthly, 88 1981, 748-753

From a reference in Tilings and Patterns. Academic, of no practical use.


Burgiel, Heidi. ‘Logarithmic Spirals and Projective Geometry in M.C. Escher’s Path of Life III’

Journal of Humanistic Mathematics. Vol. 2 No.1, January 2012


————. ‘How to lose at Tetris’. The Mathematical Gazette. Vol. 81, 491 July 1997, 194-200 (12 April 2013)

Of general interest, of both popular and an academic nature.


Burn, Bob. Note 74.45. ‘The Orton-Flower tessellation’. The Mathematical Gazette 372-373 (23 February 2013)

Burn brief follow-up comments on Orton and Flower’s article, itself in the Gazette


Burt, M. ‘The wandering vertex method’. Structural Topology 6 1982, 5-12.

One of four references in Structural Topology as mentioned in the bibliography of Tilings and Patterns; the others being Danzer, Gilbert and Lalvani, although there are indeed other tiling references in the journal.


Buseck, Peter R. ‘From 2D to 3D: I Escher Drawings Crystallography, Crystal Chemistry, and Crystal ‘Defects’’ (9 December 2014)

Use of Escher’s plane tilings, namely 43 (shells and starfish), 55 (fish), 70 (butterfly), 78 (unicorn), and Birds in Space, as crystallographic principles. Such crystallographic aspects lie outside my main interest, and so the ‘article’ has little of direct interest. I say ‘article’ in quotation marks, as I am not certain if this is indeed so. No bibliographic detail is give, although the indication that this is indeed so, as the pagination begins at p. 213.




Callingham, Rosemary. ‘Primary Students Understanding of Tessellation: An Initial Exploration. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education’, 2004. Vol 2 183-190 (10 February 2012)

Serendipitously, this contains a Cairo tiling, and even more serendipitously, the Cairo tiling and what I believe to be the Rice derivation is side by side, but without realisation!


Carnow, Bertram W. ‘Pollution Invites Disease’. The Saturday Review. August 24, 1974, 38-40 (29 July 2015)

Use of Escher's Rind, p. 38. No other references are mentioned in the article.


Casselman, Bill. ‘On the Dissecting Table’. +Plus Magazine Issue 16. 1 December 2000? (26 January 2012)

Is this a strictly on-line journal? I’m uncertain. On Henry Perigal’s Pythagoras dissections.


Chavey, D. ‘Tilings by Regular Polygons’ – II. A Catalog of Tilings. Comp. & Maths. With Appls. Vol. 17, No. 1-3. 147-165. 1989 (9 September 2010).


Chen, Elizabeth R. ‘A Dense Packing of Regular Tetrahedra’. Discrete & Computational Geometry (2008) 407: 214-240. (5 November 2012)

Of an academic nature throughout; the tone of the paper is way beyond me. Of no practical use.


Chernikov, A. A, R.Z Sagdeev, D. A. Usikov, G. M. Zaslavsky. ‘Symmetry and Chaos’. Computers and Mathematics with Applications. Vol. 17, No. 1-3, 17-32, 1989 (27 September 2013)



Chmelnizkij, Sergei. ‘Methods of Constructing Geometric Ornamental Systems in the Cupola of the Alhambra’.


Chorbachi, W. K. ‘The Tower of Babel: Beyond Symmetry In Islamic Design’. Computers and Mathematics with Applications. Vol. 17, No. 4-6, pp 751-789, 1989 (reprinted in I. Hargittai, ed Symmetry 2: Unifying Human Understanding, Pergamon, New York, 1989 (6 April 2011).

Has interesting Cairo tiling references, pp. 783-784, and quotes James Dunn’s 1971 article


Chow, W. ‘Automatic Generation of Interlocking Shapes’. Computer Graphics and Image Processing 9 (1979), 333-353 (10 April 2014)

Somewhat obscure, and dated, due to computer program used of the day. Leans heavily on Heesch’s work for the generation of tiles. Recreates Escher’s Pegasus tiling. However, the text is heavy going, and is of little to no practical use.


Christensen, A. H. J. ‘Recursive Patterns or the Garden of Forking Paths’. Leonardo 15 1982 177-182 (7 March 2013)

From a reference in Tilings and Patterns. Academic, of no practical use.


Chu, I-Ping. ‘Tiling Deficient Boards with Trominoes’. Mathematics Magazine 34-* Vol. 59, No. 1, February 1986 (11 April 2013)

Begins simply, becomes academic.


Chung, Ping Ngai, et al. ‘Isoperimetric Pentagonal Tilings’. 2011. (9 December 2011)

Contains a very interesting off shoot of the Cairo tiling, with tilings in combination with the ‘prismatic’ tile


Cibulis, Andris, and Andy Liu. ‘Packing Rectangles with the L and P Pentominoes’. Math Horizons, November 2001, 30-31 (11 April 2013)

Recreational polyominoes


Cipra, Barry A. ‘Packing Pyramids: Is the Space Race Over?’. Siam News, Volume 43, Number 4 May 2010. (13 December 2010)

On packing tetrahedral, largely a popular account


Clason, Robert G. ‘A Family of Golden Triangle Tile Patterns’. The Mathematical Gazette 130- (March 2013)

Many interesting ‘simple’ diagrams


Clauss, Judith Enz. ‘Pentagonal Tessellations’. The Arithmetic Teacher (NCTM), 38(5): 52-56, January 1991. (3 May 2012)

Nothing really of any originality here, it merely goes over old ground on the 14 types of convex pentagon


Clemens, Stanley R. ‘Tessellations of Pentagons’. Mathematics Teaching, No. 67 (June), 18-19, 1974 (3 May 2012)

Cairo diagram p. 18, and much interesting discussion arising from this


Coffin, Stewart. ‘Polyomino Problems to Confuse Computers’. The College Mathematics Journal Vol. 40, No.3, May 2009 169-173 (9 April 2013)

Packing rectangular trays, non orthogonally.


Coleman, A. J. ‘The Greatest Mathematical Paper of All Time’. The Mathematical Intelligencer (Vol. 11, No. 3, 1989, 29-38 (13 December 2011)

General interest, academic


Conlan, James P. ‘Derived Tilings’. Journal of Combinatorial Theory, Series A 20, 34-40, 1976 (24 September 2013)

From a reference in Tilings and Patterns. Of an academic nature throughout! Loosely described as of tiling polyhedra, despite the tiling title. Of no practical use.


Conway, J. H. Solution to Problem 5328. D. C. Kay. 1966 903-904 (7 March 2013)


Costello, John. ‘Dissection strategies’. Mathematics Teaching 112. September 1985. 28-29.

Not mentioned in any of Frederickson’s three books


Coxeter H. S. M. ‘The four-color map problem, 1840-1890’. The Mathematics Teacher April 1959. 283-289. (28 February 2013)

General historic interest


————. ‘The Problem of Apollonius’. The American Mathematical Monthly, Vol. 75, No. 1 Jan, 1968, 5-15 (30 January 2012)

Academic. Of no real interest; far too advanced for me. The opportunity of obtaining this arose as a result of Coxeter searching; I thought I may as well have it, if only for having ‘seen and noted it’.


————. ‘Frieze Patterns’. Acta Arithmetica XVIII 297-310. 1971

(30 January 2012)

‘Typical Coxeter’, too advanced.


————. ‘The Trigonometry of Escher’s Woodcut Circle Limit III’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration First edition 2003, and second 2005. 297-304 (31 August 2005)
‘Typical Coxeter’, too advanced. Essentially a republication of an article in Hyperspace (1997) and Mathematical Intelligencer (1996)


————. ‘The Mathematical Implications of Escher’s Prints’. The World of M. C. Escher. 51-54. Abradale Press 1988 (9 April 1993)

A brief, popular account (albeit with brief digression to typical advanced Coxeter talk), discussing, some of just a single line, of Escher’s more obvious mathematical prints: Moebius Strip I, Tetrahedral Planetoid, Flatworms, Stars, Cube with Magic Ribbons, Cubic Space Division, Order and Chaos, Gravity, Smaller and Smaller I, Whirlpools, Circle Limit I, III, IV, Belvedere, Ascending and Descending, Waterfall


————. ‘The Non-Euclidean Symmetry of Escher’s Picture ‘Circle Limit III’’. Leonardo, Vol. 12.No. 1 (Winter 1979) 1979. pp.19-25 (9 September 2010)


————. ‘Virus Macromolecules and Geodesic Domes’. In A Spectrum of Mathematics (19 June 19 2011)


————. ‘Regular and Semi-Regular Polytopes II’. Mathematische Zeitschrift 188, 559-591 (1985) (24 October 2012)

Of an academic nature throughout, minimal diagram! Of no practical use. From a reference in Tilings and Patterns.


————. ‘The Seventeen Black and White Frieze Types’. Comptes Rendus Mathematiques De L’Academie des Sciences. Vol. VII, No. 5, October 1985 327- (January 2016)

‘Typical Coxeter’, too advanced.


————. ‘Escher’s Lizards’. In: Structural Topology No.15 1988. 23-30 (Escher special edition). (28 March 2011)

An analysis of two of Escher's lizard tessellation, academic from start to finish.


————. ‘Cyclomic integers, nondiscrete tessellations, and quaiscrystals’. Indagationes Mathematicae, N. S. 4 (1), 27 -38, March 1993 (27 September 2013)

Academic, of no use.


————. ‘Escher’s Fondness for Animals’. M.C. Escher’s Legacy. A Centennial Celebration. Springer. First edition 2003, and second 2005. 297-304 (31 August 2005)
Essentially a reprint of the catalogue of the 1985 congress, with George Escher giving additional comments in an addendum


Coxeter, H. S. M; M. S. Longuet-Higgins and J. C. P. Miller. ‘Uniform Polyhedra’. Ph 246, 1953/54, 401-450

From a reference in Tilings and Patterns. Academic, of no practical use.


Crompton, Andrew. ‘Lifelike Tessellations’. In ‘Manchester Architectural Papers’ 2000, edited by Geoff McKennan, pp. 17-24

Pleasing, although brief. Begins with a history, his own thoughts with illustrations of three of his works, of badgers and birds, of isohedral and anisohedral patterns.  A chart of 49 ways of drawing lifelike tessellations is given, of which with correspondence with various parties (including Crompton), is shown to be wrong; it should be 47,


Cromwell, Peter R. ‘The Search for Quasi-Periodicity in Islamic 5–fold Ornament’. The Mathematical Intelligencer. Vol. 31, No. 1, 36-56, 2009 (25 November 2011)


————. ‘Celtic Knotwork: Mathematical Art’. The Mathematical Intelligencer. Vol. 15, No. 1, 1993 36-47 (15 December 2011)


————. ‘Islamic Geometric Designs from the Topkapi Scroll I: unusual arrangements of stars’. Journal of Mathematics and the Arts. Vol. 4, No. 2, June 2010, 73-85 (10 April 2013)

General interest


————. ‘Islamic Geometric Designs from the Topkapi Scroll II: a modular design system’. Journal of Mathematics and the Arts. Vol. 4, No. 2, June 2010, 73-85 (10 April 2013)

General interest


Cromwell, Peter and Elisabetta Beltrami ‘The Whirling Kites of Isfahan: Geometric Variations on a Theme’. The Mathematical Intelligencer. Volume 33, No. 3, 84-93, 2011 (29 December 2011)


Crowe, D. W. ‘The Geometry of African Art II. A Catalog of Benin Patterns’. Historia Mathematica 2 1975, 253-271 (1 October 2013)

From a reference in Tilings and Patterns. Of little interest!


————. ‘The Mosaic Patterns of H. J. Woods’. Comp. & Maths. With Appls. Vol. 12B, Nos. 1/2. 407-411, 1986 (9 September 2010)


Cruikshank, Garry. ‘The Bizarre History of Tessellated Tiles’. Ceramic Tiles Today, Autumn 1994, 18-19 (10 February 2012)

Potted account of tile history.


Cundy, H. Martyn. ‘A Souvenir from Paris’. The Mathematical Gazette. Vol. 55 No. 393, June 1971. 310-312. (2010)

Polyhedral lampshade


————. ‘Unitary Construction of Certain Polyhedra’. The Mathematical Gazette, Vol. 40, No. 334. (Dec., 1956), 280-282. (2010)


————. ‘Deltahedra’. The Mathematical Gazette Vol. 36,  No. 318. 263-266 December 1952 (28 February 2013)

Largely popular account


————. Notes 63.20 p3ml or p31m? The Mathematical Gazette 192-193 (28 February 2013)


Also see letters and reviews


Curl, Robert F and Richard E Smalley. Fullerenenes. Scientific American October 1991. 32-41.




Daems, Jeanine. ‘Escher for the mathematician’ (as in original). NAW 5/9 nr.2 June 2008. An interview with N.G. de Bruijn and Hendrik Lenstra. (15 December 2009)

Primarily concerning aspects of Escher’s print ‘Print Gallery’.



Danzer, Ludwig, Branko Grünbaum, G. C. Shephard. ‘Can All Tiles of a Tiling Have Five-Fold Symmetry?’ American Mathematical Monthly 89 (1982) 568-570 and 583-585. (17 February 2013)

Leans towards the academic, but some aspects are understandable. Kepler’s diagrams are used. Interesting pentagon tiling, Cairo-like in that it is from a subdivided par hexagon p. 570. tThe first of three articles of Danzer, Grünbaum and Shephard collaboration


————. ‘Does Every Type of Polyhedron Tile Three-Space?’ Structural Topology 8, 3-11, 1983 (c. 2008)

Of no real interest. One of four references in Structural Topology as mentioned in the bibliography of Tilings and Patterns; the others being Burt, Gilbert and Lalvani, although there are indeed other tiling references in the journal.


————. ‘Equitransitive Tilings, or How to Discover New Mathematics’. Mathematics Magazine, Vol. 60, No. 2, April 1987. 67-89 (17 February 2013)

Largely academic, with occasional diagrams at a popular level


Danzer, Ludwig, Grattan Murphy and John Reay. ‘Translational Prototiles on a Lattice’. Mathematics Magazine Vol. 64, No. 1, February 1991. 3-12 (18 February 2013)

Largely academic, of no practical use.


Dauben, Joseph W. ‘Personal Reflections of Dirk Jan Struik’. The Mathematical Intelligencer. (Years Ago column) Volume 33, No. 2, 23-33, 2011, 36-43 (29 December 2011)


David, Guy, and Carlos Tomei. ‘The Problem of the Calissons’. American Mathematical Monthly Vol. 96 no. 5 (May 1989) pp 429-431(22 November 2010)

Packing problem caused by a French sweet of the name (calissons)


Davies, Roy. O. ‘Replicating Boots’. The Mathematical Gazette, 50, 1966, 175 (7 March 2013)

From a reference in Tilings and Patterns. Academic, of no practical use.


Dawson, T. R. ‘Ornamental Squares and  Triangles’. The Mathematical Gazette, 19-20? (25 March 2013)

Of number theory rather than of tiles!


Dehn, M. ‘Ueber den Rauminhalt’. Mathematische Annalen 55 1902 465-478 (14 January 2015)

From a reference in Dissections: Plane & Fancy. Academic, of no practical use


Dejean, Françoise. ‘Sur un Théorème de Thue’. Journal of Combinatorial Theory A, 13, 90-99, 1972 (in French) (25 September 2013)

From a reference in Tilings and Patterns. Of an academic nature throughout!. Of no practical use.


Dekking, F. M. ‘Replicating Superfigures and Endomorphisms of Free Groups’. Journal of Combinatorial Theory, Series A 32, 315-320, 1982 (24 September 2013)

From a reference in Tilings and Patterns. Of an academic nature throughout! Too advanced for me. Of no practical use.


Delgado, Olaf, Daniel Huson and Elizaveta Zamorzaeva.‘The Classification of 2-Isohedral Tilings of The Plane’. Geometriae Dedicata 42: 43-117, 1992 (1 December 2014)

Academic in tenure, but replete with figures of interest pp.53-116, albeit to what all this means and indeed to what purpose I can use remains to be seen. P. 102 has a Cairo-like pentagon overlaid with squares, similar to Adrian Fisher’s patent


Delone, B. N. (also known as Delaunay). ‘The Theory of Planigons’ (in Russian)  Izv. Akad. Nauk SSSR  Ser  Mat 23 1959, 365-386. (23 June 2011)

Liberally illustrated with numerous tiles and tilings, of which prominent is a Cairo type tiling, pp. 371, 374, P5B1, and  a Kreugel pentagon Diagram 5a, 4. Full of interest that requires studying. Also note a much later 1978 paper by Delone, Dolbilin, and Shtogrin, which uses most of the diagrams here


Delone, B. N., N. P. Dolbilin, M. I Shtogrin. ‘Combinatorial and metric theory of planigons’ (in Russian) Trudy Mat Inst. Steklov., 1978 Volume 148, 109-140 (30 April 2015)

Many instances of Cairo tiling, pp. 115-116, 120, 127, 129, 134, 136 (skew), 137. Also has many interesting diagrams. Being in Russian, one cannot even guess as to the text, and so I only printed out the diagrams, save for the first page. also see earlier 1959 article by Delone by himself, with this apparently based on that.


Demaine, Erik D. et al. ‘Hinged dissection of polyominoes and polyforms’. Computational Geometry 31 (3) 2005 237-262 (27 January 2015)

From a reference in Piano-hinged Dissections. Academic  in tenure, occasional diagram that is understandable, but the article is of no practical use.


Dendra, Daniel. ‘Augmented Culture’. Oz. Volume 33 Article 7 1 January 2011 (28 May 2015)

Dendra’s use of the Cairo tiling in an architectural context, with his table design.


Dewdney, A. K. Computer Recreations. ‘Imagination meets geometry in the crystalline realm of latticeworks’. Scientific American, June 1988 100-103. Hard copy only

Composing Islamic patterns by means of lattices.


Deza, M. et al. ‘Fullerenes as Tilings of Surfaces’. Journal of Chemical Inf. Computer Science 2000, 40, 550-558 (4 July 2011)

The subject is too obscure for me; the only aspect of interest is a minor reference to the Cairo tiling, given as the dual, illustrated with the ‘basket weave minimum’, p. 554.


Ding, Ren; Schattschneider, Doris; Zamfirescu, Tudor. ‘Tiling the Pentagon’. Discrete Mathematics. 221 (2000)113-124. (8 September 2010)

Academic. Dissections (subdivisions) of the pentagon by pentagons. Highly technical. Of limited interest


Dixon, Robert. ‘Pentasnow’. Mathematics Teaching. 110. March 1985. 17-19?

Fractal-type patterns as formed by successive divisions of a regular pentagon, giving rise to snowflake appearance, hence the title.


Doczi, György. Seen and Unseen Symmetries: A Picture Essay. Computers and Mathematics with Applications. Vol. 12B, Nos 1/2, pp39-62, 1986 (2 October 2013)

From the symmetry ‘special edition’ of the journal. Obscure. Overlaying grids on to pictures.


Dodgson, N. A. ‘Mathematical characterization of Bridget Riley’s stripe paintings’. Journal of the Mathematics and the Arts. Vol. 6, Nos. 203, June-September 2012, 89-106 (10 April 2013)


Dolbilin, N. ‘The Countability of a Tiling Family and The Periodicity of a Tiling’. Discrete & Computational. Geometry. 13:405-414 (1995) (30 May 2012)

Academic; not a single diagram. Of no practical use

Donnay, Victor J. ‘Chaotic Geodesic Motion: An Extension of M.C. Escher’s Circle Limit Designs’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration (31 August 2005) 318-333

Dorwart, Harold L. Configurations: ‘A Case Study in Mathematical Beauty’. The Mathematical Intelligencer. Volume 7, No. 3, 39-48, 1985 (9 December 2011)



Dostal, Milos and Ralph Tindell ‘The Jordan curve theorem revisited’. Jahresbericht der Deutschen Mathematiker-Vereinigung 80, 111-128, 1978 (2 January 2015)

From a reference in Tilings and Patterns. Academic, of no practical use. Not a single diagram!


Dress, Andreas W. M. and Daniel H. Huson. ‘Heaven and Hell Tilings’. In Structural Topology, 17, 1991, 25-42 (26 August 2008 and 26 March 2011?)

Academic, with occasional simple tiling diagrams. Escher Heaven and Hell p. 26.


Driver, Denis. ‘Edging Towards Escher’. Mathematics in School, Vol. 22, No. 1, January, 1993

11-15? (17 February 2013)

A little obscure at times


Drost, John P. ‘The Vortex Tessellation’. The Mathematics Teacher, Vol. 92, No. 4, April 1994 286-289 (25 February 2013)

Tessellations in a ‘vortex’ configuration, similar to Escher’s *.

Various birds designed by Drost are shown, the premise of which I am uncertain. Whatever, of little direct interest.


Dudeney, Henry. E. All columns from Perplexity. In  The Strand Magazine


Duijvestijn, A. J. W., P.J. Federicko and P. Leeuw. Compound Perfect Squares’. American Mathematical Monthly. 89 1982 15-32

From a reference in Tilings and Patterns. Academic, of no practical use.


Dunham, Douglas. ‘Hyperbolic Symmetry’. Computers and Mathematics with Applications. Vol. 12B, Nos 1/2, pp.139-153, 1986 (1 October 2013)

From the symmetry ‘special edition’ of the journal. Academic.


————. ‘A Tale Both Shocking and Hyperbolic’. Math Horizons April 2003, 22-26


————. ‘A Family of Circle Limit III Escher Patterns’


————. ‘Artistic Patterns in Hyperbolic Geometry’. In Bridges 1999. 239-250

————. ‘Families of Escher Patterns’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration (31 August 2005) 286-296

Dunn, J. A. ‘Tessellations with Pentagons’. The Mathematical Gazette, Vol. 55, No. 394 (Dec. 1971) pp. 366-369 (17 August 2010)

Of the utmost significance in regards to the Cairo tiling; the first reference to the pentagon tiling being associated with Cairo, but with an illustration, not a photograph. Some additional correspondence generated by the above article in The Mathematical Gazette; Letters by P. Nsanda Eba Vol. 56, No. 398 (December 1972), 332-335 and M. M. Risueno Vo. 56, No. 398 (December 1972) 332




Eba, P. N. ‘Space-Filling with Solid Polyominoes’. Mathematics in School, p. 2-5 (9 April 2013)


Edmonds, Allan L, John H. Ewing, and Ravi S. Kulkarni. ‘Torsion Free Subgroups of Fuchsian Groups and Tessellations of Surfaces’. Inventiones Mathematicae. 69, 331-346 (1982). (24 October 2012)

From a reference in Tilings and Patterns. Of an academic nature throughout, with not a single diagram! Of no practical use.


Eggleton, R. B. ‘Tiling the Plane with Triangles’. Discrete Mathematics 7 1974 53-56

Academic, of no use (27 September 2013)

Einsenstein, Jane and Arthur L. Loeb. ‘Rotations and Notations’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration (31 August 2005) 334-342

Ellard, David. ‘Poly-iamond enumeration’. Mathematical Gazette, 66 1982, 310-314 (7 March 2013)

From a reference in Tilings and Patterns. Popular/academic, but of no practical use.


Ellers, Erich W. et al. ‘H. S. M. Coxeter (1907-2003)’. Notices of the AMS. Volume 50, Number 10 1234-1240. (30 January 2012)

Tributes to Coxeter by Ellers, Grünbaum, McMullen and Weiss on his death. One small paragraph on Escher, of no consequence.


Emmer, M. ‘Visual Art and Mathematics: The Moebius Band’. Leonardo. Vol. 13, pp. 108-111, 1980 (15 April 2013)

Largely a historical account


————. ‘Comments on the Note by Jean C. Rush on the Appeal of M. C. Escher’s Pictures’.

Leonardo, Vol. 13, pp 209-210, 1980 (17 February 2013)


————. ‘Art and Mathematics: The Platonic Solids’. Leonardo. Vol. 15, No. 4 pp 277-282, 1982 (15 April 2013)

The title is a little imprecise; this is a historical account, rather than a general discussion, from Uccello (1397-1475) onwards


————. ‘Comments on A. L. Loeb’s Correspondence with the Graphic Artist M. C. Escher’. Leonardo Vol. 17, No. 3, pp. 200-201, 1984 (17 February 2013)

Largely concerns impossible objects, rather than tessellations. Shows Necker’s original ‘cube’ (a parallelepiped)

————. ‘M.C. Escher: Art, Math, and Cinema’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration (31 August 2005)

————. ‘Soap Bubbles in Art and Science: From the Past to the Future of Math Art’. Leonardo. Vol. 20, No. 4 pp. 327-334, 1989 (15 April 2013)

Largely a historical account


————. ‘The ‘Belvedere’ by Escher: A Modest Hypothesis’. In Structural Topology No.17 5-10 1991. (an ‘overflow’ of the Escher Special edition of 1988) (26 August 2008 and 28 March 2011)

Speculations as to the source of Belvedere inspiration; Emmer conjectures this was as a result of Escher’s stay in Rome, with the architecture providing the source.


————. ‘Mathematics and Art: Bill and Escher’, Bridges 2000, 353-362


————. ‘Homage to Escher’. Leonardo, Vol. 33 No. 1 pp. 3-16, 2000 (17 February 2013)

Escher-inspired works from artists at the 1998 Escher conference: Victor Avecedo, Jos De Mey, Sandro Del-Prete, Valentina Barucci, Robert Fathauer, Helaman Ferguson, Kelly M. Houle, Matuska Teja Krasek, Makoto Nakamura, István Orosz, Peter Raedschelders, Dick Termes


————. ‘Ravello: An Escherian Place’. In Coxeter et al, Eds. M.C. Escher: Art and Science. Amsterdam: North-Holland 1986. (30 April 1994)


Note that I also have other various papers by Emmer, but these are largely of an inconsequential nature, such as announcements, and that for reasons of conciseness I have decided not to list here.


Emmerich, D. G. Polyèdres composites. Structural Topology  5-32 13, 1986 (c. 2008?)

Of little direct interest.


Erickson, Ralph O. ‘Tubular Packing of Spheres in Biological Fine Structure’. Science 24 August 1973, Volume 181, Number 4101.

From a reference in Tilings and Patterns. Academic, of no practical use.


Ehrlich, Paul. ‘Eco-Catastrophe!’ Ramparts Magazine. 24-28 September 1969 (29 July 2015)

Use of Escher’s Sky and Water I. No other references in the articles.


Ernst, Bruno. ‘Selection is Distortion’. In Coxeter et al, Eds. M.C. Escher: Art and Science. Amsterdam: North-Holland 1986. (30 April 1994)


Eperson, Canon D. B. ‘Lewis Carroll – mathematician and teacher of children’. The Mathematical Gazette 9-13. (19 March 2013)

General interest


————. ‘Educating a Mathematical Genius: Alan Turing at Sherborne School’. Mathematics in School, May 1994, 44-45. (19 March 2013)

General interest


Escher, George A. 'Letter to the Editor.' Scientific American 232 no. 1 (January 1975): 8-9. (27 April 1993)

G. A. Escher rebuts Teuber’s article, and Teuber’s response to that reply

Escher, George. ‘Folding Rings of Eight Cubes’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration (31 August 2005) 343-352

Escher, M. C. ‘Approaches to Infinity’. In The World of M. C. Escher. Abradale Press 1988, 39-42 (9 April 1993)




Faddeev,  D. K.  Boris Nikolaevič Delone (on the occasion of his 70th birthday [In Russian] (7 May 2015)

From a reference in Tilings and Patterns. Some doubts as to the reference here. The paper I have states 60th birthday and author, and. Grünbaum quotes  anonymous and  70th birthday, so…

Whatever, the version I  have is of no consequence; all in Russian text


Falbo, Clement. ‘The Golden Ratio - A Contrary Viewpoint’. The College Mathematics Journal Vol. 36, No. 2 (March 2005), 123-134 (2 September 2014)

A classic paper, with Falbo debunking the numerous spurious golden ratio claims made

Farkás, Tamas F. ‘Organic Structures Related to M.C. Escher’s Work’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration  150-153 (31 August 2005)

Farrell, Margaret A and Ernest R. Ranucci. ‘On the Occasional Incompatibility of Algebra and Geometry’. Mathematics Teacher 1974 491-497.

Mooting semi-regular tessellations; somewhat advanced, of little direct interest.


Fathauer, Robert W. ‘Recognizable Motif tilings Based on Post-Escher Mathematics’. In Bridges 1999 291-292.

Fractal tilings with Escher-like motifs


————. ‘Self similar tilings based on Prototiles Constructed from Segments of Regular Polygons’. In Bridges 2000, 285-292

————. ‘Extending Escher’s Recognizable-Motif Tilings to Multiple-Solution Tilings and Fractal Tilings’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration 154-165 (31 August 2005)

Fedorov, E. S. ‘Systèmes des planygones. Bulletin de l’Académe Imperiale des Scences’ de St.-Petersbourg VI serie’, volume 10 issue 16, 1523-1534 1916 (30 April 2015)
Of limited interest, the premise appears to be finding mid-points of tilings and joining to form new tilings.

Feijs, Loe M.G. ‘Geometry and Computation of  Houndstooth (Pied-de-Poule)’. In Proceedings of Bridges 2012, 299-306
A most pleasing discussion, although technical in places, with some pleasing innovations, such as the houndstooth appearing in different orientations.

Feijs, Loe M.G. and Marina Toeters. ‘Constructing and Applying the Fractal Pied de Poule (Houndstooth)’. In Proceedings of Bridges 2013, 429-432
A most pleasing discussion, although technical in places

Ferguson, Helaman. ‘A Circle Limit in Stone’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration (31 August 2005)

Field, J. V. ‘Kepler’s Star Polyhedra’. Vistas in Astronomy, Vol. 23, 1979. 109-141. (c. 10 March 1993)

Contains Kepler’s tilings, pp. 126-128


————. ‘Rediscovering the Archimedean Polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler’. 241- ? (5 April 2013)

At a broadly popular level.


Findeli, Alain. ‘Rhythm, Symmetry and Ornament’. Structural Topology 12, 37-56, 1986 (c. 2008?)

Symmetry groups illustrated with real like objects. Of very little use. Only obtained because I could… Has a sort of loose cluster puzzle photo, p. 52


Fishler, R. ‘How to Find the Golden Number Without Really Trying’ 19, 1981 406-410The Fibonacci Quarterly, (9 September 2014)

Re golden section. From a reference in Livio


Fontaine, A. and G. E. Martin. 'Tetramorphic and Pentamorphic Prototiles.' Journal of Combinatorial Theory, Series A 34, 115-118 1983 (13 January 1996)


————. ‘An Enneamorphic Prototile’. ('A note') Journal of Combinatorial Theory, vol. 37 No. 2 September 1984. 195-196. (2 February 1998)


————. 'Polymorphic Polyominoes’. Mathematics Magazine, Vol. 57, No. 5, November 1984. 119-121 (hardcopy 13 January 1996 and 2 February 1998 and 9 April 2013 )


————. 'Polymorphic Prototiles’. Journal of Combinatorial Theory, Series A 34 (1983) 119-121


Forseth, Scott L. ‘Solid Polyomino Constructions’. Mathematics Magazine Vol. 49, No. 3, May 1976, 137-138 (9 April 2013)


Fosnaugh, Linda S. and Marvin E. Harrell. ‘Covering the Plane with Rep-Tiles’. Mathematics Teaching in the Middle Schools 666-670 (13 March 2013)


Fowler, D. H. ‘A Generalisation of the Golden Section’. The Fibonacci Quarterly, 20  1982 146-158 (9 September 2014)

Re golden section. From a reference in Livio


Fraser, James A. ‘A New Visual Illusion of Direction’. Journal of Psychology, Vol. II 307-320. (10 August 1993)

Not really mathematics per se


Frederickson, Greg N. ‘Geometric Dissections Now Swing and Twist’. The Mathematical Intelligencer. Volume 23, No. 3, 9-20, 2001 (25 November 2011)


————. ‘A New Wrinkle on an Old Folding Problem’. The College Mathematics Journal. Vol. 34, No. 4, September 2003, 258-263? (21 March 2013)

Heavily academic, of little direct interest


————. ‘The Heptagon to the Square, and Other Wild Twists’ (Mathematical Entertainments) The Mathematical Intelligencer. Volume 29, No. 4, 23-33, 2007 (29 November 2011)


————. ‘Designing a Table Both Swinging and Stable’. The College Mathematics Journal. Vol. 39, No. 4, September 2008, 258-266. (21 March 2013)

Both popular and academic. Builds upon Dudeney’s dissection of triangle to square


————. ‘Casting Light on Cube Dissections’. Mathematics Magazine. Vol. 82, No. 5, December 2009, 323-331 (21 March 2013)

Heavily academic, of little direct interest


————. ‘The Manifold Beauty of Piano-hinged Dissections’. In Bridges Renaissance Banff 2005, 1-8


Also see various reviews and letters of Frederickson’s three books, by Cromwell, Eisenberg, Sykes, Orton, Pargeter, Ruane, Schattschneider


Freiling, Chris et al. ‘Tiling with Squares and Anti-Squares’ 195-?

Academic, of no practical use.


Friedichs, Olaf Delgado et al. ‘What do we know about three periodic nets?’ Journal of Solid State Chemistry 178 (2005) 2533-2554. (13 December 2012)

Chemistry inclined, polyhedral packing, loosely described


Fukada, Hiroshi, et al (Including Schattschneider). ‘Polyominoes and Polyiamonds as Fundamental Domains of Isohedral Tilings with Rotational Symmetry’ (submitted to Discrete and Computational Geometry 2010 (2010)

Somewhat advanced


Fulton, Chandler. ‘Tessellations’. The American Mathematical Monthly Vol. 99, No. 5 May 1992 442- 445. (20 February 2013)

Academic, of no practical use.




Gailunas, Paul and John Sharp. ‘Duality of Polyhedra’. International Journal of Mathematical Education in Science and Technology, Vol. 36, No. 6, 2006, 617-642 (29 April 2013)


Gailunas, Paul. ‘Spiral Tilings’. In Bridges 2000, 133-140 (1 March 2006)

Nice indeed


————. ‘Some Unusual Space-Filling Solids’. The Mathematical Gazette Vol. 88, No. 512 July 2004 230-241 (28 March 2013)


Garcia, Paul. ‘The Mathematical Pastimes of Major Percy Alexander MacMahon’. Part 1 - Slab Stacking. Mathematics in Schools, March 2005, 23-25. (18 February 2013)

Popular account of MacMahon’s slab ‘stacking’ puzzles, and brief background as to MacMahon.


————. ‘The Mathematical Pastimes of Major Percy Alexander MacMahon’. Part 2 - Triangles and beyond’. Mathematics in Schools, September 2005, 20-22. (17 February 2013)

Contains a Cairo tiling of sorts, p. 22


Note that I also have various papers from Garcia’s bibliography of MacMahon, such as from MacMahon himself, and Alder, Andrews, Cayley, Kempner, Putnam, Subbarao, Sylvester, but these are all of an academic nature, of no practical use, and so are not listed in detail here


Gardner, M. ‘The Eerie Mathematical Art of Maurits C. Escher’. Scientific American, Vol. 214, No.4 (April 1966), pp. 110-121. Reprinted in Mathematical Carnival as ‘The Art of M.C. Escher’, Chapter 8, pp. 89-102

Popular discussion. Reptiles, Day and Night 92, Angels and Devils 93, Belvedere 94, Ascending and Descending 95, Order and Chaos, 97, Hand with Reflecting Globe, 99, Knots, 100, Three Spheres, 101


————. ‘On tessellating the plane with convex polygon tiles’. Scientific American June 1975 112-117. Note that this is repeated and updated in Gardner’s Time Travel and Other Mathematical Bewilderments. W. H. Freeman and Co pp. 174-175.

Contains the second recorded reference to the Cairo pentagon. Much discussion on pentagon and hexagon tiles.


————. ‘More about tiling the plane: the possibilities of polyominoes, polyiamonds, and polyhexes’. Scientific American. August 1975 112-115. (11 February 1988)

Mentions the Conway criterion. Penrose loaded wheelbarrow tile mentioned, and illustrated.


————. ‘Extraordinary nonperiodic tiling that enriches the theory of tiles’. Scientific American.  January 1977 110-121. (January 1988)

The most popular account of the Penrose tiles.


————. Interview with Martin Gardner. Notices of the AMS 602-611 June/July 2005  (2010)


————. ‘A Quarter Century of Recreational Mathematics’. Scientific American, August 1998

A May 2010 reprint in honour of Martin Gardner. Includes reference to Polyominoes and Penrose tiles


————. ‘Is Mathematics ‘Out There’?, The Mathematical Intelligencer Volume 23, No. 1, 2001, 7-8 (25 November 2011)

Also see Reuben Hersch for a rebuttal of this piece.


————. ‘Around the Solar System’. Math Horizons. April 1995. Vol. 2 No. 4 22-23 (February 2013)

General math puzzle column. I had no inkling of this journal to as late as February 2013!


————. ‘The Game of Hip’. Math Horizons. November 1995. Vol. 2 No. 4 22-23 (February 2013)


————. ‘The Ant on a 1 x 1 x 2’. Math Horizons. February 1996. Vol. 2 No. 4 22-23 (February 2013)


————. ‘Talkative Eve’. Math Horizons. April 1996. Vol. 2 No. 4 22-23 (February 2013)


————. ‘Some New Discoveries About 3 x 3 Magic Squares’. Math Horizons. February 1998. Vol. 2 No. 4 22-23 (23 February 2013)


————. ‘Ten Amazing Mathematical Tricks’. Math Horizons. September 1998. Vol. 2 No. 4 22-23 (February 2013)


————. ‘Chess Queens and Maximum Unattacked Cells’.  Math Horizons. November 1999. Vol. 2 No. 4 22-23 (23 February 2013)


————. ‘Curious Counts’. Math Horizons. February 2003. Vol. 2 No. 4 22-23 (23 February 2013)


Also see Anthony Barcellos and Don Albers, of separate interviews with Gardner.

Also see Marjorie Senechal for tribute


Gavezzotti, A and M. Simonetta. ‘On the Symmetry of Periodic Structures in Two Dimensions’. Computers and Mathematics with Applications. Vol. 12B, Nos 1/2, pp. 465-476, 1986 (1 October 2013)

From the symmetry ‘special edition’ of the journal. Obscure, although does have a small Escher-like tessellation ‘section’, p. 469, with four instances, with a quite respectable dog, and lesser in quality  birds and human figures and another dog, but how does the latter tile?


Gelbrich, G and K. Giesche. ‘Fractal Escher Salamanders and other Animals’. The Mathematical Intelligencer Vol. 20 No. 2, 31-35. (25 November 2011)


Gerdes, Paulus. ‘On the Geometry of Celtic knots and their Lunda-designs’. Mathematics in School. May 1999, 29-33. (12 April 2013)

General interest


Gerwein, P. ‘Zerschneidung jeder beliebigen Anzahl von gleichen geradlininngen Figuren in dieselben Stücke’. Journal für dei riene und angewandte Mathematik (Crelle’s Journal). 10 1833, 228-234 and Taff III, which appears to be supplementary (23 January 2015)


Gethner, Ellen, Doris Schattschneider, Steve Passiouras, J. Joseph Fowler. ‘Combinatorial Enumeration of 2 x 2 Ribbon Patterns’. European Journal of Combinatorics 28 (2007) 1276-1311. (8 December 2014)

As inspired by Escher's own ribbon patterns. Largely of an academic nature, really only of interest per se in a personal sense (to Escher), although obviously Gethner et al seem enamoured by the premise.


Gibbs, William. ‘Paper Patterns 1 – With Metric Paper’. Mathematics in School. March 1990 pp. 24-28 (22 February 2013)

The first in a series of four articles of ‘paper patterns’. Note that I did not see these en masse, of which I first saw 3 and 4 in 1991, and 1 and 2 much later, in 2013. Of these, ‘Paper Patterns 3’ in particular is of most interest, this leaning towards tessellation, whilst 4 is also of interest in this regard, albeit decidedly less so.

Simply stated, this (1) concerns mathematically folding A4 paper (and gives a brief history of its introduction, of Germany, in 1930s) of a variety of geometric shapes.


————. ‘Paper Patterns 2 – Solid Shapes From Metric Paper’. Mathematics in School. May 1990, pp.2-4 (22 February 2013)

Folding A4 paper to polyhedra; only of minor interest.


————. ‘Paper Patterns 3 – With Circles’. Mathematics in School. September 1990, pp. 2-8 (22 February 2013)

This is by far the most interesting of the four-article series, consisting of tessellations formed by cut-out overlapping and interweaving circles.

At the time I first saw this (1991), I made extensive studies of the tilings here, the best perhaps of Fig. 22, suitable for a bird.


————. ‘Paper Patterns 4 – Paper Weaving’. Mathematics in School. November 1990, pp.16-19 (22 February 2013)

Again, another article of note, albeit not strictly on tessellation, but rather weaving, in which I tried out the weavings given. Of note is the reference here to the tiling Fig. 4, as ‘Shepherds check’, which is where I almost certainly began my references to this tile as such.


————. ‘Three Directional Weaving’. Mathematics in School. March 1992, pp. 2-4 (22 February 2013)

Further tessellation arising from weaving


————. ‘Mathematics in a Matchbox’ Part 1. Mathematics in School. May 1992, pp. 25-28 (22 February 2013)

Comments on the size of matchboxes being the same throughout the world, with proportions of 1: 2: 3, something I had not thought about


————. ‘Mathematics in a Matchbox’ Part 2. Mathematics in School. September 1992, pp. 2-4 (22 February 2013)

‘Squashed matchbox’ observations


————. ‘Polyhedra from A Sized Paper’. Mathematics in School. September 1994, pp.7-11 (22 February 2013)

Folding polyhedra from A sized paper, tetrahedron, Pelican cube, truncated icosahedron (football). Of general interest


————. ‘Window Patterns’. Mathematics in School. January 1995, pp.18-25 (22 February 2013)

From overlapping rectangles, no tessellation


————. ‘Using Books to Construct Shapes on the Blackboard’. Mathematics in School. March 1999, pp. 28-31

Novel idea!


————. ‘Tangrami Square – A paper folding puzzle’. Mathematics in School. September 1999, pp.18-19 (22 February 2013)

Folding shapes, no tessellation; lightweight


————. The Paper Roller’. Mathematics in School. January 2000, pp.12-13

Premise of ‘Owzat’, which I recall from my school days


Gibbs, William and Mphrolet Sihlabela. ‘String Figures’. Mathematics in School. May 1996, pp. 24-27 (22 February 2013)

String figures from around the world. General interest


Giganti, Paul, and Mary Jo Cittadino. ‘The Art of Tessellation’. The Arithmetic Teacher (NCTM), 37(7): 6-16, 1990 (18 February 2013)

Typical teacher efforts of creating Escher-like tesselation; shows no understanding at all. Horrendous


Gilbert, William J. ‘An easy way to construct spacefillings’. Structural Topology No. 8, 1983, 25- 32 (c. 2008)

One of four references in Structural Topology as mentioned in the bibliography of Tilings and Patterns; the others being Burt, Danzer and Lalvani, although there are indeed other tiling references in the journal.

This discusses both the (geometric) filling of the plane and three dimensions. Assessment is pending.


Giles, Jack, Jr. ‘Infinite-Level Replicating Dissections of Plane Figures’. Journal of Combinatorial Theory, Series A 26, 319-327, 1979 (24 September 2013)

From a reference in Tilings and Patterns. Largely of an academic nature throughout. Occasional aspect of broad interest and understanding, but is largely too advanced for me. Of little to no practical use. Note that this is  the first of three conjoined articles.


————. ‘Construction of Replicating Superfigures’. Journal of Combinatorial Theory, Series A 26, 328-334, 1979 (24 September 2013)

From a reference in Tilings and Patterns. Largely of an academic nature throughout. Occasional aspect of broad interest and understanding, but is largely too advanced for me. Of little to no practical use.


————. ‘Superfigures Replicating with Polar Symmetry’. Journal of Combinatorial Theory, Series A 26, 335-337, 1979 (24 September 2013)

From a reference in Tilings and Patterns. Largely of an academic nature throughout Occasional aspect of broad interest and understanding, but is largely too advanced for me. Of little to no practical use.



Glasser, L. ‘Teaching Symmetry The use of decorations’. Journal of Chemical Education 44, no 9 (September 1967) 502-511. Hard Copy (30 May 2012)

Heavy use is made of Escher's prints, in relation to chemical/crystallography-like relations. Schattsneider briefly discusses this paper on p. 277


Glickman, Michael. ‘The G-Block of Vertically Interlocking Paving’. Second International Conference of Concrete Block Paving Delft April 10-12, 1984 (29 July 2014)

Paving speculations, of general interest. Also see his patents, with many interesting tilings, especially of a modified hexagon, forming a chevron with many different placements. Does anyone know of Glickman? Contact details appear unavailable.


Tetrahedra equivalent  to cubes by dissection. Elemente der Mathematik


————. ‘Pattern, Texture and Geometry in the Paved Surface’. Journal unknown, 71-74 (29 July 2014)

Paving speculations


Goldberg, Michael. ‘Two More Tetrahedra Equivalent to Cubes by Dissection’. Elemente der Mathematik  24 304-305 1958 (15 January 2015)

From a reference in Dissections: Plane & Fancy. Academic throughout, of no practical use


————. 3162. ‘A duplication of the cube by dissection and a hinged linkage’. Mathematical Gazette 50 1966 304-305

From a reference in Dissections: Plane & Fancy.  Largely above me


————. ‘On the Original Malfatti Problem’. Mathematical Gazette Vol. 40, No. 5, 241-247 November 1967 (27 February 2013)

Of an academic nature throughout; of no practical use.


————. On the Densest Packing of Equal Spheres in a Cube. Mathematics Magazine.199-208 September 1971 (27 February 2013)

Of an academic nature throughout; of no practical use.


————. ‘Maximising the Smallest Triangle Made by N Points in a Square’. Mathematics Magazine. May 1972 135-144 (27 February 2013)

Of an academic nature throughout; of no practical use.


————. ‘New Rectifiable Tetrahedra’. Elemente der Mathematik  29, 85-89 1974 (15 January 2015)

From a reference in Dissections: Plane & Fancy . Academic throughout, of no practical use


————. ‘Proof Without Words: Trisecting the Angles of a Regular N-gon’. 283. Vol 51, No.5 November 1978 (27 February 2013)

Of an academic nature throughout; of no practical use.


————. ‘Unstable Polyhedral Structures’. Mathematics Magazine. 165-170. Vol. 51, No.3, May 1978. (27 February 2013)

Largely of an academic nature throughout; of no practical use.


————. ‘On the Space filling Decahedra’. Structural Topology 7 1982 (14 November 2012)


————. ‘On the Minimum Track of a Moved Line Segment’. Mathematics Magazine, 257- ? (27 February 2013)

Of an academic nature throughout; of no practical use.


————. ‘The Packing of Equal Circles in a Square’. Mathematics Magazine, 24-30. (27 February 2013)

Of an mostly academic nature throughout; some straightforward circle diagrams


Goldstein, Laurence. ‘Reflexivity, Contradiction, Paradox and M.C. Escher’. Leonardo, Vol. 29, No. 4, pp. 299-308, 1996. (17 February 2013)

Largely philosophical commentaries way beyond me. Profusely illustrated with Escher's non tessellation prints.


Golomb, Solomon W. ‘Replicating Figures in the Plane’. The Mathematical Gazette 403-412 (18 February 2013)

Generally a popular account, drifts towards academic. C. Dudley Langford inspired


————. ‘Tiling with Polyominoes’. Journal of Combinatorial Theory 1, 280-296, 1966 (25 September 2013)

From a reference in Tilings and Patterns. Polyominoes. Mostly of an academic nature throughout, albeit profusely illustrated with diagrams I can understand!


————. ‘Tiling with Sets of Polyominoes’. Journal of Combinatorial Theory, 9, 60-71, 1970 (25 September 2013)

From a reference in Tilings and Patterns. Polyominoes. Mostly of an academic nature throughout!. Of little to no practical use.


————. ‘Checker Boards and Polyominoes’. 675- December (9 April 2013)



Golomb, Solomon W and Lloyd R. Welch. ‘Perfect Codes in the Lee Metric and the Packing of Polyominoes’. SIAM Journal of Applied Mathematics Vol. 18, No. 2, January 1970



Gómez, R. Pérez-. ‘The Four Regular Mosaics Missing in the Alhambra’. Computer math Applications. Vol.14, No.2 pp 133-137, 1987 (16 December 2010)


Gordon, Basil. ‘Tilings of Lattice Points in Euclidean n-Space’. Discrete Mathematics 29 1989 169-174 (5 January 2015)

From a reference in Tilings and Patterns. Of an academic nature throughout; of no practical use.


Granger, Tim. ‘Math Is Art’. Teaching Children Mathematics. 10-13 (23 February 2013)

‘How to’ on Escher-like art, with all the usual teacher shortcomings


's-Gravesande, G. H. ‘Nieuw werk van M. C. Escher’. Elsevier's geillustreerd mandschrift 48, deel 96 (1938): 312-314 (in Dutch). (15 September 2014)


Green, P. J. and R. Sibson. ‘Computing Dirichlet tessellations in the plane’. The Computer Journal. 21 (1978) 168-173 (31 October 2012)

From a reference in Tilings and Patterns. Of an academic nature throughout; of no practical use.


Gregory, Richard L. and Priscilla Heard. ‘Border locking and the Café Wall illusion’ [sic]. Perception, 1979, volume 8, 365-380. (30 October 2012)

Not really mathematical, but rather of perception, concerning the well-known café wall illusion, but of considerable interest nonetheless. The café wall illusion a long-standing interest. I also have 21 other s of Gregory’s, not recorded here, taken from his website. These are really of perception, largely academic, the merits of which including here are dubious, hence their exclusion.


Gridgeman, N. T. ‘The 23 Colored Cubes’. Mathematics Magazine Vol. 44 No. 5 November 1971, 243-252 (26 March 2013)

From reference in Garcia


Groemer, H. ‘On Multiple Space Subdivisions by Zonotopes’. Monatshefte für Mathematik 86, 185-188 1978 (5 January 2015)

From a reference in Tilings and Patterns. Of an academic nature throughout; of no practical use.


Grünbaum and Shepherd:


Grünbaum, B. and G. C. Shephard. ‘Satins and Twills: An Introduction to the Geometry of Fabrics’. Mathematics Magazine. Vol. 53, No. 3 May 1980. 139-161. (13 February 1996, hard copy)

Somewhat of a let down as regards content, the subject matter is of no real interest. No reference to tessellation.


————. ‘The 2-Homeotaxal Tilings of the Plane and the 2-Sphere’. Journal of Combinatorial Theory B 34, 113-150 1983. (26 September 2013)

From a reference in Tilings and Patterns. Of an academic nature throughout; profusely illustrated with simple tilings, but otherwise of academic text . Cairo tiling in premise on p.138


————. ‘A Generalisation of Theorems of Kirszbraun and Minty’. 812- 1961 (26 February 2013)

Of an academic nature throughout; of no use


————. ‘A Variant of Helly’s Theorem’. 517-  August (26 February 2013)

Of an academic nature throughout; of no use


————. ‘How to Cut All Edges of a Polytope? ‘890-

Of an academic nature throughout; of no use


————. ‘Some results on the Upper Bound Conjecture for Convex Polytopes’

Siam Journal of Applied Mathematics Vol. 17, No.6, November 1969

Of an academic nature throughout; of no use


————. On Venn Diagrams and the Counting of Regions (26 February 2013)


Of an academic nature throughout; of no use


————. Venn Diagrams and Independent Families of Sets (26 February 2013) Mathematics Magazine


————. Do normal Line Generated Triangulations of the Plane Exist?


Of an academic nature throughout; of no use


————. ‘A dual for Descartes theorem on polyhedra’. The Mathematical Gazette 214-

Of an academic nature throughout; of no use


————. ‘Rotation and Winding Numbers For Planar Polygons and Curves’. Transactions of the American Mathematical Society, Vol. 322, Number 1, November 1990

Of an academic nature throughout; of no use


————. ‘A new Ceva-type theorem’. The Mathematical Gazette 492-

Of an academic nature throughout; of no use


Branko Grünbaum only


————. ‘A problem in Graph Coloring’ 1088- n

Of an academic nature throughout; of no use


————. ‘Polygons in Arrangements Generated by n points’ 113- Mathematics Magazine May-June


————. ‘Musings on an example of Danzer’s’. European Journal of Combinatorics 29 (2008) 1910-1918 (9 December 2014)

Academic, not of tiling, of limited interest


Branko Grünbaum and others


————. BG. TM On components in some families of Sers  607-


————. BG. MK Euler’s Ratio Sum Theorem and Generalisations Mathematics Magazine 122-

BG JM Some Models of Plane Geometries  The Teaching of Mathematics 1990 839-


BG GC Is Self duality Involutory? 729-


————. ‘The Emperor’s New Clothes: Full Regalia, G String, or Nothing?’ The Mathematical Intelligencer Vol. 6, No.4, 1984 (24 November 2011)

Accessible. Interestingly, has a type 13 Rice pentagon, of which I’ve been studying lately, but no connection is made with the Cairo tiling!


————. ‘Geometry Strikes Again’. Mathematics Magazine Vol. 58, No. 1 January 1985. 12-17. (18 September 1989)

As kindly supplied by Grünbaum, 18 September 1989


————. ‘Hypersymmetric Tiles’. Congressus Numerantium 50 (1985). 17-24. (18 September 1989)

As kindly supplied by Grünbaum following correspondence, 18 September 1989


————. Art and Science. ‘Mathematical Challenges in Escher's Geometry’. (Supplement to article).19 September 1989.

Contains Koloman Moser’s tilings, MacMahon, Pólya, Delone, Heesch


————. ‘The Bilinski Dodecahedron, and Assorted Parallelohedra, Zonohedra, Monohedra, Isozonohedra, and Otherhedra’. The Mathematical Intelligencer .Vol 32, Number 4, 2010 (24 November 2011)

Leaning heavily towards the academic. Polyhedra, no tiling as such


————. ‘A Relative of Napoleon’s Theorem’. Geombinatorics 10 (2001) 116-121 (30 January 2012)



————. ‘A starshaped [sic] polyhedron with no net’. Geombinatorics 11 (2001) 43-48 (30 January 2012)



————. ‘Families of point-touching squares’. Geombinatorics 12 (2003) 167-174 (30 January 2012)



————. ‘Tilings with Congruent Tiles’. Bulletin (new Series) of the American Mathematical Society Volume 3, Number 3, November 1980 (2010).

Tilings arising from the second part of Hilbert’s eighteenth problem. Although largely academic, with diversions into polyhedra, there is occasional recreational use, notably with convex pentagons, with Kershner, James, and Rice, pp. 956-957


Grünbaum, Branko, and G. C. Shephard. ‘Perfect Colorings of Transitive Tilings and Patterns in the Plane’. Discrete Mathematics 20 (1977). 235-247 7 (9 December 2014)

Largely academic, occasional Escher reference, shows a different presentation of Laves diagrams.


————. ‘Tilings by Regular Polygons’. Math Magazine 50 (1977), 227-247

Broadly accessible, with reservation! A ‘follow up’, which I don’t have, is 51, (1978), 205-206)


————. ‘Regular polyhedra – old and new’. Aequationes mathematicae. 16 (1977) 1-20 (28 May 2012)



 ————. ‘The Ninety-One types of Isogonal Tilings in the Plane’. Transactions of the American Mathematical Society. Volume 242, 335-353 August 1978. (23 May 2011)

Academic. Mostly too technical for me, despite noticeable numerous diagrams.


————. ‘Isohedral tilings of the plane by polygons’. Comment. Math. Helvetici 53 (1978) 542-571 (30 May 2012)

Largely of an academic nature, but broadly accessible on occasions. A profusion of diagrams, has a Cairo tiling, and skew variation, p. 568, the context of which requires examination. Interesting discussion on ‘straightening’ procedure, p. 556, that requires examination.


————. ‘Isotoxal Tilings’. Pacific Journal of Mathematics. Vol. 76, No. 2. 1978 407-430 (20 September 2012)

Largely academic throughout. Many ‘simple’ diagrams; however, all of this no practical use, being academic


————. ‘Is there an all-purpose tile?’ American. Mathematical Monthly 93 545-551 (1986) (18 September 1989)

As kindly supplied by Grünbaum, 18 September 1989. The premise is of tiling as according to each of the 17 wallpaper groups, using a  triangular tile. Largely of academic interest only; certainly, it has not impacted on my studies.


————. ‘Ceva, Menelaus, and the Area Principle’. Mathematics Magazine, Vol. 68, No. 4 Oct. 1995) pp. 254-268 (30 January 2012)



————. ‘What Symmetry Groups are Present in the Alhambra?’ Notices of the AMS, Vol. 53, Number 6, 670-673, June/July 2006 (4 July 2011)

Popular account


————. ‘Unambiguous Polyhedral Graphs’. Journal unknown. 235-238.1963 (24 October 2012)

Of an academic nature throughout. Of no practical use.


————. ‘Patch Determined Tilings’. The Mathematical Gazette. 31-38. (18 February 2013)

Has dimorphic tiling and spiral tiling forerunners to appearing in Tilings and Patterns. Also see follow up article in the Gazette by Heiko Horbath 61.25 Prescribed number of tiles and tilings


Grünbaum, B. and Z. Grünbaum, G. C. Shephard. ‘Symmetry in Moorish and other Ornaments’. Comp. & Maths. With Appls. Vol. 12B, Nos. 3/4. 641-653. (18 September 1989) (30 April 2012)

As kindly supplied by Grünbaum, 18 September 1989. An examination of how many of the 17 wallpaper groups are present  in the Alhambra, concluding that 13 are present, this being in contrast to the widely repeated claim that all 17 are to be found.


Gupta, Madhu S. ‘Escher’s Art, Smith Chart and Hyperbolic Geometry’. IEEE Microwave Magazine October 2006, pp. 66-76 (11 October 2011)


Gutiérrez, Angel. ‘An Experience with M. C. Escher and the Tessellations’, Mathematics in School, March 1983, 17-21 (17 February 2013)

Largely an analysis of the underlying symmetry of Escher's tessellations; not a ‘how-to’ guide. Not particularly impressed.




Haag, F. ‘Die regelmässigen Planteilungen’. Zeitschrift fur Kristallographie 49 (1911): 360-369.

(24 April 2012)

Although this has what can be interpreted as ‘skewed Cairos’, there is not a standard Cairo tile here. Note that this article was the first of three by Haag on a listing of B.G. Escher as given to M. C. Escher (as documented in Visions…, p. 357


 ————. ‘Die regelmässigen Planteilungen und Punktsysteme’. Zeitschrift fur Kristallographie 58 (1923): 478-488. (24 April 2012)

This is the article Doris Schattschneider quoted me as a Cairo tiling, fig 13, p. 487 in her tiling listserver response to my posting. However, after a translation was obtained, this is not so, Haag is referring to a quadrilateral tile, and not a pentagon; the pentagon ‘appears’ incidentally, upon a misinterpretation of the diagram. Note that this article was the second of three by Haag on a listing of B.G. Escher as given to M. C. Escher (as documented in Visions…, p. 357


————. ‘Die 17 regelmässigen Planteilungen und Punktsysteme’. Zeitschrift fur Kristallographie 61,  (1925) 339 (23 December 2014)

Essentially, just as note rather than an article, one diagram, of staggered rectangles.


————. ‘Die pentagonale Anordung von sich berührenden Kriesen in der Ebene’. Zeitschrift fur Kristallographie 61  (1925), 339-340 (3 August 2012, 23 December 2014)

(Quoted in Bradley, Schattschneider)

Has Cairo tiling in the form of circle packing


————. ‘Die Planigone von Fedorow’ (Federov?). Zeitschrift fur Kristallographie 63 (1926).179-186. (24 April 2012)

Note that this article was the third of three by Haag on a listing of B.G. Escher as given to M. C. Escher (as documented in Visions…, p. 357


————. ’Die Symmetrie verhältnisse einer regelmässigen Planteilung’. Zeitschrift für mathematischen und naturwissenschaftlichen Unter-richt, Band 57 (1926), 262-263 (3 August 2012, 17 December 2014)

Quoted in Bradley, Schattschneider. Two diagrams, of ‘arrowhead’ and a small patch Cairo tiling, albeit the Cairo tiling arises as a result of a disc packing as according to the sqaure and equlaiteral triangle tiling.


Haak, Sheila. ‘Transformation Geometry and the artwork of M.C. Escher’. Mathematics Teacher (2010). December 1976. 647-652.


Haake, A. ‘The Role of Symmetry in Javanese Batik Patterns’. Computers Math Applic. Vol. 17, No. 4-6, pp 815-826, 1989.

Of limited interest.


Hadwiger, H. Z’eregungsgleichheit ebener polygone’. Elemente der Mathematik 6 97-106 1951


Hales, Thomas C. ‘The Status of the Kepler Conjecture’. The Mathematical Intelligencer. Vol. 16, No. 3, 1994, 47-58 (15 December 2011)



————. ‘Sphere packing, 1. Discrete & Computational Geometry (1992) 17: 1-51. (5 November 2102)

Of an academic nature throughout; the tone of the paper is way beyond me. Of no practical use.


————. ‘Sphere Packing 3. ‘Extremal Cases’. Discrete & Computational Geometry (2006) 36: 71-110. (5 November 2012)

Of an academic nature throughout; the tone of the paper is way beyond me. Of no practical use.


————. ‘Sphere Packings, 4. Detailed Bounds’. Discrete & Computational Geometry (2006) 36: 111-166. (5 November 2012)

Of an academic nature throughout; the tone of the paper is way beyond me. Of no practical use.


————. ‘Sphere Packings 6. ‘Tame Graphs and Linear Programs’. Discrete & Computational Geometry (2006) 36: 205-265. (5 November 2012)

Of an academic nature throughout; the tone of the paper is way beyond me. Of no practical use.


————. ‘Historical Overview of the Kepler Conjecture’. Discrete & Computational Geometry (2006) 36: 5-20. (2 November 2012)

Although largely of an academic nature throughout, this has various aspects of interest, such as the history of the problem, of which given the Kepler connection is of interest.


Hales, Thomas C. and Samuel P. Ferguson. ‘A Formulation of the Kepler Conjecture’. Discrete & Computational Geometry (2006) 36: 21-69. (5 November 2012)

Of an academic nature throughout; the tone of the paper is way beyond me. Of no practical use.


Hales, Thomas C. et al. ‘A Revision of the Proof of the Kepler Conjecture. Discrete & Computational Geometry (2010) 44: 1-34. (5 November 2012)

Of an academic nature throughout; the tone of the paper is way beyond me. Of no practical use.


Hall, Kelli. ‘Escher Tilings and Ribbons: A Mathematical Look’. 1-19. Paper source not known, possibly Bridges. (12 December 2012)

Of an academic nature throughout; the tone of the paper is way beyond me. Of no practical use.


Halsey, George D and Edwin Hewitt. ‘Eine Gruppentheoretische Methode in der Musiktheorie’ Jahresbericht der Deutschen Mathematiker-Vereinigung 80 1978 151-207 (31 December 2014)

From a reference in Tilings and Patterns. Academic throughout, no diagrams. Of no practical use.


Hankin, E. Hanbury. ‘On Some Discoveries of the Methods of Design Employed in Mahommedan Art’. Journal of the Society of Arts, 1905, Vol. LIII, 461-477 (25 February 2013)

Hankin’s ‘polygons in contact’ method.


————. ‘The Drawing of Geometric Patterns in Saracenic Art’. Memoirs of the Archaeological Survey of India. No. 15. Calcutta: Government of India Central Publication Branch 1925 (25 September 2010).

Islamic style patterns, a ‘how it was done.’


————. ‘Examples of Methods of Drawing Geometrical Arabesque Patterns’. Math. Gazette 12 (1925), 370-373 (25 February 2013)

In effect this is I of a unstated series of three related articles. Only much later, nine years, did Hankin decide to continue, with II, and later with III


————. ‘Some Difficult Saracenic Designs II’. A Pattern Containing Seven-Rayed Stars. Math. Gazette 18 (1934), 165-168 (25 February 2013)


————. ‘Some Difficult Saracenic Designs III’. A Pattern Containing Fifteen-Rayed Stars Math. Gazette 20 (1936), 318-319 (25 February 2013)


Hansen, Vagn Lundsgaard. ‘From Figure to Form’. Math Horizons November 1999 8-11



Harborth, H. 'Prescribed numbers of tiles and tilings'. The Mathematical Gazette 61 (1977): 296-299. (18 February 2013)

n-morphic tilings, in response to Grünbaum and Shepherd’s patch determined article.


Hargittai, István. ‘Limits of Perfection’. Computers and Mathematics with Applications. Vol. 12B, Nos 1/2, pp.1-17, 1986 (1 October 2013)

From the symmetry ‘special edition’ of the journal. Obscure.

————. ‘Dethronement of the Symmetry Plane’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration (31 August 2005) 353-365

————. ‘Quasicrystal Sculpture in Bad Ragaz’. (Mathematical Tourist) The Mathematical Intelligencer Vol. 14 Number 3, 1992, 58-59 (14 December 2011)


————. ‘Octagons Abound’.  (Mathematical Tourist) The Mathematical Intelligencer Vol. 17 Number 2, 1995, 52-54 (28 November 2011)

Includes pavement


————. ‘Fullerene Geometry under the Lion's Paw’. (Mathematical Tourist) The Mathematical Intelligencer Vol. 17 Number 3, 1995, 34-36 (28 November 2011)

Chinese theme


————. ‘Lifelong Symmetry: A Conversation with H. S. M. Coxeter’. The Mathematical Intelligencer Vol. 18 Number 4, 1996, 35-41 (28 November 2011)

Interview with Coxeter. Minor Escher references, pp. 36, 38-39


————. ‘Sacred Star Polyhedron’. (Mathematical Tourist) The Mathematical Intelligencer Vol. 18 Number 3, 1996, 52-54 (14 December 2011)


Hargittai, István. ‘John Conway – Mathematician of Symmetry and Everything Else’. The Mathematical Intelligencer Vol. 23 Number 2, 2001, 6-14 (28 November 2011)



Hargittai, István and Magdolna Hargittai. ‘Symmetries of Opposites: Antisymmetry’ The Mathematical Intelligencer Vol. 16 Number 2, 1994, 60-66 (28 November 2011)

Occasional tessellation with Mamedov p. 65, also in Symmetry book


Hargittai, Magdolna and István Hargittai. ‘Symmetry and Perception: Logos of Rotational Point-Groups Induce the Feeling of Motion’. The Mathematical Intelligencer Vol. 19 Number 3, 1997, 55-58 (28 November 2011)


Harries, John, G. ‘Symmetry and Notation: Regularity  and Symmetry in Notated Computer Graphics’. Computers and Mathematics with Applications. Vol. 12B, Nos 1/2, 303-314, 1986 (1 October 2013)

From the symmetry ‘special edition’ of the journal. Obscure.


Hart, George W. All papers from the Bridges archive, from 1998; with his interest in polyhedra to the fore, the more important of which:


————. ‘Creating a Mathematical Museum on your Desktop’. The Mathematical Intelligencer Vol 27 No. 4, 2005 14-17

On solid freeform fabrication


————. ‘Bringing M.C. Escher’s Planaria to Life’. Bridges , 2012, 57-64

Begins with a brief discussion on Escher’s planaria, then concentrates on the polyhedral aspect per se.


Hart, Harry. ‘Geometrical dissections  and transpositions’. Messenger of Mathematics, Vol. 6 1877,150-151 (12 January 2015)

From a reference in Dissections: Plane & Fancy. All text, no diagrams


Harrower, M. R. ‘Some Factors Determining Figure-Ground Articulation’. 407-424. (29 November 1993)


Hedian, H. ‘The Golden Section and the Artist’. The Fibonacci Quarterly, 14, 1976 406-418 (9 September 2014)

Re golden section. From a reference in Livio.

Heesch, H. ‘Über topologisch gleichwertige Kristallbindungen’. (About topologically equivalent crystal bonds) Zeitschrift für Kristallographie 84 1933: 399-407. (23 April 2015)
From a reference in Schattsneider. 25 figures, but only of a simple nature; nothing of any real interest

————. ‘Aufbau der Ebene aus kongruenten Bereichen’ (Tiling the Plane with Congruent Tiles). Nachr. Grees Wiss. Gott NF 1 (1935, 115-117) see online translation (2006, 2010)

Simply stated, tiling with a decagon, largely of a popular level.


Hekking, Sjoerd. ‘Zuilen van Escher Gered!’. (In Dutch) From TEGEL journal issue 41. (16 May 2014) date of article is not stated

A popular discussion on Escher’s columns he did for Baarn Lyceum and Joanna Westerman schools Has many pictures not seen before, including single tiles and the installation.


Hemmings, Ray. ‘Lobachevsky on a Micro’. Mathematics Teaching 111. June 1985. 23-27.

Somewhat advanced concepts for the intended audience! Uses Escher print Circle Limit, p. 27


Hensley, Douglas. ‘Fibonacci Tiling and Hyperbolas’. Fibonacci Quarterly 16 (1978) 37-40

From a reference in Tilings and Patterns. Of an academic nature throughout, with not a single diagram! Of no practical use.


Henry, Richard. ‘Pattern and Contemplation: Exploring the Geometric Art of Iran’ (2010)  

Public lecture given by Richard Henry at the Middle East Association on 27 April 2007. Published in the Journal of the Iran Society, September 2007.


Henry, Bruce. ‘Polyominoes’. Mathematics in School .Vol. 7 No. 2, March 1978 p. 13 (9 April 2013)

Brief look at combining polyominoes into pre-determined configurations.


Heppes, A. ‘Solid Circle-Packings in the Euclidean Plane’. Discrete & Computational Geometry (1992) 7: 29-43. (5 November 2012)

Of an academic nature throughout; the tone of the paper is way beyond me. Of no practical use.


Herda, Hans. ‘Tiling the Plane With Incongruent Regular Polygons’. Fibonacci Quarterly 19 (1981) 437-439

Square packing. Largely of an academic nature throughout, with not a single diagram! Of no practical use. From a reference in Tilings and Patterns.


Herrmann, Heinz. ‘Asymmetry and Symmetry in Cellular Organization’ Computers and Mathematics with Applications. Vol. 12B, Nos 1/2, 155-167, 1986 (1 October 2013)

From the symmetry ‘special edition’ of the journal. Obscure.


Hersh, Reuben. Reply to Martin Gardner. Opinions Column. The Mathematical Intelligencer

An open letter to Martin Gardner (25 November 2011) Vol. 23, No. 2, 2001 3-5


Hilbert, David. ‘Mathematical Problems’. Bulletin of the American Mathematical Society.  8 1901-1902, 437-479 (2 January 2015)


Hill, Anthony. ‘Art and Mathesis: Mondrian’s Structures’. Leonardo Vol I 233-242, 1968 (18 April 2013)

Philosophical musings; obscure


————. ‘A View of Non-Figurative Art and Mathematics and an Analysis of a Structural Relief’. Leonardo Vol 10 pp. 7-12, 1977 (18 April 2013)

Philosophical musings; obscure


Hilton, Peter, and Jean Pedersen. ‘Comments on Grünbaum’s Article’. The Mathematical Intelligencer Vol. 6, No.4 1984 (24 November 2011)


————. ‘Symmetry in Mathematics’. Computers and Mathematics with Applications. Vol. 12B, Nos 1/2, 155-167, 1986 (1 October 2013)

From the symmetry ‘special edition’ of the journal. Obscure.


Hippel, von Frank N. ‘Arthur von Hippel: The Scientist and the Man’. MRS Bulletin, Volume 30, November 2005 (29 May 2014)

Upon reading Cyndie Campbell’s book ‘M. C. Escher Letters to Canada, 1958-1972’, I noticed a reference to von Hippel, page 65, whose name I was unfamiliar with. Upon looking on the web for him, I found various papers, with this, containing  the background to Escher’s  ‘Man with Cuboid’ print, of which the background, with the von Hippel connection, was unknown to me. Page 840 titles this as ‘The Thinker’. For more on von Hippel, see the article by Markus Zahn.


Hirschhorn, M. D. and D. C Hunt, ‘Equilateral Convex Pentagons Which Tile the Plane’. Journal of Combinatorial Theory, Series A. Vol. 39, No.1, May 1985. (22 May 1996) (PDF 24 September 2013)

Somewhat technical in places, but still of considerable interest in regard of Cairo-type tiling


Hirschhorn, M. D. ‘Tessellations with Convex Equilateral Pentagons’. Parabola 2-5, 18, 23, 36. Vol.13, No.1 February/March 1977 (22 May 1996)

Considerable Cairo-esque type pentagons


————. ‘More Tessellations with Convex Equilateral Pentagons’. Parabola 20-22, 36. Vol.13, No.2  May/June 1977 (22 May 1996)

Considerable Cairo-esque type pentagons


————. Parabola 20-22, 36. Vol.13, No.3 August/September 1977 (22 May 1996)

Limited to a single pentagon patch tiling, pp. 14 and 17, by Hirschhorn junior!


Hofstadter, Douglas R. ‘Metamagical Themas 'Parquet Deformations: patterns of tiles that shift gradually in one dimension’. Scientific American (July 1983): 12-18.

William Huff’s student-inspired parquet deformations  Absolutely delightful!


————. ‘Mystery, Classicism, Elegance: an Endless Chase After Magic’. In Coxeter, et al, Eds. M.C. Escher: Art and Science. Amsterdam: North-Holland 1986. (30 April 1994)


Hoogewerff, G. J. 'M. C. Escher, grafisch kunstenaar.' Elsevier’s geiliustreerd maandschrift, Vol. 40, no. 10 (1931),  225-235.  (In Dutch) (18 December 2014)


Hoggatt, V. E. Jr. and M. Bicknell-Johnson. ‘Reflection across Two and Three Glass Plates’. The Fibonacci Quarterly, 17 1979, 118-142 (9 September 2014)

Re golden section. From a reference in Livio.


Holden, Herbert L. ‘Fibonacci Tiles’. Fibonacci Quarterly 13 (1975), 45-49 (26 October 2012)

From a reference in Tilings and Patterns. Of an academic nature throughout, albeit with the occasional diagram ‘understandable! Of no practical use.


Hollingsworth, Caroline. ‘Polyominoes: An Unsolved Problem’. Mathematics Teacher, May 1985 364-365 (9 April 2013)

Determining the numbers possible.


Hollist, J. Taylor. ‘Escher Correspondence in the Roosevelt Collection’. Leonardo Vol. 24, No.3. 329-331 1991 (17 February 2013)

On: ‘Impossible figures with Penroses’, ‘Coxeter and the Circle Limit Prints’, ‘Reproduction of Prints’, and ‘Influence of George Pólya’. A misnomer of ‘Penrose wheelbarrow’ is shown.


————. ‘M.C. Escher’s Association with Scientists’. In Bridges 2000. 45-52 (1 March 2006)

Roosevelt collection, Coxeter, impossible figures, Scientific American, Pólya, Crystallographers, other scientists

Hollist. J. Taylor and Doris Schattschneider. ‘M.C. Escher and C.v.S. Roosevelt’. In Coxeter, et al, Eds. M.C. Escher: Art and Science. Amsterdam: North-Holland 1986. 52-62 (30 April 1994)

Houle, Kelly. ‘Portrait of Escher: Behind the Mirror’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration 175-188 (31 August 2005)

Huff, William S. ‘Simulacra in non-reorientable surfaces-experienced in timing’. In Chapter 4, ‘Spatial Lines’, Patricia Muñoz, compiler
Minor reference to parquet deformations.

Hughes, Anne. ‘Escher’s Sense of Wonder’. In Coxeter, et al, Eds. M.C. Escher: Art and Science. Amsterdam: North-Holland 1986. 63-68 (30 April 1994)

Huson, H. Daniel. ‘The Generation and Classification of Tile-k-Transitive Tilings Of The Euclidean Plane, the Sphere And The Hyperbolic Plane’. Geometriae Dedicata 47: 269-296, 1993. (9 September 2010).

Highly technical, of limited interest




Inchbald, Guy. ‘Five space-filling polyhedra’, The Mathematical Gazette. 466-475,

(18 February 2013)

Cairo-esque aspects with bisymmetric hendecahedron


İldeş, Güslseren. ‘An Analysis for the works of Escher and Their Use In Art Education’. Procedia - Social and Behavioural Sciences 141 (2014) 1196-1202 (8 December 2014)




Jansen, René. ‘Polycairos in Disguise’. Newsletter Nederlandske Cube Club’, CFF 63, March 2004 (June 2011)

Of note is that Jansen has a Cairo tiling article in the form of Polycairos, and with a request for an in situ tiling picture.


Jaworski, J. ‘A Mathematician’s Guide to the Alhambra’. Second revised edition 2006 (25 October 2012)


Jendrol, S. and E. Jucovic. ‘On a Conjecture by B. Grünbaum’. Discrete Mathematics Vol. 2, No. 1 1972 35-49 (6 January 2015)

From a reference in Tilings and Patterns. Of an academic nature throughout, albeit with the occasional simple diagram ‘understandable! Includes a Moore pentagon inside a regular hexagon, but not as a tiling, 42-43. Of no practical use.


Johnson, Paul B. ‘Stacking Colored Cubes’. The American Mathematical Monthly Vol. 63, No. 6 June-July 1956 392-395 (26 March 2013)
From reference in Garcia. Really only of interest to those involved in the premise; one diagram


Jones, Christopher B. ‘Periodic tilings with vertices of species number 3’. Structural Topology 20, 1993  (c. 2008?)

A whole host of ‘demi-regular’ tilings of squares and triangles


Jucovic, E. ‘Analogues of Eberhard’s Theorem fir 4-Valent 3-Polytopes with Involutory Automorphisms’. Discrete Mathematics Vol. 6, No. 1 1973 249-254 (6 January 2015)

From a reference in Tilings and Patterns. Of an academic nature throughout, with the occasional diagram of an obscure nature!. Of no practical use.



Juhel, Alain. ‘Prince of Samarqand Stars’. The Mathematical Intelligencer. (The Mathematical Tourist) Volume 29, Number 4, 44-50, 2007 (30 April 2012)

On Ulugh-Beg


Kahan, Steven J. ‘Eight blocks to Madness’ – A logical solution. Mathematics Magazine March-April 57-?
From reference in Garcia

Kaiser, Barbara. ‘Explorations with Tessellating Polygons’. The Arithmetic Teacher (NCTM), 36(4):19-24, 1988 (18 February 2013)
Mostly of elementary tiling theory per se, for children. Contains tessellating letters of the alphabet S, T, C, L. One page of Escher-like tesselaltions, with usual teacher shortcomings

Klamkin, and A. Liu. ‘A Note on a result of Niven on Impossible Tesselations’. American Mathematical Monthly 87, October 1980, pp. 651-653


Kanon, Joseph. ‘The Saturday Review December’ 16 1972  ** (Sphere Spirals) (29 July 2015)


Kaplan, C. S. ‘Computer Generated Islamic Star Patterns’. In Bridges 2000 105-112 (and Visual Mathematics, 2, (3) 2000)


————. ‘Escherization’. In proceedings of Siggraph 2000

Most informative, with a considered approach to life-like tiling


————. ‘Islamic Star Patterns from Polygons in Contact’

Building upon Hankin’s ‘polygons in contact’ method


————. ‘The trouble with five’. December 2007.

For Plus magazine, and written for a less academic audience than with most of his papers. Tiling with a five theme. No Cairo tiling


————. ‘Metamorphosis in Escher’s Art’, Bridges 2008 (Leeuwarden), 39-46

Parquet deformation pp.43-45, based on arbitrary isohedral tiles. Deformations between the Laves tilings


————. ‘Curve Evolution Schemes for Parquet Deformations’. Bridges 2009


————. ‘Patterns on Surfaces’

Various tilings applied to an arbitrary 3D model, here a rabbit. Includes Escher’s Shells and Starfish drawing, no. 42


Kaplan, C. S. and David H. Salesin. ‘Islamic Star Patterns in Absolute Geometry’. ACM Transactions on Graphics, Vol. 23, No. 2 April 2004, pp. 97-119


Kaplan, C. S. and Robert Bosch. ‘TSP Art’. In Bridges 2005 Renaissance Banff, 301-308


Kappraff, Jay. ‘A Course in the Mathematics of Design’. Computers and Mathematics with Applications Vol. 12B, Nos. 3/4, 913-948, 1986 (27 September 2013)

Cairo tiling p. 923 in the context of the Laves tiling; but as such, inconsequential.


————. ‘The Geometry of Coastlines’. Computers and Mathematics with Applications. Vol. 12B, Nos 3/4, pp 655-671, 1986 (1 October 2013)

From the symmetry ‘special edition’ of the journal. Obscure.


Kazarinoff, N. D and Roger Weitzenkamp. ‘Squaring Rectangles and Squares’. The American Mathematical Monthly, Vol. 80 No. 8 (October 1973), 877-888 (12 March 2013)

From a reference in Tilings and Patterns. Academic, of no practical use


Kazancigil, Ali ed. By Emerita S. Quito: ‘Value as a factor in social action’, p. 605. International Social Science Journal Epistemology of Social Science 102 Unesco Vol. XXXV1 No. 4, 1984.

Use of Escher’s print Relativity, p. 605. Note that this is in isolation to the article, there is no accompanying text (Indirectly from a reference by Ken Wilkie, in Holland Herald)


Keeton, Greg. 'The Artist who Aims to Tease.' Reader’s Digest (March 1981): 37-41.

This is, I believe to the best of my dim and distant recollection (but still clear enough to plainly recall), my first encounter with Escher’s work, in c. 1983, but I didn’t do anything about it at the time. I inscribed on the front cover, likely in 1990 ‘saw first prob(bably) (19)83, rediscovered January (19)90’. Uses Escher’s prints: Hand with Reflecting Globe,37; Three Worlds, 38; Bond of Union, 38; Day and Night, 39, Belvedere, 40; Mobius Strip II, 41. Also of note in that no-one has referenced this article! Sent whole journal to Jeffrey Price upon request, 16 April 2010.


Kelly, J. B. ‘Polynomials and Polyominoes’. The American Mathematical Monthly Vol. 73 No. 5 464-471, May 1966 (9 April 2013)

Academic, of no use.


Kendall, M. G. ‘Who Discovered the Latin Square?’ The American Statistician. Vol. 2 No. 4 Aug. 1948 p.13 (20 March 2013)

Minor Dudeney reference


Kershner, R. B. ‘On Paving the Plane’. The American Mathematical Monthly. Vol. 75, No. 8 (October 1968) 839-844. (12 December 2010)

Of significance re the distinct convex pentagon types. Gives eight of the convex pentagon types then known

————. ‘On Paving the Plane’. APL Technical Digest. 8 No. 6 pp. 4-10 1969 (6 January 2016)
Note that there are slight, but subtle differences to the two papers of a like name by Kershner

————. ‘The Laws of Sines and Cosines for Polygons’. Mathematics Magazine Vol. 44  No.3 May 1971. 150-153
Academic in tenure, of no practical use

Kim, Scot. ‘Computer Games Based on Escher’s Spatial Illusions’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration (31 August 2005) 366-378

Kingston, J. Maurice. ‘Mosaics by Reflection’. The Mathematics Teacher (NCTM), 50(4): 280-286, April 1957 (18 February 2013)
Semi-popular, but a little obscure.

Klamkin, M. S and A. Liu. ‘Polyominoes on the Infinite Checkerboard’. Journal of Combinatorial Theory, Series A, 28 7-16, 1980 (25 September 2013)

From a reference in Tilings and Patterns. Polyominoes. Mostly of an academic nature throughout!. Of little to no practical use.

Klarner, David A. ‘Some Results Concerning Polyominoes’. Fibonacci Quarterly 3 (1965), 9-20 (26 October 2012)
Of an largely academic nature throughout, with occasional diagram ‘understandable! That said, still of no practical use. From a reference in Tilings and Patterns.

————. ‘Packing a Rectangle with Congruent N-ominoes’. Journal of Combinatorial Theory, 7, 107-115, 1980 (26 September 2013)

From a reference in Tilings and Patterns. Of an academic nature throughout; of no use.


Klarner, David A. and R. L. Rivest. ‘Asymptotic Bounds for the Number of Convex n-Ominoes’. Discrete Mathematics 8 (1974) 31-40 (23 September 2013)

From a reference in Tilings and Patterns. Of an academic nature throughout, no diagrams! Of no practical use.


Klein, Felix. ‘Vergleichende Betrachtungen uber neuere geometrische Forschungen’. Mathematische Annalen 43 1893 63-100 (31 December 2014)

From a reference in Tilings and Patterns. Of an academic nature throughout, no diagrams! Of no practical use. See below for translation


————. ‘A Comparative Review of Recent Researches in Geometry’. Bulletin of New York Mathematics Society 2 1893, 215-249 (2 January 2015)

From a reference in Tilings and Patterns. Text throughout, of no practical use. Translation of the above in Mathematische Annalen

Knoll, Eva. ‘Life After Escher: A (Young) Artist’s Journey’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration 189-198 (31 August 2005)

Koizumi, Hiroshi and Kokichi Sugihara. ‘Maximum Eigenvalue Problem for Escherization’.

The authors’ own Escherization program. (2010)

Koptsik, Vladimir A. ‘Escher’s World: Structure, Symmetry, Sense’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration (31 August 2005) 379-392

Krašek, Matjuška Teja. ‘Sharing some Common Interests of M.C. Escher’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration 199-206 (31 August 2005)


Krishnamurti, R and P. H. O’N. Roe. ‘On the generation and enumeration of tessellation designs’. Environment and Planning B, volume 6, pp.191-260 (October 2015)

From a reference in Tilings and Patterns. Wholly academic, of no use


Krizic, Michal, Jakub Solz, Alena Solkova. ‘Is There a Crystal Lattice Possessing Five-Fold Symmetry?’ Notices of the AMS Vol. 59 No. 1 22-30

References to Kepler and Penrose


Kulpa, Zenon. ‘Are Impossible Figures Possible?’ Signal Processing 5 (1983) 201-220. (17 July 1993) Hard Copy

From a reference in The Eye Beguiled


Kuratowski, Casimir. ‘Sur les coupures irréductibles du plan’. Fundamenta Mathematicae 6 1924, 130-145

From a reference in Tilings and Patterns. Text throughout, of no practical use.


Kvern, Olav, M. ‘Eschersketch – An Adventure in the World of Tessellations’. Desktop Science. Adobe magazine 43-46 Winter 1998-1999. (10 September 2007)

Tessellation tutorial




Lagae, Ares and Philip Dutré. ‘Tile Packing Problems’. 2006

Broadly, edge tile colouring, Wang tiles, only of peripheral interest


Lalvani, Haresh. ‘Structures on hyper-structures’. Structural Topology 6 1982, 13-20

One of four references in Structural Topology as mentioned in the bibliography of Tilings and Patterns; the others being Burt, Danzer and Gilbert, although there are indeed other tiling references in the journal.


Langford, C. Dudley. ‘Correspondence’. The Mathematical Gazette, Vol. 40, No. 332 May 1956 p. 97 (1 March 2013)

Drawing readers attention to MacMahon’s Cairo tiling picture in New Mathematical Pastimes. Of importance, due to Cairo tiling reference, refereeing to Rollett’s piece in the Gazette (Note 2530). Also of not in that Langford gives a different construction to MacMahon’s. Also see T. Bakos, which completes a non stated ‘trilogy’ of writings of the day


————. ‘To pentasect a pentagon’. The Mathematical Gazette, Vol. 40, No. 218 May 1956 p. 105-110 (1 March 2013)

From a reference in Dissections: Plane and Fancy


————. ‘Tiling patterns for regular polygons’. The Mathematical Gazette, Vol. 44, No. 332 May 1960 p. 105-110 (1 March 2013)

From a reference in Dissections: Plane and Fancy. The title is a little misleading, it consists of dissections of polygons


————. ‘On dissecting the dodecahedron’. The Mathematical Gazette, Vol. 51, No. 218 May 1967 p. 139-141 (1 March 2013)

From a reference in Dissections: Plane and Fancy


————. ‘Polygon dissections’. The Mathematical Gazette, Vol. 51, No. 332 May 1960 p. 139-141 (1 March 2013)

From a reference in Dissections: Plane and Fancy


————. 1538. ‘Tangrams and incommensurables’ The Mathematical Gazette 233-235(1 March 2013)


————. ‘Super Magic Squares’. The Mathematical Gazette 86-97 (1 March 2013)


————. 3133. ‘Some teaching points’. The Mathematical Gazette 155-156 (1 March 2013)


————. ‘Some missing figure problems and coded sums’. The Mathematical Gazette 247--? (1 March 2013)


Also see obituary by E.A. Maxwell. The Mathematical Gazette 314


Langford also has many other articles listed in the Gazette, mostly of a brief (few lines) nature, concerning ‘calculations’ or ‘hard’ geometry, that are strictly out of my ‘easy’ geometric remit here, and so are not listed here.


Lamontagne, Claude. ‘In Search of M.C. Escher’s Metaphysical Unconscious’. In Coxeter, et al, Eds. M.C. Escher: Art and Science. Amsterdam: North-Holland 1986. 69-82 (30 April 1994)

Lansdown, John. In ‘Escher revisited’ in ‘Not only computing - but also art’ column. Computer Bulletin. April/May 1992 (23 April 2015)

On using a computer to create Escher-like tilings, with reference to Heesch type, Visions of Symmetry by Schattsneider.


Larson, P. ‘The Golden Section in Earliest Notated Western Music’. The Fibonacci Quarterly, 16 1978 513-515 (9 September 2014)

Re golden section. From a reference in Livio


Laves, F. ‘Ebenenteilung in Wirkungsbereiche’. Zeitschrift für Kristallographie 76 (1931): 277-284. (10 December 2014)

Has Cairo tiling on p. 280, fig. 4.

Note that this article was the first of two by Laves on a listing of B.G. Escher as given to M. C. Escher (as documented in Visions…, p. 357

‘Ebenenteilung in Wirkungsbereiche’ = Level division in impact areas

————. ‘Ebenenteilung und Koordinationszahl’. Zeitschrift für Kristallographie 78 (1931): 208-241. (5 December 2014)

Somewhat disappointing, the article is mostly text, with only a few diagrams, and furthermore what there is of little consequence. Note that this article was the second of two by Laves on a listing of B.G. Escher as given to M. C. Escher (as documented in Visions…, p. 357


Le, San. ‘The Art of Space Filling in Penrose Tilings and Fractals’. On-line article, pending print

The title is somewhat misleading, in that other, non Penrose tilings feature. Escher is prominently mentioned. Le makes uses of what I term as ‘placements’.


Lee, A. J. ‘Islamic Star Patterns’. Muqarnas 182-197 (2010)


————. ‘Islamic Star Patterns’ – Notes from 1975 A. J Lee’. (10 May 2013)

A series of handwritten notes and diagrams assembled as a single document

Lee, Kevin. ‘Adapting Escher’s Rules for “Regular Division of the Plane” to Create TesselMania!’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration (31 August 2005) 393-407

Léger, Jean-François. ‘M.C. Escher at the Museum: An Educator’s Perspective’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration (31 August 2005) 408-419

Lenngren, Nils. ‘k-uniform tilings by regular polygons’. 1-23, Uppsala University report. November 2009 (September 2015)
An interesting discussion on Kepler’s tilings, with conjectures offered for Kepler’s ‘analomies’

Levy, Silvio. ‘Automatic Generation of Hyperbolic Tilings’. Leonardo, Vol. 25, No. 3 (1992) pp. 349-354 (March 2013)



Lindgren, Harry. ‘Dissecting the Decagon’. Mathematical Gazette 46 305-306 1962 (20 February 2013)

From a reference in Dissections: Plane & Fancy. Accessible.


————. ‘Going One Better in Geometric Dissections’. Mathematical Gazette 1961 94-97 (20 February 2013)

From a reference in Dissections: Plane & Fancy. Accessible.


Liu, Yang and Godfried Toussaint. ‘Uravelling Roman mosaic meander patterns: a simple algorithm for their generation’. Journal of Mathematics and the Arts. Vol. 4, No. 1, September 2010, 1-11 (10 April 2013)

Of general interest.


Liu, Yang and Godfried T. Toussaint. ‘The marble frieze patterns of the cathedral of Siena: geometric structure, multi-stable perception and types of repetition’. Journal of Mathematics and the Arts. Vol. 5, No. 3, September 2011, 115-127 (16 April 2013)

Of general interest.


Liversidge, Anthony. Interview with Roger Penrose. Omni, Vol 8 No. 9, June 1986, 66-67, 70, 73 106, 108 (29 October 2014)

Minor references to Penrose tiles in an article/interview mostly about cosmological matters


Lloyd, D. R. ‘How old are the Platonic Solids?’ BSHM Bulletin Volume 27 (2012), 131-140 (29 April 2013)


Locher, J. L. ‘The Work of M. C. Escher’. In The World of M. C. Escher. 7-29. Abradale Press 1988 (9 April 1993)


Locher, G. W. ‘Structural Sensation’. In The World of M. C. Escher. 7-29. Abradale Press 1988, 43-50 (9 April 1993)


Lockwood, E. H. ‘Colouring the faces of a Cube’. The Mathematical Gazette 180-?

(26 March 2013)
From a reference in Garcia


Loe, Brian J. ‘Penrose Tiling in Northfield, Minnesota’. (Mathematical Tourist) The Mathematical Intelligencer Vol. 17 Number 2, 1995, 54 (28 November 2011)


Loeb, A. L.  ‘Structure and patterns in science and art’. Leonardo 4 1971, 339-346


————. ‘On My Meetings and Correspondence between 1960 and 1971 with the Graphic Artist M.C. Escher’. Leonardo, Vol. 15, No. 1 (Winter, 1982) 23-27 (9 September 2010)

Most interesting. Contains reference to Von Hippel, p. 24. Donald Smits, p. 24, who I have not been able to find anything as regards his interaction with Escher. David Hawkins, p. 25, Wagenaar, p. 26


————. ‘Symmetry and Modularity’. Computers Maths with Applications Vol. 12B, Nos.1/2, 63-75, 1986 (27 September 2013)

Interesting pentagon and skew pentagon tiling discussion, 67-69


————. ‘Symmetry in Court and Country Dance’. Computers Maths with Applications Vol. 12B, Nos.3/4, pp 629-639, 1986 (27 September 2013)

General interest.


————. ‘Some Personal Recollection of M.C. Escher’. Leonardo, 318-319


————. ‘The Magic of the Pentangle: Dynamic Symmetry From Merlin to Penrose’. Computers Maths with Applications Vol. 17, 1-3, 33-48, 1989 (27 September 2013)



Lowman, E. A. ‘Some Striking Proportions in the Music of Bela Bartók’. The Fibonacci Quarterly, 9 1971 527-537 (9 September 2014)

Re golden section. From a reference in Livio


Luecking, Monica. ‘Polycairo Tiling as a Motif for Land Design’. ISAMA 2007 57-60 (9 April 2014)

Cairo tiling supposedly set in a park in downtown Austin, Texas. Actual reference to the Cairo aspect, with an equilateral pentagon. I asked for more detail in a 2014 mail, but didn’t receive a reply




Macaulay, W. H. ‘The Dissection of Rectilinear Figures’. Mathematical Gazette Vol.7 No. 113 381-388, 1914 (4 March 2013)


————. ‘The Dissection of Rectilinear Figures’. Mathematical Gazette Vol.8 No. 117, May 1915 (4 March 2013)


————. ‘The Dissection of Rectilinear Figures concluded’. Mathematical Gazette Vol.8 No. 117, May 1915 109-115(4 March 2013)


————. ‘The Dissection of Rectilinear Figures’. Messenger of Mathematics, Volume 48, 1919a. 159-165

This continues in two further volumes. In truth, there is very little here (a) of direct interest, and (b) that I can actually follow (or at least have the inclination to pursue)! Nonetheless, it is indeed gratifying in that these articles can at least be put aside knowing that there is nothing of importance that I may be missing out on.


 ‘The Dissection of Rectilinear Figures (continued)’. Messenger of Mathematics, Volume 49, 1919b. 111-121 (29 July 2011)


————. ‘The Dissection of Rectilinear Figures (continued)’. Messenger of Mathematics Volume 52, 1919b. 53-56


Macbeath, A.M. ‘The classification of non-Euclidean plane crystallographic groups’.

Canadian Journal of Mathematics 19 (1967), 1192-1205 (1 November 2013)

From a reference in Tilings and Patterns. Academic throughout, no diagrams whatsoever!. Of no practical use.


MacGillavry, Caroline H. ‘The Symmetry of M. C. Escher’s ‘Impossible’ Images’. Computers and Mathematics with Applications Vol. 12B, Nos.1/2, pp 123-138, 1986 (2 October 2013)

Popular account of Escher's prints as regards symmetry.


Macmillan, R. H. ‘Pyramids and Pavements: some thoughts from Cairo’. Mathematical Gazette. 63 251-255, 1979 (10 August 2010)  

Highly significant as regards the Cairo tiling, Cairo pentagon discussion, in depth, the fourth discussion (1979), after Dunn (1971), Gardner (1975) and  Schattschneider (1978), and the second in situ account.


Mackay, Alan L. ‘Bending the Rules. Crystallography, Art and Design’ (lecture 1997/98) (2010)

Also see his review of Visions of Symmetry.


————. ‘But What is Symmetry?’ Computers and Mathematics with Applications. Vol. 12B, Nos 1/2, 19-20, 1986 (2 October 2013)

From the symmetry ‘special edition’ of the journal. Obscure.


————. ‘Generalised Crystallography’. Computers and Mathematics with Applications. Vol. 12B, Nos 1/2, 21-37, 1986 (2 October 2013)

From the symmetry ‘special edition’ of the journal. Obscure.


————. ‘De Nive Quinquangula: on the pentagonal snowflake’. [in Russian] Kristallografiya, 909-918. English version: Soviet Physics–Crystallography 26 (1981) 517-522 (31 January 2011)

Mentioned in Grünbaum bibliography


MacMahon, P. A. ‘The design of repeating patterns for decorative work’. Journal of the Royal Society Arts. 70 Friday, June 30, 1922, pp. 567-578. Related discussion ibid. 578-582 (3 May 2012)

As received from George Andrews. Of note is that MacMahon refers to a ‘haystack’, meaning a Cairo tile, p. 573, after fig. 13, and gives the construction, as in Pastimes. This term is also interestingly used by him in Pastimes (1921). W. P. D. MacMahon (1925) also uses this word in ‘The theory of closed repeating polygons…’ . So who devised it?!

As is typical of MacMahon, his favoured method of showing tessellations is of a single diagram, notably pp.576-577, rather than, as he puts it, ‘assemblages’. p.577 also has tiles derived from the Cairo tilings


MacMahon, P. A. and W. P. D. MacMahon. ‘The Design of Repeating Patterns’. Part I. N.B. There is no Part II) Proceedings of the Royal Society A. London 101 (1922), 80-94 (30 April 2012)

The article is mostly of text, with the only diagrams on p. 89. Although the text is not particularly advanced, the lack of diagrams is a major hindrance to understanding the text. Indeed, so much so that I have not truly studied this paper.


MacMahon, W. P. D. ‘The theory of closed repeating polygons in Euclidean space of two dimensions’. Proc. London Math Society (2) 23 (1925) 75-93 (30 April 2012)

Of note is that (WPD) MacMahon refers to a ‘haystack’, meaning a Cairo tile, p. 89, after fig. 6, although this is not shown. As this term is also used in New Mathematical Pastimes, by PA, and so which of the MacMahons devised this is unclear.

This is noteworthy on account that although a tiling paper, no tilings are actually shown! instead, single diagrams are shown, with  the presumption of a tiling. MacMahon refers to ‘contact system of assemblages’, of which this is presumed to tessellate. the text in general is too difficult for me to follow.


Note that I have other MacMahon papers from Garcia’s bibliography, but these are of an academic nature of no practical use, and so are not listed here. Also see others from Garcia reference


‘MacMahon mentions’ in: Nature: October 18, 1906. Nature B, 1906, 1908, September 10, 1908? (April 2012)


Mackinnon, Nick. ‘Some thoughts on polyomino tiles’. The Mathematical Gazette 31-33 (9 April 2013)

‘Idiot-proof tiles’. See the Grünbaum follow-up on the concept


————. ‘The Portrait of Fra Luca Pacioli’. The Mathematical Gazette Vol. 77 No. 479 (July 1993)  130-219 (28 March 2013)

The defining work on the well-known painting


————. ‘Polyomino Tessellations: A Class project’. Mathematics in Schools. Vol. 18 No. 3 May 1989 8-9 (11 April 2013)

School children project


Makovicky, E. ‘Ornamental Brickwork. Theoretical and applied symmetrology and classification of patterns’. Computers and  Maths  with  Applications. Vol. 17, No. 4-6, 955-999, 1989. (30 April 2012)

Of limited, but still general interest. Of note is an interesting tessellation, with many stackings, p. 962


————. ‘Symmetrology of Art: Coloured and Generalised Symmetries’. Computers and Mathematics with Applications. Vol. 12B, Nos 1/2, pp 949-980, 1986 (8 December 2014)


Maletsky, Evan, M. Activities: ‘Designs with Tessellations’. The Mathematics Teacher, April 1974, Volume 67, Number 4, 335-338 and continued 360. (Confusingly, inside the book this is also titled as ‘Mathematics Teacher’ (first saw 1988, hard copy of  magazine of 18 June 2011, and individual articles 18 February 2013)

‘Special edition’ on tessellations, specifically concerning three Escher-inspired tessellation articles. Typical teacher attempt at the Escher-like aspect, showing no understanding of the matter.


Malcolm, Paul S. ‘Braided Polyhedra’. The Arithmetic Teacher. 386-388 (28 March 2013)

Simple braiding


Mallinson, Philip R. ‘Geometry and its Applications. Tessellations’. 1-74. (2010)

Seems to be excerpted from a book. Very pleasing, in that the basics are covered succinctly, and then it moves on to tilings by Rice, and others, and pentagons.


Mamedov, Kh. S. ‘Crystallographic Patterns’. Computers and Mathematics with Applications. Vol. 12B, Nos 1/2, 511-529, 1986 (1 October 2013)

From the symmetry ‘special edition’ of the journal. Largely obscure, but with a small tessellation section pp 527-529, of which although in general the tessellations are not particularly good, an exception is that of his ‘Unity’, of what appears to be a jailer and prisoner theme, of which has much to recommend it.


Mann, Casey. ‘A Tile with Surround Number 2’. The American Mathematical Monthly. Vol. 109 No. 4 (April 2002) pp. 383-388 (24 August 2015)

On coronas, something of which I am not particularly interested in. Voderberg spiral discussion.


————. ‘Heesch’s Tiling Problem’. The American Mathematical Monthly. Vol. 111 No. 6 June-July 2004, 509-517


————.  ‘Hyperbolic Regular Polygons with Notched Edges’. Discrete & Computational Geometry (2005) 34: 143-166 (5 November 2012)

Of an academic nature throughout; the tone of the paper is way beyond me. Of no practical use.


Mani, P. ‘Automorphismen von polyedrischen Graphen’. Mathematische Annalen 192 1971 279-303 (31 December 2014)

From a reference in Tilings and Patterns. Of an academic nature throughout, no diagrams! Of no practical use.


March, L. and R. Matela. ‘The animals of architecture: some census results on N-omino populations for N = 6. 7. 8’. Environment and Planning B, 1974, volume 1,193-216 (October 2015)

From a reference in Tilings and Patterns. Begins at a popular level, but then quickly becomes academic, of no practical use.


Marck, K. W. ‘Enkele Overeenkomsten tussen het werk van M. C. Escher en de plastiche chirugie’ (in Dutch). Ned Tindschr Geneeskd 2002 21 December 146 51, 2498-2503.  (17 June 2011).

An English abstract is given – ‘Some similarities between the work on M.C. Escher and plastic surgery’.


Marley, Gerald C. ‘Multiple Subdivisions of the Plane’. Mathematical Magazine, 47, 1974, 202-206 (8 March 2013)

From a reference in Tilings and Patterns. Academic throughout, no practical use


Markowsky, George. ‘Misconceptions about the Golden Ratio’. The College Mathematics Journal Vol. 23, No. 1 (Jan. 1992), 2-19 (2 September 2014)

Also see for contrary opinion!


Martin, G. E. ‘Polytaxic Polygons’. Structural Topology No.12 (1986) pp. 5-10. (2 February 1998)  (Also see Fontaine, A; Martin, G. E. Polymorphic Polyominoes)

Mey, Josh De. ‘Painting After M.C. Escher’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration 130-141 (31 August 2005)

Maynard, Phillip M. ‘Isohedrally compatible tilings’. (16 April 2014)

References to isohedral aspects, with many morphic tilings, albeit the text is largely beyond me. Includes Escher-like tilings of such variations


McConnell, James V. ‘Worm-Breeding Tongue in Cheek’. In Worm Runner’s Digest Vol. XVI No. 2, December 1974, p. 12-15.

Light-hearted article on Flatworms (Planarians), with Escher’s print Flatworms used. Note that this is the amended version; the piece originally from the Unesco Courier


McKennan, Geoff, Editor. ‘Lifelike Tessellations’ by Andrew Crompton. Manchester Architectural Papers 2000 pp. 17-24. (30 May 2006)

Some hints and tips on drawing lifelike tessellations. A listing of the ‘permissible’ isohedral tilings is given, as well as a brief history of lifelike tiling.


McLean, Robin K. ‘Dungeons, dragons and dice’. The Mathematical Gazette 243-256 Vol 74 469 October 1990 (22 March 2013)


————. ‘Loops of Regular Polygons’.  June-July 2000 500-510 (12 March 2013)

Largely academic


————. ‘The tiling conjecture for equiangular polygons’. The Mathematical Gazette March 2005, 28-34 (5 March 2013)

Academic, of no practical use


Mielke, Paul T. ‘A Tiling of the Plane with Triangles’. Two-Year College Math Journal.  14 1983 (8 March 2013)

From a reference in Tilings and Patterns. Academic throughout, no practical use


Miller, William. ‘Pentagons and Golden Triangles’. Mathematics in School, 2-4 September 1996


Miller, A. William. ‘Golden Triangles, Pentagons, and Pentagrams’. The Mathematics Teacher 338-341 Vol. 87, No.5, May 1994 (5 March 2013)


Millington, W. ‘Polyominoes’. Mathematics in School. 20-21 (9 April 2013)

Friezes, poster design, the cube and Pentominoes


Moore, Calvin, C. ‘Mathematical Sciences Research Institute Berkeley, California’. The Mathematical Intelligencer Vol. 6, No. 1, 1984, 59-64 (9 December 2011)

General interest


Molnar, V. and F. Molnar. ‘Symmetry-Making and – Breaking in Visual Art’. In Computers and Mathematics with Applications. Vol. 12B, Nos 1/2, pp. 291-301, 1986 (1 October 2013)

From the symmetry ‘special edition’ of the journal. Obscure.


Mozes, Shahar. ‘Aperiodic tilings’. Inventiones Mathematicae. 128, 603-611 (1997). (24 October 2012)

Largely of an academic nature throughout, with not a single diagram! Of no practical use.


Muirhead, R. F. ‘On superposition by the aid of Dissection’. Publication not known.109-112

How I came about this reference is forgotten. whatever, it is not of any greatsignificance, and is academic in tenure, with not a single diagram!


Muscat, Jean-Paul, ‘Polygons & Stars’. Mathematics in School, March 1992 25-28 (19 February 2013)

LOGO type instructions/diagrams


Myers, Joseph. ‘Tiling with Regular Star Polygons’. Publisher Unknown (24 February 2011) pp. 20-27.

A part work apparently taken from a book, but which is not stated?




Naylor, Michael. ‘Nonperiodic tiling: the Irrational Numbers of the Tiling World’. The Mathematics Teacher (5 March 2013)


Nakamura, Makoto. ‘New Expressions in Tessellating Art: Layered Three-Dimensional Tessellations’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration (31 August 2005) 207-214

Necefoğlu, Hacali. ‘Turkish Crystallographic Patterns: From Ancient to Present’

Includes Escher-like tessellations from Imameddin Amiraslan and Khudu S. Mamedov


Nemerov, Howard. ‘The Miraculous Transformations of Maurits Cornelis Escher’. Artist's Proof 3, no. 6 (Fall-Winter 1963-1964): 32-39. (7 April 2014)

Fairly lightweight treatment, no new insight gained. Shows seven prints (six from Mickelson Galleries, one from Roosevelt) Castrovalva, Day and Night, Reptiles, Three Worlds, Another World, Relativity, Three Spheres


Nelson, David R. and Betrand I. Halperin. ‘Pentagonal and Icosahedral Order in Rapidly Cooled Metals’. Science. 19 July 1985 Vol. 229, No. 4710 (March 2013)

Quasicrystal, academic


Niman, John and Jane Norman. ‘Mathematics and Islamic Art’. 1978 489-490 (18 February 2013)

Discussion at a popular level, no drawings or pictures


Niven, Ivan. ‘Convex Polygons that Cannot Tile the Plane’. The American Mathematical Monthly 85, December 1978 785-792 (8 March 2013)

Mostly academic, a few simple diagrams. The premise of this article is the old chestnut of convex polygon of seven or more sides cannot tile, widely quoted but never shown, and put succinctly Niven proves it.


Norgate, M. ‘Non-Convex Pentahedra’. The Mathematical Gazette (18 February 2013) 115-124

Academic nature throughout




O’Beirne, T. H. ‘Puzzles and Paradoxes’ (?): New Scientist. (No.258) 26 October 1961. 261 (7 February 1998)


————. ‘Puzzles and Paradoxes 44: Pentominoes and Hexiamonds’. New Scientist (No.259) 2 November 1961. 316-317 (7 February 1998)


————. ‘Puzzles and Paradoxes 45: Some Hexiamond solutions: and an introduction to a set of 25 remarkable points’. New Scientist. (No.260) 9 November 1961. 379 (7 February 1998)


————. ‘Puzzles and Paradoxes 50: Thirty-six triangles make six hexiamonds make one triangle’. New Scientist (No.265) 14 December 1961. 706-707 (7 February 1998)


————. ‘Puzzles and Paradoxes 51: Christmas Puzzles and Paradoxes’. New Scientist (No.266) 21 December 1961. 752-753 (7 February 1998)


————. ‘Puzzles and Paradoxes 55: Some tetrabolical difficulties’. New Scientist. (No. 270) 18 January 1962. 158-159. (27 March 1993)


O’Keefe, M and B. G. Hyde. ‘Plane Nets in Crystal Chemistry’. Philosophical Transactions Royal Society London. Series A, 295 1980, 553-618 (8 March 2013)

From a reference in Tilings and Patterns. Academic throughout, of no practical use


Oliver, June. ‘Symmetries and Tessellations’. Mathematics in School, Vol. 8 No.1 January, 2-5, 1979 (18 February 2013)

How to Escher-guide, typical teacher lack of understanding of issues, own illustrations belying lack of knowledge. That said, a ‘spider and web’ tessellation does indeed show a little imagination, albeit of a novelty level.


Ollerton, Mike. ‘Dual Tessellations’. Mathematics in School. January 2000. p. 9 (20 February 2013)

This seems to be a small innovation of Ollerton’s devising. Mostly duals are taken from the semi-regular tilings, but here he uses a dual of a right-angled triangle tessellation, with his own notation to describe this.

Orosz, István. ‘The Mirrors of the Master’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration (31 August 2005) 215-229

Orton, Tony. ‘From Tessellations to Fractals’. Mathematics in School. March 1991 30-31

Fractals based on a equilateral triangle. This lead to an extensive study, but somewhat overblown on my part.


————. ‘Tessellations in the curriculum’. Mathematics in School. Vol. 24, No.4. September 1994 12-15 (19 April 2013)

Lightweight in the extreme.


————. ‘Circle Tessellations’. Mathematics in School. May 1998 13-21 (19 April 2013)

Tessellations derived from various grids of circles. Master copies of grids are provided


Orton, T. and S. M. Flower. ‘Analysis of an ancient tessellation’. The Mathematical Gazette December 1989, Vol 73, No. 466, 297-301. (2010)  and (25 February 2013)

The ‘Hammerhead’ tessellation, as described by the authors, of interest. Bob Burn comments on this in…


Osborn, J. A. L. ‘Amphography: The Art of Figurative Tiling’. Leonardo, Vol. 26 No. 4 289-291, 1993 (9 September 2010).

A brief exemplar of Osborn’s philosophy, with his own term of ‘amphography’. Usual shortcoming as to pretentious. For instance, a claim is made for ‘ultra realistic’ as regards a sub species of bats, but no corroborating real life picture is shown.


————. ‘Diminishing Opportunity in Amphography’. In Symmetry: Culture and Science,  editors Gyorgy Darvas and Denes Nagy, Vol. 6, Number 3, 418-421, 1995 (17 September 2013)

Broadly philosophical speculations by Osborn on the reducing numbers of possible future life-like tessellation. This is noteworthy for the ridiculous statement by Osborn ‘…Escher foreclosed forever the possibility of any subsequent artist employing this user friendly geometry for the Amphographic depiction of any even remotely related subject matter without incurring the epithet ‘imitator, or ‘derivative’, or even plagiarist’. Absurd. So every painter copies from caveman art? Every mathematician copies from Euclid? Every artist is copying Escher? No, no, no; people build on cavemen, build on Euclid, and build on Escher, not plagiarising.

Also see his two patents’ ‘Variably Assemblable [sic] Figurative Tiles for Games, Puzzles, And For Covering Surfaces’ and Single-Shape Variably Assemblable [sic] Figurative Tiles for Games, Puzzles, And For Covering Surfaces’. And also a self-published 14 page booklet concerning his ‘The Bats and Lizards How-To-Play book’, a guideline to ‘his’ Bats and Lizards tiles


Osborne, Harold. ‘Symmetry as an Aesthetic Factor’. Computers and Mathematics with Applications. Vol. 12B, Nos 1/2, 77-82, 1986 (2 October 2013)

From the symmetry ‘special edition’ of the journal. Obscure.


Ostromoukhov, Victor. ‘Multi-Color and Artistic Dithering’. Siggraph 1999.  (2010)

Of no interest beyond a Cairo tile reference that cannot be seen! A sample image produced using a threshold matrix inspired by the Cairo tessellation. Figures (a)-(e) show the building process of the threshold matrix’.


Özdural, Alpay. ‘On Interlocking Similar or Corresponding Figures and Ornamental Patterns of Cubic Equations’. 191-211 (2010).

No citation for this article. Mostly of geometric constructions rather than interlocking figures. The original Arabic manuscripts are shown at the end of the article. Like most of Özdural’s writings, not a particularly easy read.


————. ‘Mathematics and Arts: Connections between Theory and Practice in the Medieval Islamic World’. Historia Mathematica 27 (2000), 171-201. (30 April 2012)

Discusses Arabic geometers of yesteryear, notably Abu’l-Wafa. Of little practical use.


————. ‘The Use of Cubic Equations in Islamic Art and Architecture’. Source unknown (30 April 2012)




Palmer, Chris K. ‘Spiral Tilings with C-curves Using Combinatorics to Augment Tradition’. In Bridges Renaissance Banff 2005, pp. 37-46


Pargeter, A. R. ‘Plaited Polyhedra’. The Mathematical Gazette, Vol. 43, No. 344 (May, 1959) 88-101

As quoted in Mathematical Models by H. Martyn Cundy and A. P. Rollett, p. 152


Parker, John. ‘Tessellations’, Topics, Mathematics Teaching 70, 1975, p. 34 (3 May 2012)

Cairo-esque pentagon, p. 34. Parker states that this is a footnote to the article by Clemens of the same journal


————. ‘Dissections’. Mathematics in School, September 1992 p. 5-9. (20 February 2013)

Begins of a accessible level, then moves onto academic ground


————. ‘The Ratchet’. Mathematics in School, November 1998 p. 36-38. (25 February 2013)

Discusses the various and numerous ways in which a ‘ratchet’ tessellation as shown can be composed.


————. ‘The rhomboid and its parts’. Mathematics in School, January 1998 p. 13-15. (19 March 2013)


Parviainan, Robert. ‘Connectivity Properties of Archimedean and Laves Lattices’. Uppsala Dissertations in Mathematics 34. p. 9. 2004.

Quote: The lattice D (32. 4. 3. 4) is sometimes called the Cairo lattice, as the pattern occurs frequently as tilings on the streets of Cairo.

A fleeting mention of the Cairo tiling in the context of a study on Laves tilings.


Penrose, L. S. and Penrose R. ‘Impossible Objects: A Special Type of Visual Illusion’. British. Journal of Psychology. Vol.49, No.1, (1958). 31-33

(Reprinted in The Eye Beguiled by Bruno Ernst, p. 72-73).


Penrose, R. ‘On the Cohomology of Impossible Figures’. Structural Topology 17, 1991 11-16. (c. 2008?)

Largely academic. Impossible tribars, p. 12


————. ‘Pentaplexity. A Class of Non-Periodic Tilings of the Plane’. The Mathematical Intelligencer Vol. 2 Number ?, 1?, 4-11 or 32-37, 1979 (25 November 2011)

Penrose tiles, a popular account. The article is a reprint from the ‘Archimedeans’ of Cambridge University, which first appeared in Eureka No. 39 (1978), 16-22.


Perigal, Henry. ‘On Geometric Dissections and Transformations’. Messenger of Mathematics, Volume 2, 1873. 103-106

From a  reference in Dissections: Plane & Fancy


————. ‘Geometrical Dissections and Transformations. No. II’. Messenger of Mathematics, Volume 4, 1875. 103-104

From a  reference in Dissections: Plane & Fancy


Perisho, Clarence R. ‘Colored polyhedra: a permutation problem’. The Mathematics Teacher April 1960, 253-255. (26 March 2013)

From reference in Garcia


Petersen, Mark A. ‘The Geometry of Piero della Francesca’. The Mathematical Intelligencer. Vol. 19, No. 3, 1997, 33-40 (2 December 2011)

General and academic


Peterson, Ivars. ‘The Fivefold Way for Crystals’. Science News. Vol. 127 23 March 1985. (March 2013)

Crystallography inclined, a little obscure in places

(Reference in Frederickson)


————. ‘Shadows and Symmetries’. Science News. Vol. 140 408-410. (13 March 2013)


————. ‘The Honeycomb Conjecture’. Science News. Vol. 156 60-61. July 24 1999 (13 March 2013)


————. Pieces of a polyomino puzzle. Science News. Vol. 132 310. (9 April 2013)

On Karl A. Dahlke


Platt, Charles. ‘Expressing the Abstract’. In New Worlds Speculative Fiction. July 1967  44-49 (February 2016)

Illustrated with nine artworks: Relativity (cover), Mobius Strip, Horseman, Pegasus, Reptiles, Dragon, Liberation, Three Worlds, High and Low. An intersecting aside (p. 46) is of the Reptile print ‘as an optical illusion  in a colour supplement feature’. Does anyone know what this is referring to?


Pomerance, Carl. ‘On a Tiling Problem of R. B. Eggleton’. Discrete Mathematics 18 (1977) 63-70 (23 September 2013)

From a reference in Tilings and Patterns. Of an academic nature throughout, no diagrams! Of no practical use.


Post, K. A. ‘Regular Polygons with Rational Vertices’. Mathematical Gazette 62 (1978) 205-206

(8 March 2013)

From a reference in Tilings and Patterns. Academic throughout, no practical use


Post, Diana and Munro Meyersburg. ‘Celebrating Rachel Carson (1907-1964) In Her Centennial Year’. Rachel Carson Council Inc. March 2007 (October 2015)

Use of Escher’s Metamorphosis print throughout discussion.

Prete, Sandro Del. ‘Between Illusion and Reality’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration (31 August 2005)

Propp, James. ‘A Pedestrian Approach To a Method of Conway, or, A Tale of Two Cities’. Mathematics Magazine, Vol. 70, No.5 December 1997, 327- (12 April 2013)

Polyominoes, begins at a popular level, then becomes academic




Radin, Charles. ‘Symmetry of Tilings of the Plane’. AMS Bulletin Volume 29 No. 2 October 1993 (12 December 2012)

Radin’s numerous papers are typically highly academic, way beyond me. As he pleasingly makes these available for download, I here only record only the more, in relative terms, accessible instances.


————. ‘The Pinwheel tilings of the plane’. Annals of Mathematics.  139, (1994) 661-702 (12 December 2012)

Of an academic nature throughout; the tone of the paper is way beyond me. Of no practical use.

Raedschelders, Peter. ‘Tilings and Other Unusual Escher-Related Prints’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration (31 August 2005) 230-243

Ranucci, Ernest R. ‘Tiny Treasury of Tessellations’. The Mathematics Teacher (NCTM) 61(2): 114-117, 1968 (15 February 2013)

A simple article on tessellation per se, without Escher-like aspects. Of note is that this has his own version of a Cairo tiling, albeit of different sized pentagons. Also, he uses the term ‘par hexagon’, perhaps taken for one of his references, by Kasner and Newman.


————. ‘Master of Tessellations: M.C. Escher, 1898-1972’. The Mathematics Teacher, April 1974, Volume 67, Number 4, 299-306. (Confusingly, inside the book this is also titled plainly as ‘Mathematics Teacher’ (18 June 2011)

A brief discussion on Escher’s tessellations and prints: Coast of Amalfi, Castrovalva, Eight Heads, Day and Night, Fish and Scales, Convex and Concave and Belvedere, both tessellation based and non tessellation, nothing of any significance, the article consists mostly of pictures.

Note that this articles part of a ‘special edition’ on Escher-like tessellations, by Ranucci, Teeters, and Maletsky


————. ‘Space-Filling in Two Dimensions’. The Mathematics Teacher (NCTM) November 1971 64 (11): 587-593 (16 February 2013)

Somewhat advanced, of little direct interest. no Escher like tessellations


————. ‘Cutting Candles’. Mathematics in School Vol. 2 No. 6 (Nov.1992 pp. 24-25. (16 February 2013)

General interest, from an idea in Mathematical Snapshots, by Hugo Steinhaus


————. ‘Function follows form’. The Arithmetic Teacher 278-282.

General interest.


————. ‘The World of Buckminster Fuller’. The Mathematics Teacher October 1978, 568-577

(16 February 2013)

General interest.


————. ‘On Skewed Regular Polygons’. The Mathematics Teacher, March 1970, 219-222 (16 February 2013)

General interest.


————. ‘Fruitful Mathematics’. Mathematics Teacher January 1974, 5-14 (16 February 2013)

Sphere packing


Note that Ranucci was a prolific author, of mostly school-orientated material that is generally understandable, and so I trawled the JSTOR archives for any such articles, although the distinction between usefulness and those of lesser interest is no easily demarked. I have many other articles by Ranucci of no special importance to me (being non-tessellation and polyhedra), and so for this reason they are not listed in any great detail here in the main listing, instead I simply list the titles: Jungle-Gym Geometry, Isosceles, Permutation Patterns, On the Occasional Incompatibility of Algebra and Geometry, Of Shoes-and Ships-And Sealing Wax-of Barber Poles and Things, Tantalizing ternary, applications


Rawsthorne, Daniel A. ‘Tiling Complexity of small n-ominoes’. Discrete Mathematics 70 (1998 71-75 (27 September 2013)

Fairly popular level, of general interest


Reyes, Encarnación.  Cuando la geometría se hace arte in Enfoques actuales en la didáctica de las Matemáticas 9-31 (26 January 2016) Formas estrelladas y de otros tipos en elementos artísticos 31

Inmaculada Fernández Benito


Redondo Buitrago, Antonia and Encarnación Reyes. ‘The Geometry of the Cordovan Polygons’, Visual Mathematics 2008b 10, 4.

Cairo-like tiles p. 12, based on the ‘Cordovan proportion’. Also see p. 14. Also of note is that this paper mentions a par hexagon.


Reeve, J. E. and J. A. Tyrrell. ‘Maestro Puzzles’. The Mathematical Gazette 97-99 (20 March 2013)

Polyominoes, polyiamonds


Reid, Michael. ‘Tiling with Similar Polyominoes’. Journal of Recreational Mathematics. 31 (2002-2003) No 1 15-24 (8 November 2102)

Very accessible.


Reinhardt, Karl ‘Über die Zerlegung der hyperbolischen Ebene in konvexe Polygone’ Jahresbericht der Deutschen Mathematiker-Vereinigung. 37 330-332 1928 (31 December 2014)

From a reference in Tilings and Patterns. Academic throughout, no diagrams, of no practical use


Rhoads, Glenn C. ‘Planar tilings by polyominoes, polyhexes, and polyiamonds’. Journal of Computational and Applied Mathematics. 174 (2005) 329-353 (8 January 2015)

Of particular interest in regards of polyhexes, of pp.336-338.

Rice, Marjorie. ‘Escher-Like Patterns from Pentagonal Tilings’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration (31 August 2005) 244-251
Of the utmost interest, of pentagon tilings, one of my own research interests, if I can describe my efforts worthy of the title

Richardson, Bill. ‘A short tale on two small tiles’. Mathematics in Schools, Vol. 29, No. 1, January 2000, 16-17 (2 April 2013)

Quadrilateral tiling leading to Penrose tiles.


Richmond, C. A. ‘Repeating Designs in Surfaces of Negative Curvature’. American Mathematical Monthly, 44, 1937, 33-35 (8 March 2013)

From a reference in Tilings and Patterns. Academic throughout, no practical use


Richmond, H. W. 1282. ‘A topological puzzle’. The Mathematical Gazette

Dudeney utilities reference, not in Frederickson


Rigby, John F. ‘Napoleon, Escher, and Tessellations’. Structural Topology. 17, 1991, 17-23. (C. 2008?) Also appeared in Mathematics Magazine 64 (1991) 242-246

On Escher’s ‘pure tiling’ conjecture. Of limited interest.

————. ‘Escher, Napoleon, Fermat, and the Nine-point Centre’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration (31 August 2005) 420-426

————. ‘Precise Colourings of Regular Triangular Tilings’. The Mathematical Intelligencer. Vol 20, Number 1, 1998, 4-11 (25 November 2011)


————. ‘A Turkish interlacing pattern and the golden ratio. Whirling dervishes and a geometry lecture in Konya’. Mathematics in Schools. January 2005 16-24 (18 February 2013)

Begins at a popular level, and then become progressively academic. Still much of interest though


————. 79.51 ‘Tiling the plane with similar polygons of two sizes’. The Mathematical Gazette 560-561. (18 February 2013)

Largely academic throughout, no practical use


Richert, Michael and Franz Gähler. ‘Cluster Models of Decagonal Tilings’. (2010)

Penrose-like material, somewhat advanced, but of some interesting diagrams 2003. 


Roberts, David L. ‘Albert Harry Wheeler (1873-1950): A Case Study in the Stratification of American Mathematical Activity’. Historia Mathematica 23 (1996), 269-287 Article No. 0028

On Harry Wheeler’s polyhedra.. This appears to be the main reference to Wheeler. I believe he first came under my orbit from Frederickson, ‘Dissections….’ pp. 141,145 (biography), 236-235. Oddly no mention is made of dissection in the Roberts article.


Roberts, Siobhan. ‘A Reclusive Artist Meets Minds with a World-Famous Geometer: George Odom and H. S. M. (Donald) Coxeter’. Leonardo Vol. 40 No. 2, 175-177, 2007 (28 March 2013)

On George Odom’s polyhedra, and interaction with Coxeter. Odom himself is a most interesting character.


Roberts, Siobhan and Asia Ivić Weiss. ‘Donald in Wonderland: The Many Faceted Life of H. S. M. Coxeter’. The Mathematical Intelligencer Vol. 26 Number 3, 2004 (25 November 2011)


Robinson, E. Arthur. ‘The Dynamical Properties of Penrose Tiling’. Transactions of the American Mathematical Society Vol. 348, Number 11, November 1996

Of an academic nature throughout, of no practical use


Robinson, J. O. and J. A Wilson. ‘The Impossible Colonnade and Other Variations of a Well-Known Figure’. Brit. Journal of Psychology 64, 3 (1973) 363-365. (10 August 1993). Hard Copy


Robinson, Raphael M. ‘Undecidability and Nonperiodicity for Tilings of the Plane’. Inventiones Mathematicae. 12, 177-209 (1971). (24 October 2012)

From a reference in Tilings and Patterns. Of an academic nature, of which some parts are broadly readable, but of no practical use. The premise is of the Hao Wang conjecture.


————. ‘Multiple Tilings on n-Dimensional Space by Unit Cubes’. Mathematische Zeitschrift 166, 225-264 (1979) (24 October 2012)

From a reference in Tilings and Patterns. Of an academic nature throughout, with not a single diagram! Of no practical use.


————. ‘Undecidable Tiling Problems in the Hyperbolic Plane’. Inventiones Mathematicae. 44, 259-264 (1978). (24 October 2012)

From a reference in Tilings and Patterns. Largely of an academic nature throughout, with not a single diagram! Has minor recreational references to Gardner, Penrose and Ammann, but is still of no practical use.


Robinson, Sara. ‘M.C Escher: More Mathematics Than Meets the Eye’. SIAM news, Volume 35, Number 8, 1-4, 2002 (11 July 2011)

Examination of ‘Print Gallery’ type effect, with H. Lenstra quoted


Robinson, S. A. ‘Classifying triangles and quadrilaterals’. The Mathematical Gazette 38-.(19 February 2013)

Of an academic nature throughout, of no practical use

Roelofs, Rinus.  ‘Not the Tiles, but the Joints: A little Bridge Between M.C. Escher and Leonardo da Vinci’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration (31 August 2005) 252-264

Rollett, A. P. ‘A Pentagonal Tessellation’. The Mathematical Gazette, Vol 39, No. 329 (Sep. 1955) p. 209, Note 2530 (2 April 2012).

Cairo-like diagram p. 209, but without the attribution, of interest due to so early an instance. Also included are other references to its sighting; a school in Germany (speculating, of Villeroy and Bosch?), and The Listener. Also see C. Dudley Langford and (Correspondence) T. Bakos 2801 for follow-up to this article


Rollings, Robert Wheadon. ‘Polyhedra expressed through the beauty of wood’. Journal of Mathematics and the Arts. Vol. 4, No. 4 December 2010, 191-199 (10 April 2013)

Of general interest re polyhedra.


Rose, Bruce I. and Robert D. Stafford. ‘An Elementary Course in Mathematical Symmetry’. American Mathematical Monthly, 88, 1981, 59-64 (8 March 2013)

From a reference in Tilings and Patterns. Academic throughout, no practical use


Rosenbaum, Joseph. Problem E721?



Roth, Richard L. ‘Color Symmetry and Group Theory’. Discrete Mathematics 38 (1982) 273-2963 (23 September 2013)

From a reference in Tilings and Patterns. Of an academic nature throughout, no diagrams! Of no practical use.


Rowe, David E. ‘Coxeter on People and Polytopes’. (In ‘Years Ago’ column). The Mathematical Intelligencer Vol. 26, Number 3, 2004, 26-30 (22 December 2011)

Minor Escher reference p. 30


————. ‘Herman Weyl, the Reluctant Revolutionary’. (In ‘Years Ago’ column). The Mathematical Intelligencer Vol. 25, Number 1, 2003, 61-70 (22 December 2011)

General interest


————. ‘Puzzles and Paradoxes and Their (Sometimes) Profounder Implications’. The Mathematical Intelligencer Vol. 33, Number 1, 2011, 55-? (In ‘Years Ago’ column). (29 December 2011)

In remembrance of Martin Gardner. Minor Martin Gardner reference, square to rectangle paradox.


————. ‘From Königsberg to Göttingen: A Sketch of Hilbert’s Early Career’. (In ‘Years Ago’ column). The Mathematical Intelligencer Vol. 25, Number 2, 2003, 61-70 (22 December 2011)

General interest


————. ‘On Projecting the Future and Assessing the Past – the 1946 Princeton Bicentennial Conference’ The Mathematical Intelligencer Vol. 25, Number 4, 2003, 8-15 (31 December 2011)

Limited interest


————. Euclidean Geometry and Physical Space. The Mathematical Intelligencer Vol. 28, Number 2, 2006, 51-59 (31 December 2011)


————. Felix Klein, Adolf Hurwitz, and the Jewish Question in German Academia The Mathematical Intelligencer Vol. 29, Number 2, 2007, 18-? (31 December 2011)

December 2011) In ‘Years Ago’ column).

General interest


Rózca, Erzsébet. ‘Symmetry in Muslim Arts’. Computers and Mathematics with Applications. Vol. 12B, Nos 3/4, pp725-750, 1986 (2 October 2013)

From the symmetry ‘special edition’ of the journal. Obscure.


Ruane, P.N. ‘The curious rectangles of Rollett and Rees’. (2010)

Of limited interest


Rush, Jean C. ‘On the Appeal of M. C. Escher’s Pictures’. Leonardo, Vol. 12, No. 1 (Winter, 1979), pp. 48-50 (9 September 2010).

Also see another letter in Leonardo ‘On the Appeal of M. C. Escher’s Pictures cont.’


————. ‘On the Appeal of M. C. Escher’s Pictures’ (Cont). Leonardo p. 174 (and reply by Emmer below) (17 February 2013)

Debating (among other matters) on who devised the impossible triangle; Escher or Penrose




Sadahiro, Yukio. ‘An exploratory method for analyzing a spatial tessellation in relation to a set of other spatial tessellations’. Environment and Planning A, 2002, volume 34, pp. 1037-1058 (October 2015)

Unlike other articles from Environment and Planning mentioned in Tilings and Patterns, this one is not given. I suspect I found it upon a general search in their archives for ‘tessellation’. Whatever, it is of no practical use.


Sallows, Lee. ‘The Lost Theorem’. The Mathematical Intelligencer. Vol. 19, No. 4, 1997, 51-54 (3 January 2012)

Magic squares


————. ‘More on Self-Tiling Tile Sets’. Mathematics Magazine. Vol. 87, No. 2 April 2014 100-112. (2015)

On rep-tiles.


Sallows, Lee and Martin Gardner, Richard K. Guy, Donald Knuth. ‘Serial Isogons of 90 Degrees’. Mathematics Magazine 315-324 (5 March 2013)

Begins at a popular level, then turns academic


Samyn, Phillipe and partners. Hotex – Village de Toile. Lacs de l’Eau d’Heure (Belgiqiue)

c. 28 May 2010 (18 December 2012)

Cairo tiling architecture


Sands, A. D and S. Swierczkowski. ‘Decomposition of the line in isometric three-point sets’. Fundamenta Mathematicae 48 1960 361-362 (7 January 2015)

From a reference in Tilings and Patterns. Of an academic nature throughout, no diagrams, of no practical use


Sarhangi, Rezi. ‘The Sky Within: Mathematical Aesthetics of Persian Dome Interiors’ pp.145-156. In Bridges 1999 (7 March 2006)


————. All Bridges articles, from 1998 to present day


The Geometry and Art of Tesselation ISAMA 2007 223- (9 April 2014)

Cairo tiling p. 226, albeit in the context of pentagon tiling possibilities; no reference is made to the Cairo aspect.


Sauer, Robert. ‘Ebene gleicheckige Polygongitter’ Jahresbericht der Deutschen Mathematiker-Vereinigung. 47 115-124 1937 (31 December 2014)

From a reference in Tilings and Patterns. Academic throughout, occasional simple diagrams, of no practical use


Sawada, Daiyo. ‘Symmetry and Tessellations from Rotational Transformations on Transparencies’. Arithmetic Teacher. December 1985 12-13. (23 February 2013)


Schattschneider, Doris. ‘Tiling the Plane with Congruent Pentagons’. Mathematics Magazine Vol.1, 51, No.1 January 1978. 29-44. (13 February 1996 and 2010)

Of fundamental importance concerning tiling with pentagons, full of interest, and all largely accessible. ‘Cairo tiling’ as a term is mentioned, as an Archimedean dual, p. 30, with three references: to likely Gardner’s article (as Macmillan does not get a mention in the bibliography, but it could be Dunn), Coxeter’s cover, and Escher’s usage of the tiling


————. ‘The Plane Symmetry Groups: Their Recognition and Notation’. American. Mathematical. Monthly June-July 1978. 439-450. (3 October 1996)

Largely of an academic nature, and the subject itself is of limited interest. Two uses of Escher's tessellations, p. 440. Tilings in the form of Chinese lattice designs, pp. 444-445


————. ‘Will it Tile? Try the Conway Criterion!’ Mathematics Magazine. 53, No. 4, 224-233. 1980 (13 February 1996)

Of both academic and popular nature. Rightly or wrongly, it has had no practical application in my studies. Figure 6 is an obvious fish, of which I haven’t found the time to compose. Needs a re-read


————. ‘In Black And White: How To Create Perfectly Colored Symmetric Patterns’. Computers and Mathematics. with Applications. Vol. 12B, Nos. 3/4. 673-695, 1986. (9 September 2010)

As such, of limited interest as regards tessellation. This borders on the popular and academic, and in relation to tessellation per se is of little value.


————. ‘The Pólya-Escher Connection’. Mathematics Magazine Vol. 60, No. 5. (Dec. 1987)  292-298 (17 March 2010)

Contains a previously unpublished page from Escher's sketchbook, which is of some significance, in that it shows how Escher formed his Eagle motif (PD 17), by fusing two tiles. This formation had previously escaped me. This is all the more galling, in that the information was available from as far back as 1987 with this article, but it took me 23 years to find it! Much of the material here later appears in Visions. The background to the creation of the Eagle motif is discussed in Visions, p. 289, of which I was aware of previously, but here the fusing is not mentioned, and so I couldn’t understand Schattsneider’s belief at the time as to the accreditation. This article reveals it.


————. ‘Escher: A Mathematician In Spite Of Himself’. In: Structural Topology No.15 1988. 9-22 (Escher special edition). (28 March 2011)

This largely features aspects arising from Escher’s notebooks of 1941-1942, in which Schattschneider examines his mathematics.


————. ‘The Fascination of Tiling’. Leonardo, Vol. 25, No. 3/4, pp. 341-348, 1992 (15 September 2010).

Full of interest; various aspects; Escher, Rice, pentagons, Penrose, kites and darts, rep-tiles


————. ‘Escher’s Metaphors’. Scientific American 271, No. 5 66-71 November 1994 (13 June 2011)

Somewhat curious; the premise here is unclear, and there is nothing that has not been discussed before in Visions.


————. ‘Math and Art in the Mountains’. The Mathematical Intelligencer (Mathematical Communities column) Vol. 28 31-37, Number 3, 2006 (24 November 2011)

Talking about the Banff Bridges Conference of 2005


————. ‘The Mathematical Side of M. C. Escher’. Notices of the AMS, Vol. 57, Number 6, June/July 2010. 706-718 (2010)

Although full of interest, this largely covers ground already discussed in Visions, but new is Escher analysis of Coxeter’s diagram, and occasional snippets of interest, such as with Speiser.


————. Afterword. On pentagon article. source, date not recorded (24 November 2009)


————. ‘Escher’s Combinatorial Patterns’. The Electronic Journal of Combinatorics 4 (NO.2) (1997), #R17. ‘Potato printing game’ of Escher updated. (29 May 2007)

An examination of Escher’s ‘combinatory tile’ problem. Of very little interest in itself; it’s really an instance of personal study to the person concerned.


————. ‘Mathematics and Art’. Math Awareness Month – April 2003. Web. (6 November 2007)


————. ‘The Mystery of the MAA Logo’. Mathematics Magazine 18 (25 April 2013)

On the icosahedron logo of the journal


N.B. Also see Ding, Ren; Doris Schattschneider, Tudor Zamfirescu. ‘Tiling the Pentagon’. Discrete Mathematics. 221 (2000)113-124. (8 September 2010)

Academic. Dissections (subdivisions) of the pentagon by pentagons; highly technical. Of limited interest

Also see Review section


Scher, Daniel. ‘Lifting the Curtain: The Evolution of the Geometer’s Sketchpad’. The Mathematics Educator 42-48 (14 August 2014)

General interest of the development


Scherer, Karl. ‘The impossibility of a tesselation of the plane into equilateral triangles whose sidelengths are mutually different, one of them being minimal’. Elemente der Mathematik 38 1983, 1-4 (2 January 2015)

From a reference in Tilings and Patterns. Of an academic nature throughout, diagrams, but too advanced!


Schulte, Egon. ‘The Existence of Non-tiles and Non-facets in Three Dimensions’. Journal of Combinatorial Theory, Series A 38, 75-81, 1985 (24 September 2013)

From a reference in Tilings and Patterns. Of an academic nature throughout! Too advanced for me. Of no practical use.


Schwarzenberger, R.L. E. ‘The 17 Plane Symmetry Groups’. Mathematical Gazette 58 1974 123-131 (11 March 2013)

From a reference in Tilings and Patterns. Of an academic nature throughout, with not a single diagram! (albeit the author states why) Of no practical use.

Seccaroni, Claudio and Marco Spesso. ‘Architecture, Perspective and Scenography in the Graphic Work of M.C. Escher: From Vredeman de Vries to Luca Ronconi’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration (31 August 2005) 265-274

Senechal, Marjorie. ‘Color Groups’. Discrete Applied Mathematics 1 (1979) 51-73 (23 September 2013)

From a reference in Tilings and Patterns. Of an academic nature throughout, minimal diagrams! Of no practical use.


————. ‘Which Tetrahedra Fill Space?’ Mathematics Magazine, Vol 54, No. 5, November 1981 227-243 (18 February 2013)

Largely popular account; some academic


————. ‘Coloring Symmetrical Objects Symmetrically’. Mathematics Magazine, Vol. 56, No. 1, January 1983, 3-16 (19 February 2013)

From a reference in Tilings and Patterns. Of an academic nature throughout, no practical use


————. ‘Geometry and Crystal Symmetry’. Computers and Mathematics with Applications. Vol. 12B, Nos 3/4, pp 565-578, 1986 (1 October 2013)

From the symmetry ‘special edition’ of the journal. Obscure.


————. ‘The Algebraic Escher’. In Structural Topology No.15 31-42 1988. Escher Special edition (28 March 2011)

Largely of group theory, academic, of little practical use


————. ‘Tiling the Torus and Other Space Forms’. Discrete Computer Geometry  3:55-72 (1988)



————. ‘Orderly Dispositions in Space’. March 1988. A report on a workshop meeting. (13 December 2012)


(18 February 2013)


————. ‘Symmetry Revisited’. Computers and Mathematics with Applications. Vol 17, No. 1-3, 1-12. 1989 (27 September 2013)

Cairo diagram as in the context of the set of 11 Laves diagrams, p. 9; as such per se, inconsequential. Of note is the poor accuracy of the Cairo drawing - it appears to show an equilateral pentagon (or is at least intended), and not the Archimedean dual!


————. ‘Tilings, quasicrystals, and Hilbert’s 18th problem’ (lower case as in article). Structural Topology No. 20 7-26. 1993.  (2008?)

No Cairo tiles. Escher tiling E128 (ghosts) p. 9. Mostly of two rhomb tiling

Senechal, Marjorie. ‘The Symmetry Mystique’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration (31 August 2005) 427-441

————. ‘Coxeter and Friends’. The Mathematical Intelligencer Vol. ? Number ?, 2004, 16 ‘Mathematical Communities’ column. (28 November 2011)


————.Parallel Worlds: Escher and Mathematics, Revisited’. The Mathematical Intelligencer Vol. 21 Number 1, 1999, 13-19 (24 November 2011)

Note that this is reprinted in M.C. Escher’s Legacy


————. ‘The Mysterious Mr. Ammann’. (Mathematical Communities) The Mathematical Intelligencer. Vol. 26 Number 4, 2004, 10-21 (25 November 2011)


————. Martin Gardner tribute (1914-2010). The Mathematical Intelligencer. Vol. 33 Number 1, 2011, 51-54 (25 November 2011)


————. ‘What is… a Quasicrystal?’ Notices of the AMS. September 2006, 886-887. (2010)

Somewhat advanced.

———— ‘Parallel Worlds: Escher and Mathematics, Revisited’. In Coxeter, et al, Eds. M.C. Escher: Art and Science. Amsterdam: North-Holland 1986. 83-91(30 April 1994)

Senechal, Marjorie and Jean Taylor. ‘Quasicrystals: The View from Les Houches’. The Mathematical Intelligencer. Vol. 12 Number 2, 1990, 54-64 (28 November 2011)


Senechal, Marjorie and R. V. Galiulan. ‘An Introduction to the Theory of Figures: the Geometry of E. S. Federov’. Structural Topology No. 10 1984 5-20 (2010)


Sequin, Carlo, H. ‘Topological tori as abstract art’. Journal of the Mathematics and the Arts Vol 6, No. 4, December 2012, 191-209 (8 April 2013)

Largely academic.


————. All papers available on the web from the Bridges archive, from 1998.


Sharp, John. ‘Dürer’s Melancholy Octahedron’. Mathematics in School. September 1994 18-20 (21 February 2013)

For general interest


————. ‘The Circular Tractrix and Trudix’ Mathematics in School. January 1997 10-13 (21 February 2013)

Advanced. For general interest


————. ‘Have you seen this number?’ Mathematical Gazette, 203-214

Fibonacci sequence. Academic


————. ‘Pictures inspired by Theo van Doesburg’. 18-19


————. All papers from the Bridges archive, from 1998.


————. ‘Golden Section Spirals’. Mathematics in School. November 1997 8-12 (21 February 2013)

For general interest


————. ‘Beyond the Golden Section – the Golden tip of the iceberg’. Bridges 2000, 87-98

General interest


————. ‘Fraudulent Dissection puzzles – a tour of the mathematics of bamboozlement’. Mathematics in School. September 2002 7-13 (21 February 2013)


————. ‘Sliceform Craters’. An exploration in equations’. Mathematics in School. March 2004 12-21 (21 February 2013)

For general interest


————. ‘Parabolic Tiling’. Mathematics in School. March 2005 9-11 (21 February 2013)

For general interest


————. ‘D-forms and developable surfaces’. Bridges Renaissance Banff, 2005 121-128


————. ‘Beyond Su Doku’. Mathematics Teaching in the Middle Years. Vol. 12, No. 3 October 2006 pp. 165-169 (21 February 2013)

Cairo tiling on pp. 167-169, in the context of a ‘Cairo Su Doku’.


Shephard, Geoffrey C. ‘Super and Superb Colourings of Tilings’. In Structural Topology No.15 43-74 1988. Escher Special edition (28 March 2011)

Largely academic, of little practical use. Profusely illustrated though. Escher rd fish p. 49, lizards, birds flying fish  51


Shorter, S. A. ‘The Mathematical Theory of the Sateen Arrangement’. Mathematical Gazette, 92-97 19?? (25 March 2013)

Of what I would tem as ‘motif placement, nothing here of tessellation. Of limited interest, to put it mildly.


Sibson, R. ‘Comments on Note 1464’ (C. Dudley Langford proposition). Mathematical Gazette, 24 1990 343 (11 March 2013)


Silva, Jorge Nuno. ‘On mathematical games’ [sic]. BSHM Bulletin Volume 26 (2011), 80-104 (29 April 2013)

A survey of games throughout history that can be described as ‘mathematical’ up to and including the present day. Dice and astragal, Go,  Mancala board, Tangram, Alquerque board, Chess, to name but few. Of general interest, at a accessible level.


Simpson, R. ‘Locally equiangular triangulations’. The Computer Journal.  21 (1978) 243-245. (31 October 2012)

From a reference in Tilings and Patterns. Of an academic nature throughout; of no practical use.


Singer, Connie. ‘Front or Back, Prof. Sam Savage’s Laser-Cut Schmuzzles [sic] Make Jigsaw Puzzles Go to Pieces’. People , June 2 1980, p. 71 (4 September 2015)

From Anne Williams.


Singmaster, David. 82.42 ‘According to Cocker’ 302-303. The Mathematical Gazette (5 March 2013)

General historical interest


————. ‘Covering Deleted Chessboards With Dominoes’. Mathematics Magazine 60- (5 March 2013)



Situngkic, Hokky. ‘What is the relatedness of mathematics and art and why should we care?’  (2010)

(Escher p. 5)


Slocum, Jerry and Dieter Gebhardt. ‘Puzzles from Catel’s Cabinet  and Bestelmeier’s Magazine 1785-1823’. English translation. (11 June 2014)

General puzzle history; of historical significance, the first such catalogue of this type. Tiling negligible.


Smith, Cyril Stanley. ‘The Tiling Patterns of Sebastien Truchet and the Topology of Structural Hierarchy’. Leonardo, Vol. 20, No. 4 pp. 373-385, 1987. (13 March 2013)


Smith, Erich. ‘Puzzlemakers piece a living together from intricate work’. Newspaper Unknown (4 September 2015)

From Anne Williams. On ‘Pieceful Solutions, with Steve Shumaker and Annie Power


Sobczyk, Andrew. ‘More progress to madness via Eight Blocks’. Mathematics Magazine. 115-

(26 March 2013)
From reference in Garcia


Socolar, Joshua E. S. ‘Hexagonal Parquet Tilings k-Isohedral Monotiles with Arbitrarily Large kThe Mathematical Intelligencer Vol. 29 No.2 2007. 1-6. (2010)

Somewhat advanced. N.B. this is not parquet deformation per se!


Socolar, Joshua E. S. and Joan M. Taylor. ‘An aperiodic hexagonal tile’. 1-21 (2010)

Somewhat advanced.


Sohncke, Leonard. ‘Die regelmässig ebenen Punksysteme von umbegrenzter Ausdehung’. Crelle’s Journal für die reine und angewandte Mathematik. Berlin, 1874 (24 December 2014)


Somerville, Duncan M. Y. ‘Semi-regular Networks of the Plane in Absolute Geometry’. Transactions of Royal Society Edinburgh. Vol. 41 1905 725-747 + 12 plates (2 January 2015)

 From a reference in Tilings and Patterns. Largely academic, albeit of a rather simple premise, with occasional simple tilings


Sprague, R. ‘Beispiel einer Zerlegung des Quadrats in lauter verschiedene Quadrate’. Mathematische Zeitschrift 45 607-608 1939

From a reference in Tilings and Patterns. Academic throughout, two square packing diagrams, of no practical use


Spiro, Michel. ‘On the Golden Ratio’. 12th International Congress on Mathematical Education, 2012. (2 September 2014)

Debunks the usual nonsense about the golden ratio appearing in numerous works of art; also see Markowsky and Falbo of a like mind on the subject.


Stebbins, G. L. ‘Prospects for Spaceship Man’. Saturday Review, March 7 1970. 48-50 (29 July 2015)

Use of Bond of Union, p. 48 and Sky and Water I, p. 49. no other references are made in the article.


Steggal, J. E. A. ‘On the Number of Patterns Which Can be Derived from Certain Elements’ Messenger of Mathematics 1907-1908

From a reference in Geometry of Design for High Schools.


Stein, Sherman. ‘Tiling, Packing, and Covering by Clusters’. Rocky Mountain Journal of Mathematics Vol 16, Number 2, Spring 1986 (20 September 2012)

Academic throughout. Bare minimum of diagrams; much theory, all of no practical use.


————. ‘Algebraic Tiling’. American Mathematical Monthly 81 1974 445-462 (11 March 2013)

From a reference in Tilings and Patterns. Academic, of no practical use.


Steinhardt, Paul Joseph. ‘Quasicrystals’. American Scientist, Vol. 74. 586-596 (March 2013)

Largely popular account


Stengel, Carol Elizabeth. ‘A Look at Regular and Semiregular Polyhedra’. Mathematics Teacher 714-719, December 1972

Largely popular account as regards historical account, with Plato etc


Stewart, Ian. ‘Rep-Tiling the Plane’. Scientific American. May 2000. 84-85. (5 May 2000)


————. ‘The Art of Elegant Tiling’. Scientific American. July 1999. 96-98. (30 July 2012)

Minor instance of Cairo tiling, page 97, as devised by Rosemary Grazebrook


Stillwell, John. ‘The Tessellating Art of M.C. Escher’. Function 13-20 (16 January 2015)

Somewhat of a curious article. This begins ‘simply’ by showing ‘absurd’ overlays of grids onto Escher tilings drawn as wireframes, before then moving onto hyperbolic tilings.


Stock, Daniel L. and Brian A. Wichmann. ‘Odd Spiral tilings’. Mathematics Magazine Vol. 73, No. 5, December 2000 (4 June 2013)

Some parts are more academic than others

Sugihara, Kokichi. ‘Computer-aided generation of Escher-like Sky and Water tiling patterns’.  ‘Journal of Mathematics and the Arts’. Volume 3, Issue 4, December 2009, pages 195-207

Sugimoto, T. ‘Classification of Convex Pentagons That Can Generate Edge-to-Edge

Monohedral Tilings of The Plane’ 2012? (25 May 2012)


Sugimoto, Teruhisa and Tohru Ogawa. ‘Tiling Properties of tilings by Convex Pentagon’. Forma 21, 113-128 (25 November 2009)

Set of 14 Convex pentagons


————. ‘Convex Pentagonal Tiling Problem and Properties of Nodes in Pentagonal Tilings’, 452-455 Form and Symmetry Art and Science Buenos Aires Conference, 2007 (25 November 2009)


————. (2000a) ‘Tiling Problem of Convex Pentagon’, Forma, 15, 75–79, 2000. (25 November 2009)


————. ‘Systematic Study of Convex Pentagonal Tilings’, I: Case of Convex Pentagons with Four Equal-length Edges. Forma 20, 1-18, 2005 (25 November 2009)


Sydler, J.-P. ‘Sur les tétrahèdres  équivalents á un cube’. Elemente der Mathematik 11 78-81 (16 January 2015)

From a reference in Dissections: Plane & Fancy. Of an academic nature throughout, occasional diagram, of polyhedra, of no practical use.


————. ‘Conditiones nécessaries et suffisantes pour l’équivalence des polyhèdres de l’espace  Euclidien á trios dimensions’. Commentarii Mathematici Helvetica 40 43-80 (21 January 2015)

From a reference in Dissections: Plane & Fancy. Of an academic nature throughout, occasional diagram, of polyhedra, of no practical use.




Taalman, Laura and Eugénie Hunsiscker. ‘Simplicity is not Simple’. Math Horizons September 2002. 5-9 (19 February 2013)

Loosely polyhedra architecture


Tapson, Frank. ‘Cutting and Sticking’. Maths Resources 21-24: Mathematics in School Vol. 14 No. 1, January 1985, pp 18-23

Dissections, the originality of which is not at all clear; likely taken from Lindgren.


————. 71.25 ‘The magic hexagon: an historical note’. The Mathematical Gazette. October 1987, 217-220

Of note is that it contains an interesting discussion of Dudeney correspondence, p. 218-220


————. ‘Filling in Space’. Maths Resources 106-108: Mathematics in School. March 1989 Vol. 18, No. 2 22-25

Tessellations, many of which I studied, and used for bird motifs, in 1991


————. ‘Watch Your Mathematical Language!’ Mathematics in School. Vol. 23, No. 1 January 1997,14-15 (15 March 2013)

General interest


Note that Tapson has a whole host of articles published in the Mathematics in Schools journal, of which only a few are of direct interest i.e. geometrical, in the broadest of terms. As such, I only show those of the most general interest and geometrical examples here.


 Taylor, Steve and Matthew Taylor. ‘Does Alicante have the longest urban geometric illusion in the world?’ Perception, 2013, volume 42, pages 1362 – 1367

On geometric pavements; very good indeed


Taylor, H. M. ‘On some geometrical dissections’. Messenger of Mathematics, Volume 35, 81-101 (1 August 2011)

From a reference in Dissections: Plane & Fancy. also see Holday Inn sttrtford, London for café wall illusion


Taylor, M. V. ‘The Roman Tessellated Pavement at Stonesfield, Oxon’. Oxoniesia Vol. VI 1941. (2010)

Historical account, of limited interest.


Teeters, Joseph L. ‘How to Draw Tessellations of the Escher Type’. The Mathematics Teacher, April 1974, Volume 67, Number 4, 307-310. (Confusingly, inside the book this is also titled as ‘Mathematics Teacher’ (18 June 2011)

‘Special edition’ on tessellations, specifically concerning three Escher-inspired tessellation articles; (i) Ernest R. Ranucci, ‘Master of Tessellations M.C. Escher;’ (ii) Joseph  L. Teeters ‘How to draw tessellations of the Escher Type’,  and (iii) Evan M. Maletsky ‘Activities: Designs with Tessellations’. That by Maletsky is particularly excruciating.

Termes, Richard A. ‘Hand with Reflective Sphere to Six-Point Perspective Sphere’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration (31 August 2005) 275-285

Teuber, Marianne L. 'Sources of Ambiguity in the Prints of Maurits C. Escher.' The fascinating graphic inventions of the late Dutch artist reflect a strong mathematical and crystallographic influence. Their original inspiration, however, came from experiments on visual perception. Scientific American 231 No. 1 (July 1974): 90-104. (1987)


Thiel, Anton. ‘M. C. Escher: Treppauf und Treppab’. (26 June 2011)

There is much I am unsure of this article, in German, the references are a little unclear, hence the lack of bibliographic detail.


Thomas, B. G. and M. A. Hann. ‘Fundamental principles governing the patterning of polyhedra 2007’. IaSDR. (2010)

The Cairo tessellation gets a mention (page 6), defined incorrectly as a equilateral pentagon


————. ‘Patterned Polyhedra: Tiling the Platonic Solids’. In Bridges 2008.

Again the Cairo tessellation is mentioned, with the same incorrect definition as above. This paper seems to be derived from the above.


Thomas, Dylan and Doris Schattschneider. ‘Artist profile Dylan Thomas: Coast Salish artist’. Journal of the Mathematics and the Arts. Vol. 5, No. 4, December 2011, 199-211 (10 April 2013)


Thomassen, Carsten. ‘Planarity and Duality of Finite and Infinite Graphs’. Journal of Combinatorial Theory, Series B 29, 244-271, 1980 (26 September 2013)

From a reference in Tilings and Patterns. Academic throughout, of no practical use.


Thompson, D’Arcy Wentworth. ‘On the Thirteen Semi-regular Solids of Archimedes, and on their development  by the Transformation of certain Plane Configurations’.

Proceedings of the Royal Society London Series A, 107 (1925) 181-188 (11 March 2013)

From a reference in Tilings and Patterns. Semi-popular account


Thro, E Broydic. ‘Distinguishing two classes of impossible objects’. Perception, 1983, volume 12, pages 733-751

Numerous Escher references. Concave and Convex p. 738 and again 746. Tuning fork, necker cubes.


Thurston, W. P. ‘Three Dimensional Manifolds, Kleinian groups and Hyperbolic Geometry’. Bulletin (New Series) of American Mathematical Society, Vol. 6 No. 3, May 1982. (12 December 2012).

Reference in Grünbaum. Of an academic nature throughout; the tone of the paper is way beyond me. Of no practical use.


————. ‘Conway’s Tiling Groups’. The American Mathematical Monthly, Vol. 97, No. 8, October 1990 (12 December 2012).

See fig. 5. 21, of three fused hexagons. Of an academic nature throughout; the tone of the paper is way beyond me. Of no practical use.


Tóth, Fejes, L. ‘What the Bees Know and What They Do Not Know’. AMS Bulletin 70 (4) July 1964 468-471. (8 March 2012)

Both popular and academic in parts. Various aspects, noticeably on isoperimetric aspects.


————. ‘Symmetry Induced by Economy’. Computers and Mathematics with Applications. Vol. 12B, Nos 1/2, 83-99, 1986 (2 October 2013)

From the symmetry ‘special edition’ of the journal. Obscure.


————. ‘Tessellation of the Plane with Convex Polygons Having a Constant Number of Neighbours’. American Mathematical Monthly, 82, 1975, 273-276 (7 March 2013)

From a reference in Tilings and Patterns. Although the premise is straightforward, I do not understand the tenure of Toth’s paper. Occasionally academic .

Has a Cairo tiling diagram on p. 274, with a possible later reference by Richard K. Guy to a sighting at the Taj Mahal! However, upon enquiring with Guy, he does not exactly recall this. Likely he was mistaken.

Note that I have a whole host (15) of other papers by Toth, all of which are of an academic nature, of no practical use. Therefore, these are not listed here.


Trigg, Charles W. ‘What is Recreational Mathematics?’ Mathematics Magazine Vol. 51 No. 1, January 1978. 18-21

General interest


Trotter, Robert J. ‘Transcendental Meditation’. Society for Science & the Public. December 15 1973 Vol. 104 No. 23 376-378 (12 April 2013)

Although not a maths article it is included here as it uses Escher's print Snakes on p.376 on an article on transcendental meditation, although in the article itself there is no discussion of this, or indeed mention of Escher aside from the picture credit.


Tutte, W. T. ‘The dissection of equilateral triangles into equilateral triangles’. Proc. Cambridge Philosophy Society, 44 (1948) 463-482 (12 December 2012).

From a reference in Grünbaum. Academic throughout, three figures only


————. ‘The Quest of a Perfect Square’. American Mathematical Monthly, 72, 1965, No. 2 Part II 29-35 (11 March 2013)

From a reference in Grünbaum. Academic throughout




Usiskin, Zalman. ‘Enrichment Activities for Geometry’. Mathematics Teacher 264-266 (5 March 2013)

Minor reference to convex pentagon problem; Marjorie Rice




Vallete, G. and T. Zamfirscu. ‘Les partages d’un polygone convexe en 4 polygones semblables au premier’. Journal of Combinatorial Theory, Series B 16, 1-16, 1974 (25 September 2013)

From a reference in Tilings and Patterns. Of an academic nature throughout!. Of no practical use.

Veldhuysen. Mark. ‘M.C. Escher in Italy: The Trail Back’. In Coxeter, et al, Eds. M.C. Escher: Art and Science. Amsterdam: North-Holland 1986. 92-99 (30 April 1994).

Ver Sacrum. 1899. (4 November 2013)

A collection of articles of the Art Nouveau period; quite where to categorise this is uncertain, as book or articles.

A whole years’ worth of articles of 1899 is posted on-line, and although not in any way a mathematics book nonetheless contains Moser’s historic first true life-like tessellation, of fish, and of others of less importance, and so is of significance in that regard. On the off chance that there might be other instances from other contributors, I examined every page, but to no avail.


Villiers, Michael D. ‘An Investigation of Some Properties of the General Haag Polygon’. Mathematics in School Vol. 42, No. 3, May 2014 15-18 (12 March 2015)

Contains a discussion of the Haag hexagon, of which I believe I have shown (upon looking at the paper in depth on the 13 March) that Escher did indeed use this to at least E21 (Running Man), of which De Villiers, and John Rigby, who has done some work on this, left open-ended.


Vince, Andrew. ‘Replicating Tessellations’. Siam Journal of Discrete Math, Vol. 6 No. 3, pp. 501-521, August 1993. (2010)  

Academic. Mentioned in Schattschneider’s bibliography, of no practical use.


————. ‘Rep-tiling Euclidean space’. Aequationes Mathematicae 50 (1995) 191-213 (30 May 2012)

Academic, of no practical use


Vincent, Jill. ‘Shrine to University: Mathematics in the Constructed Environment’. 25-37.  (2010)

Penrose and pentagon paving tilings in situ in Australia. A picture this refers is on p. 35 , I believe, to a Rice type 13 pentagon,


Vincent, Jill and Claire Vincent. ‘Japanese temple geometry’. Australian Senior Mathematics Journal 18 1. 8- (16 January 2015)


Vincent, Peter. ‘Tessellating into Algebra’. Mathematics in Schools. May 1991. 29 (19 February 2013)

As the title suggest, largely algebra based; of no real interest. No diagrams


Voderburg, H. ‘Zur Zerlegung der Umgebung eines ebenen Bereiches in kongruente’.

Jahresbericht der Deutschen Mathematiker-Vereinigung 46 229-2311936 (31 December 2014)

From a reference in Tilings and Patterns. Of an academic nature throughout. Has the Voderberg spiral tiling. Of no practical use.


————. Zur Zerlegung der Ebene eines in kongruente’ Bereiche in Form einer Spirale.

Jahresbericht der Deutschen Mathematiker-Vereinigung 47 159-23160 1937 (31 December 2014)

From a reference in Tilings and Patterns. Of an academic nature throughout. Has the Voderberg spiral tiling. Of no practical use.

Vulihman, Valentin E. ‘Escher-Like Tessellations on Spherical Models’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration (31 August 2005) 442-447




Waldman, Cye H. ‘Voderberg Deconstructed & Triangle Substitution Tiling’ 2014. No article (24 August 2015)

Much of interest; spiral tilings. Both popular and academic


Walker, Jearl. ‘What explains subjective-contour illusions, those brightspots that are not really there?’ Scientific American 84-87. (1988)


Walle, John Van de. ‘Concepts, Art, and Fun from simple Tiling Patterns’. Arithmetic Teacher November 1980 4-8 (20 February 2013)

Children’s article.


Walter, Marion. ‘The day all the textbooks disappeared’. Mathematics Teaching 112. September 1985. 8-11.


Wang, David G. L. ‘Determining All Universal Tilers’. Discrete & Computational Geometry (2012). (5 November 2012)

Of an academic nature throughout; the tone of the paper is way beyond me. Of no practical use.


Wang, Hao.  ‘Proving Theorems by Pattern Recognition – II’. Bell System Technical Journal, 40: 1. January 1961 pp 1-41. (11 September 2015)

From a reference in Tilings and Patterns. Wholly academic (as expected), no diagrams, of no practical use.


————. ‘Notes on a class of tiling problems’. Fundamenta Mathematicae  82 1975, 295-305 (8 January 2015)

From a reference in Tilings and Patterns. Largely academic, occasional diagram, of no practical use.


Warner, Marina. ‘When People See My Drawings They Cannot Sleep, They Do Not Sleep’.

The Daily Telegraph Magazine Number 399 June 23 1972. 24-25, 27-28 (26 is an advert). Oversize. (2 May 2014)

Escher interview by Marina Walker, of 1968, with occasional use of Escher’s prints, in order of use, Moebius Strip, Waterfall, a picture of Escher by Patrick Thurston, Cycle, Reptiles. Contains snippets of unpublished detail of Escher that I am unaware of, such as that he never owned a car or had a TV set.


Washburn, Dorothy K. ‘Pattern Symmetry and Coloured Repetition in Cultural Contexts’. Computers and Mathematics with Applications. Vol. 12B, Nos 3/4, pp. 767-781, 1986 (1 October 2013)

From the symmetry ‘special edition’ of the journal. Obscure.


Watson, R. ‘Semi-regular Tessellations’. Mathematical Gazette. 57 (1973), 186-188 (19 February 2013)

From a reference in Tilings and Patterns. Largely academic, of no practical use.


Wheeler, Harvey. ‘The Politics of Ecology’. Saturday Review, 1970, March 7, 51-52 (29 July 2015)

Use of Escher’s Day and Night. No other references are mentioned in the article


White, William. A .


Wilkie, H.C. ‘On non-Euclidean crystallographic groups’. Mathematische Zeitschrift 91, 87-102 1966 (8 January 2015)

From a reference in Tilings and Patterns. Academic throughout, not a single diagram!. Of no practical use.


Willcocks, T. H. ‘A note on some perfect squared squares’. Canadian Journal of Mathematics 3 (1951), 304-308 (1 November 2013)

From a reference in Tilings and Patterns. Academic throughout, occasional diagrams. Of no practical use.


Wieting, Thomas. ‘Capturing Infinity’. Reed, 21-29 (March 2010)  

A layman’s guide to constructing hyperbolic tessellations using compass and straight edge


Wilker, J. B. ‘Topologically Equivalent Two-Dimensional Isometries’. Topology and its Applications. 12 1981 105-114  (26 September 2013)

From a reference in Tilings and Patterns. Academic throughout, no diagrams of no practical use.


————. ‘Open Disk Packings of a Disk’. Canadian Math Bulletin Vol. 10, No.3, 1967 (4 November 2013)

From a reference in Tilings and Patterns. Academic throughout; no diagrams whatsoever!. Of no practical use.


Wilkie, Ken. ‘The Weird World of Escher the Impossible’. Magazine of the Netherlands Holland Herald. Volume 9, Number 1 1974 3, 20-43 (28 March 2011)

Popular essay on Escher by Wilkie with many interesting sub tales, much of which is new. Perhaps of most note concerns the background to the Mick Jagger-Escher correspondence.


Willcut, Bob. ‘Triangular Tiles for Your Patio?’ Arithmetic Teacher. May 1987. 43-45 (18 February 2013)

Simple tilings. The title is a little misleading, in that the implication is on ‘practical patio concerns’, but this is put in the context of a hypothetical situation


Williams, Anne D. Jigsaw Puzzles. An illustrated History and Price Guide. Wallace-Homestead Book Company, 1990. (11 November 2015)

Although not a true maths book, included as it has tenuous connections. Primarily obtained for details of Margaret Richardson and Elspeth Eagle-Clarke cluster type puzzles, but although Richardson is mentioned, this is not in relation to cluster puzzles. Eagle-Clarke was omitted by default, this being a US focussed book. Sam Savage’s ‘Shmuzzles’ puzzles mentioned, p. 325.


Willson, John Scott. ‘Tessellated Designs in my Op Art Paintings’. Leonardo Vol. 4 pp 369-370 1971 (19 April 2013)

This must be the same Willson of Mosaic and Tessellated Patterns fame


Wood, Donald G. ‘Space Enclosure Systems’. Bulletin 203. Engineering Experiment Station, The Ohio State University, Columbus Ohio. 1967 or 1968 (Google gives 1968, * gives 1967, no date is on the booklet. (11 December 2012). (From a footnote in R. Williams).

Has occasional Cairo-tile instances (non-attributed), of an equilateral pentagon, derived from MacMahon and Cundy and Rollett’s works, pp. 3-5, 30-31.

Wood makes a curious observation as regards tilings with equal length sides, with the Cairo tiling being one of five such instances; as such, I do not recall seeing this simple observation elsewhere. Much of his work here, and elsewhere in the book, is in regards to prisms.


Wood, Elizabeth A. ‘The 80 Diperiodic Groups in Three Dimensions’. Bell System Technical Journal, 43: 1. January 1964 pp. 541-559. (11 September 2015)

From a reference in Grünbaum. Wholly academic.


Woods, Jimmy C. ‘Let the Computer Draw the Tessellations That You Design’. Mathematics Teacher. 138-141. February 1988 (23 February 2013)

A little dated, due to the computer program of the day used. Only one tessellation is shown, of a geometric bird, of a reasonable standard.


Wollny, Wolfgang. ‘Contributions to Hilbert’s Eighteenth Problem’. Pacific Journal of Mathematics Vol. 112, No 2, 1984 451-495. (20 September 2012)

From a reference in Tilings and Patterns. Academic throughout. First diagram appears on p.468, followed by a profusion of diagrams; however, all of this no practical use, being academic.


Wunderlich, Walter. ‘Starre, kippende, wackelige und begwegliche Achtfläche’. Elemente der Mathematik 20, 25-32 1965 (26 January 2015)

From a reference in Hinged Dissections: Swinging & Twisting. Academic throughout. Of no practical use.


Wykes, Sara L. ‘Stanford Professor to Market his Brain Teaser on TV’. San Jose Mercury News, December 4, 2003 (4 September 2015)

Newspaper reference on San Savage, from Anne Williams.




Yen, Jane and Carlo Séquin. ‘Escher Sphere Construction Kit’. In Proceedings  of the 2001 Symposium  on Interactive 3D Graphics, pp. 95-98, ACM Press, 2001 (16 January 2011)

Another Beyer reference… Partly on free-forming Escher's lizards; a little technical in places


Yoneyama, K. ‘Theory of continuous  set of points’. Tôhoku Mathematics Journal. 12 1917, 43-158 (9 January 2015)

From a reference in Tilings and Patterns. Academic throughout, not a single diagram!. Of no practical use.




Zahn, Markus. ‘The Contributions of Arthur Robert von Hippel to Electrical Insulation Research’. IEEE Transactions on Electrical Insulation. Vol. 23, No. 5, October 1988 pp 791-800. (29 May 2014)

Upon rereading Cyndie Campbell’s book ‘M. C. Escher Letters to Canada, 1958-1972’, I noticed a reference to von Hippel, page 65, whose name I was unfamiliar with. Upon looking on the web for him, I found various papers, with this, containing  the background to Escher’s  ‘Man with Cuboid’ print, of which the background, with the von Hippel connection, was unknown to me. Page 798  titles this as ‘The Thinker’. For more on von Hippel, see the article by Frank N. von Hippel.


Zeeman, Christopher and Ian Stewart. ‘Mathematics for Young People: The Royal Institution Masterclasses’. The Mathematical Intelligencer.  Vol. 7, No. 3, 1985, 59-64 (9 December 2011)

General interest


Zeitler H. ‘Über Netze aus regularen Polygonen in der hyperbolischen Geometrie’. Elementarie Der Mathematik 22 56-62 1967 (31 December 2014)

From a reference in Tilings and Patterns. Academic throughout; a few simple diagrams to begin with. Of no practical use.


Zitronenbaum, A. C. editor Queries. Reply to Query 23 913 (20 March 2013)

Dudeney utility reference re its history


Zucker, Andrew A. ‘Student Projects in Geometry’. Mathematics Teacher 567-570

Brief reference to Escher-like tessellation, inconsequential


Zurstadt, Betty K. ‘Tessellations and the Art of M. C. Escher’. Arithmetic Teacher, Vol. 31 No. 5 54-55, January 1984 (1 February 2013) Print Screen Copy (17 February 2013)

Child-oriented tessellation guide. Very poor indeed, typical teacher lack of understanding. Also, poor presentation; she cannot even present a print of Escher’s in full!




Britton, Jill. ‘Compromising tessellation?’ The Mathematics Teacher. 536. 602 (23 February 2013)


Cundy, H. M. ‘What is Stellation?’ Mathematics in Schools. May 1989 p. 47 (18 February 2013)


Frederickson, Greg N. ‘Problem 28’, October 1999. Mathematics Teacher. 527-528. (21 March 2013)

Correcting an attribution


Nelsen, Ethel.  ‘Quilted Escher’. The Mathematics Teacher 586


Rigby, John. ‘Correct names for polygons’. Mathematics in Schools. May 2003 (18 February 2013)


Rush, Jean C. ‘On the Appeal of M. C. Escher’s Pictures cont.’ Leonardo 174 (17 February 2013)

Replying to M. Emmer’s comments, and Emmer’s comments here thereof


Sharp, John. Mathematics in Schools. January 1998 47 (21 February 2013)

on Jacob Bernoulli spiral tombstone error inscription




George J. Paulus to Maurits C. Escher, March 20, 1963, typed (forwarded by Doris Schattschneider) (8 January 2016)


George J. Paulus to Martin Gardner, handwritten (forwarded by Doris Schattschneider) (8 January 2016)


Maurits C. Escher to George J. Paulus, 29 March 20, 1964 (or 1963), typed (forwarded by Doris Schattschneider) (8 January 2016)




Unknown Author, on Penrose tiling game: Perplexing Poultry Instructional Manual. Pentaplex Limited. 1983.


Isenberg, Cyril. Soap Film Experiments with Kubic Bubbles. 1974.





The more important ones I list individually, the lesser ones categorised under tiling, cluster puzzles, 3D puzzles, or pavements, with just the patent number and name, for the sake of a briefer page


Clerc, Daryl G. and Pamela A. Clerc. ‘Figural Puzzle’. (Cluster Puzzle). United States Patent 2001/0052670A1. 2001 8 pages (26 June 2014)

Rectangular frame, themed, with 78 animals. This seems wholly impractical.


Dodd, Mark D. ‘Toy’. (Cluster Puzzle). United States Patent 1,443,217 (Cluster Puzzle) 23 January 1923. Four pages (9 June 2014)

Human and animal themed cluster puzzle, better than most. Of note historically, in that this is among the first instances of the genre, the second after Richardson, and the earliest patent.


Eagle-Clarke, Elspeth. ‘Improvements in or relating to Jig Saw Puzzles’. (Cluster Puzzle). UK patent 407,185 15 March 1934 (22 July 2014)

Of note historically, in that this is the third instances of the genre, after Richardson and Dodd


Fisher, Adrian. ‘Tessellatable Elements and Plane Tessellations for Covering or Decoration’. United States Patent 5,945,181. 31 August 1999. 19 pages. (4 September 2014)

As ever with Fisher, some most inventive tilings. In particular of note here is Sheet 12, which  can be described as a ‘Cairo tiling hexagon’ with a ‘intruding’ square tile that changes the pentagon to a heptagon. The resulting tiling, Sheet 13 does not appear to be connected at a first glance.


Fisher, Adrian and Ed Pegg Jr. ‘Tessellation Set’. WO 01/21417 A1 (16 August 2010)


Geissler, Bernhard. United States Patent 2003/0136069, July 24, 2003 ‘Structural Elements and Tile Sets’ (6 March 2014)

Mostly concave pentagonal tiles, most pleasing indeed. N.B. Fig. 5 bears a resemblance to a dog; I might be able to do something with this


Glickman, Michael N. United States Patent 4,711,599 ‘Paving Block’ 8 December 1987 (29 July 2014)

Many different placements of a chevron block, formed by appropriate extensions of a regular hexagon


Godinet, Wayne P. ‘Bed And Mattress Formed By Animal Shaped Nesting Play Cushions’, (Cluster Puzzle) United States Patent  4,719,656. 1988 (27 June 2014)

Bed and mattress premise, themed, with 13 animals. Articulations are poor, 8 pages


Gopfert, Reinhard. ‘Composite Stone Set’. United States Patent 4,919,565. 24 April 1990 (25 August 2014)

This shows the source of one of my German tiles on my pavement page.


Graham, Stanford A. ‘Educational Puzzle Toy Set’. (Cluster Puzzle) United States Patent 5,720,481 8 pages (27 June 2014)

Circular cluster puzzles, of dinosaur and sea themed. Of note is that this quotes a reference in Business Week that I have not seen: ‘Puzzles go Full Circle’, July 10, 1965 p. 68. Whether this is a cluster puzzle of a generic puzzle remains to be seen

Note that the Graham patent shows the animals used on the Anything’s Puzzable jigsaw puzzle


Lalvani, Heresh. ‘Crescent shaped polygonal tiles’. United States Patent 4,620,998, November 4, 1986 (4 July 2011)

N.B. this patent was used on one of his ‘Tesselz’ jigsaw puzzles.


————. ‘Metamorphic Tiling Patterns Based on Zonohedra’. United States Patent 5,211,692 18 May 1993 16 pages (2 July 2014)

Broadly of a parquet deformation nature.


————. ‘Space Structures with Non-Periodic Subdivisions of Polygonal Faces’. United States Patent 5,524,396 11 June 1996 71 pages (1 September 2014)

Mostly of squares and rhombs tilings and two rhombs in combination. like all of Lalvani’s most, most interesting and innovative


————. ‘Non-Convex and Convex Tiling Kits and Building Blocks from Prismatic Nodes’.

United States Patent 5,775,040 7 July 1998. 36 Pages (10 July 2014)


Moore, Herbert C. ‘Tile’. United States Patent 928,320. 20 July 1909, 4 pages (1 July 2014)

Upon following up two references in Roger Penrose’s US patent 4,133,152, one (the other, by Muller is not of tiling) by Herbert. C. Moore, to my surprise and delight shows a pentagon (Cairo) tiling, the earliest reference I have in print, of 1909. Also see following patent, where it is repeated again, in conjunction with another pentagon tiling


————. ‘Tile’. United States Patent 928,321. 20 July 1909, 2 pages (2 July 2014)

A most pleasing tiling of pentagons formed by  subdividing a hexagon in a most simple but effective way. The Cairo tiling of patent 928,320 is also shown again


Muller, N. ‘Toy-Blocks for Object-Teaching’. United States Patent 143,835. 21 October 1873 [sic]. 2 pages (1 July 2014)

From a reference in Roger Penrose’s patent, of a loosely kite and dart nature


Hazlett, Darren G. ‘Interlocking Ground Concerning Element’. United States Patent 5,625,990 6 May 1997, 7 pages (2 September 2014)

This shows a tessellation paving of the state of Texas, US


Odier, Marc G. ‘Puzzle with Irregular Pentagonal Pieces’. United States Patent 3,981,505 21 September 1976 (2 July 2014) 4 pages

Cairo tile diagram Fig. 3, and various patches of tiles formed with the pentagons. Odier has a series of patents of a ‘game piece’ premise’.


Osborn, John ‘Single-Shape Variably Assemblable [sic] Figurative Tiles for Games, Puzzles, And For Covering Surfaces’. United States Patent 5,520,388, 28 May 1996 7 pages (3 July 2014)

Dedicated to his ‘Ozbird’. For the life of me, I cannot fathom out why I had previously neglected this; Osborn quotes it in his 1997 patent that I had in 2011. Again, as detailed above I have reservations here. However, if indeed original, this is most pleasing concept, and worthy of praise; I do not recall having seen this arrangement before.


————. ‘Variably Assemblable [sic] Figurative Tiles for Games, Puzzles, And For Covering Surfaces’. United States Patent 5,619,830, April 15 1997. 7 pages (16 June 2011)

Quite frankly, I am not sure as to what to make of this. Given his track record for outlandish claims I remain dubious as the inherent originality here.


Penrose, Roger. ‘Set of Tiles For Covering a Surface’. United States Patent 4,133,152, 9 January 1979. 13 pages (18 February 2011)

Penrose kites and darts (with chicken motifs)  and thin and thick rhombs. Of note is that this refers to a patent by Herbert C. Moore, which is of significance concerning my Cairo tiles investigations, as it shows a Cairo tiling, but despite having this in 2011 only in 2014 did I find this! Oddly, no reference is made to Moore (and another reference, N. Muller) in the patent itself


Plötner, J. C. ‘Road Surfacing Unit’. United States Patent 2,919,634, 5 January 1960 (three pages,18 November 2014).

Of significance in regards as pavements per se, with this being of a frequently seen type, albeit with little known as to history. Paul Stephenson discusses this tile in his article, but without reference to this patent


Savage, Sam L. ‘Jig-Saw Puzzle With Identically Shaped and Sized Interlocking Jig-Saw Elements’. International Publication Number WO 80/01990 2 October 1980. 18 pages (26 June 2014)

Noteworthy to the extraordinary extent Savage plagiarizes Escher’s lizards without any credit to Escher whatsoever in a 18-page document, as indeed a like situation arises on the commercial puzzle box itself, titled as ‘Shmuzzles’, with the implication that these are designs by Savage. Only on his website is reference is made to Escher (admittedly on the front line), but even here, the impression given is that these ‘salamanders’ are Savage’s original. Some minor additions are made to overlaying the lizards, and this somehow qualifies as ‘original’ art worthy of a patent! Amazingly, Savage is a maths professor at Stanford! He should surely know better; I originally thought he was some kind of amateur puzzle inventor, and with goodwill on my part was unsure of creditation responsibilities. And another annoying aspect is the way that the configurations are prefixed with Sh, to the point of ad nauseam, with ‘Shlameleons’, ‘Shmacrobats’. A reviewer in Mathematics Teacher of November 1980 finds ‘the whole idea of shmuzzles is rather silly’…, of which I agree.


Soriano, Rene. ‘Puzzle/Play Set Toy Product’. (Cluster-type Puzzle) United States Patent 5,628,513 13 May 1997 (15 July 2014)

Noah’s ark theme cluster puzzle, but generally lacking articulations


Siegel, Florence. ‘Puzzle formed of geometric pieces having an even number of equilateral sides’. United States Patent 4,561,097 (6 March 2014) an octagonal theme

Some rather trivial tilings of subdivisions of octagons; ‘seen and noted’.


Stevens, Denise M. ‘Figurine Puzzle with Display Apparatus’. (Cluster Puzzle) United States Patent 5,615,883, 1 April 1997 (18 June 2014)

Nativity themed cluster puzzle, based on Ashe’s  work, but different  


Wallace, John. ‘Surface Covering Tiles’. United States Patent 4,350,341, September 21, 1982

8 pages (8 August 2011)


Weary, E. D. ‘Mosaic Floor Covering’. United States Patent 35,949, 10 June 1902, 2 pages (1 July 2014)

From a reference in Moore’s patent, with in principle a tiling that can be construed as being ‘Cairo related’ (by subdivision)


As regards patents, I have others but have chosen not to list here in full, for a variety of reasons, due to at best tenuous connections, and at worst can only be described as related in the sense of a puzzle per se that strictly are undeserving of the time and trouble of a full listing as above. These I now list by category, with surname and patent number only. Note that some of these are a little ambiguous to category at times:

For unclear reasons, some of these patents, notably by Castonguay et al and Riccobene basically repeat themselves


Cluster/Jigsaw puzzles (cluster are boldened)

Ach 3,477,167; Asheminary 5,362,054; Barrett 6,425,581; Barry 4,365,809; Beal 4,052,072; Bernstein 6,361,045; Billow 1208277; Brown 1,532,875; Clerc et al G2001/0052670A1; Decker 1,948,962; Dodd 1,443,217; Dougan 4,092,752; Erickson 6,634,928; Gantt 1,976664; Graham 5,720,481; Hassenbach 2,953,380; Houghton 941680; Irwin 2,353,037; 3,193,294; Krase 2,011,058; Libeskind 5,957,454; Mannino D320,050; Michlin 7,383,978; Mitchell 5,368,301; Murphy 4471960; Iosla et al 3613133, Ishizuka 243736; Jacobs 4,281,425; Johenning 4,197,602; Johnson 4,835,800; Jones 228359; Marks 4,364,134; Odier, 3,608,906; Palmer 3,575,418; Rasberry 5067714; Renn et al 3,433,485; Moore? 1152,383, Salgado 8,490,976 B2; Scovell D79576, Snedeker 964,065; Soriano 5,628,513; Steiger 175519; Stein 4,361,328; Stevens 5,615,883; Van Nierck 4961708; Volkert 6,366,631; Wakeland 4,109,887; Young 721446


Dissected Maps (Jigsaw)

Adams 3,242, 591; Allen 1,597,562; Flenniken 1,228,197; Fukal D170,753; Harrison 180,476; Higgins 142,338; Hyde 266,628; Kreitler 2,199,499; Mathes 1,375,308; McCleary 6,747; McCurdy 443,373; Mayer 618,114; Rippon 2,008,189; Sanderson 952,997; Wesley 3,495,833



Games and Puzzles

Blonder US2003/0216103; Dorr 647,814; Essebaggers EP 0584883A1; Farley 2,069,106; McKenna 4,722,712; Odier 3,608,906 (Cluster), 3,687,455, 3,712,622, 3,837,651, 3,869,125, 3981,505 (Cairo). Resch 3,201,894; Richards 331,652; Stevens 5,299,804


3D puzzle (manipulative games, such as Rubik cube) patents:

Alexander et al 4,506,891; Alford 4,558,866, Ashley 4,474,377, Breslow 4,413,823, Engel 4,415,158, Goldfarb 4,496,155, Greene 4,452,454, Gustafson 4,474,376, 4,522,401; Halpern 4,453,715, Kikis 4,526,372 Hanson et al 4,373,729, Hart 4,473,228, Hewlett 4.451,039, Horvath 4,540,177; 4,405,131; Isobe 4,344,623, Keister 3,637,215, Komiya 4,377,916, Miller 4,437,667, Nichols 3,655,210, Nieto 4,500,090, Opresco 4,513,970, Rubik, 4,378,117, 4,378,117, 4,392,323, 4,410,179, 4,471,959; Sebesteny 4,421,311, Shermann Jr 4,557,484, Silbermintz 4,409,750; Titus 4,441,715, Wiggs 4,553,754; Yokoi 4,402,510, 4,376,537


Pavement or floor paving.

Patents of particular interest, in regard of ‘notable tiling’, are boldened. Patents with a suffix ‘T’ is tenuous, sometimes in the extreme, albeit from a paving reference source. Indeed some references have little to no relevance to paving!: X is ‘seen and noted’ (i.e. not downloaded), but of no relevance


Abbracati D426,897 D431,871; Abracanti D424,212; Abraham D75,761; Alberti et al 2,932,745; Alcott 929,366; Allard D256,817; Allocca 3,746,458; Alcott 929,366; Almy et al 3,056,224; Altmann et al 7,425,106; Anderson 687,106; Appleton D231,926, 3,903,702; Ardit 1,600,787X; Assanti 4,217,740; Atkinson 4,372,705

Balz 1,410,729X; Banze 5,941,657, EP 415 093B1; Barbier D480,819; Barbour 1,084,058; Barth et al D389,926, D399,978, D425,628, 4,128,357; 4,185,939, 4,583,341, 4,834,575, 5,342,142, 5,360,285, 5,797,698 (T), 5,902,069; Bartlechner 4,761,095; Baumberger 3,494,266; Bedell 696,792 (T); Beidler 106,651; Bennett 681,946; Berrie 400,997; Bichan et al 1,364,236; Bignell 382,683; Biller 3,466,986; Bolli 3,891,340(T), 3,922,105(T); Bolton D82,657, D82,970, D86,076; Bordallo 5,002,425; Borthwick 1,845,579; Bouchard et al 8,668,404, 2012/0247050; Bowes 2,101,019X; Bowman 4,105354; Brainerd 1,379,440; Bradley 1,689,107; Bresnahan 4,878,778X; Brimo 4,776,723; Brinkley D246,544X; Brock D314,829; 5,251,997; Brotsch et al 1,384,236; Bushnell 110,954


Cadjew 2,509,558X; Capen, 726,506; Carroll 1,479,647; Castonguay et al (8) 8,500,361 B2, D505,733, D522,667, D553,260, D537,959, D543,642, 2012/0189386; 20007/0217865; Childress Jr et al 4,172,344X; Ciccarello 6,073,411; Clanton D185,592X; Clarke Jr et al 4,492,065X; Clemente 3,602,111; Coleman 3,603,0590X; Collete 4,018,025X; Conners et al 6,027,280; Cowan 338,490; Curtois 2,025,496


Daigle 3,842,562X; Damianik 1,969,729; Davis et al 3,600,773 (T); Decker et al D159,255; Deslauriers D22,609X, D22,610X;  deWitt 2,321,067X; Dodino 3,897,164; DiFrancisco 5,194,969; Dubé 5,496,129; Dukart et al D330,435X; Dyer 130,027


Easy D341,218; Eggers, 953,413; Eusemann 8,875,716; Evers 2,050,299, 2,094,167X (T)


Fagan 2,138,270; Faudree 3,061,947; Fenberg 1,699,351; Fenton 3,557,669; Fifield D442,703; Fisher (1) 5,945,181*, 2,306,184*; Fisher (2) D 4,550,040; Fitzgerald 3,239,981X; Fleishman D429530, D488,566; Flood 708,470,  708,471; Fontana et al 5,046,887; Fort 1,894,584; Fresquez 4,131,406 (T); Fulton 1,622,103; Furness 565,734, 645,800


Gable Jr 4,571,353; Gabo 2,549,189; Gale 2,715,289; Galliano D97,318; Gargollo 4,850,739; Gaston 6,185,893X; Geiger 5,645,369, 6,015,243; 6,536,988; Gerald 1,456,499X; Gilbert 1,184,945; Glickman 4,711,599, 5,277,514; Göpert 4,919,565; Graham 470,377, 474,339; Grant 330,110; Green 2,340,526; Gregory 719,790; Grossman 5,230,584 (T); Guyer 244,594


Hagenah D421,135; 4,773,790, 5,224,792, 5,486,066; 5,588,775 (T), 5,597,591(T), 6,263,633; Hagopian 2,162,777; Hair D328,499 D343,238, D343,460, 4,544,305, 4,973,192; 5,108,219, 5,201,843, 5,244,303, 5,286,139; Hamel D553,759; Hartnell 3,824,755; Hass D263,082; Hayden 306251; Hazelton 449,739; Hazlett 5,625,990; Healy 16,867, 1,437,304; Henderson et al 2,346,304; Hendriks et al 4,828,426 (T); Henry 1,505,642; Higgins 510,259; Hill 722,580, 2,084,141X; Hodson 5,921,705; Hooper 1,577,165; Hopkinson 1,158,051; Hoyt et al 4,026,083; Humphreys Re 19,906; Hupp D342,528; 5,487,526; Hurlbut 456,378; Hybertson 4,921,372


Ingalls 5,840; Ito 1,456,259; Ivery 741,497X


Jacklich 5,211,895 (T); Jacobs D28,084; Jacobsen et al 3,873,225; Jennison 962,150, 1,061,296, 1,302,560 1,340,896 1,395,829; Jordan et al 3,923,410, 4,016,692; Jurik 5,466,089; Johnson (1953) 2,629,135X; Johnson 5,267,810; Johnson II D343,237; 5,054,957; Jones 447612


Kammer 1,474,779; see Zur, Kan 3,953,009; Kapusta 4,681,481X; Keller 1,895,801; Kennedy 653,515; Kertes 1,058,674; Kilburn 140,835; Klaffke et al 4,172,168; Knowles 621,100; Knudsen 4,465,398X; Koenig 2,493,458; Krauss 1,853,824X; Kreuger et al D349,967; D352,559


Lalvani, 4,620,998, 5,211,692, 5,524,396, 5,775,040; Landers 572,762; Lanz D195,460X; Larsen et al D281,505; Lavin 2,569,065X; Leary 710,062; Leeth 204,803; Levin 1,828,193; Lewis EP 0347113A; Lewis D326,530,5,275,503; Littman 230,478 (T); Lööv 4.445,802X; Lowrigkeit; 4,052,131; Ludvigsen 3,722,162; Luke et al 212,243; Lyons 198,638


MacDonald 1,976,575X; MacRae 3,267,823; Madge 1,673,630; Manderino 3,030,951X; Manico 141,283X; Mann 952,918; Mark D27,761; Marsh 313,221X; Marson 3,344,570; Martin D347,899; Mattox D439,677; Medico 3,870,422; Metten 5,400,561X; McClain 1,314,278; McCoy 5,137,392; McGiehan; McGilvray 1,174,269X; McKee  2,708,329, 5,568,391; McKeever 4,838,728 (T); Merlette 127,628; Milot D429,343, Milot et al 6,168,347; Miniere D265,689X; Monick 2,836,108  Moore 358,288; Moore, 928,320; 928,321; Morse 110,153; Mullins 4,107,894


Naether D312,881; Nash 777,858X; Newsom 5,035,098(T); Noack 4,354,773; Notari 2,210,150


Oden 1,740,110X, O’Donnell 1,700,542 (T); Ogawa et al D426,648X; Oldfather 2,606,428 (T); Osborn 5,619,830;


Pandolfi  D125,642 (T?); Parquin-Kleinerman 2,723,607X; Paulson 2,132,757; Parker D102,145; 1,694,655; Perkins 862,012; Perrody 529,747(T), Pertien 3,221,614; Penrose 4,133,152; Perry 1,595,686; Petty 2,111,003; Pietz 3,870,423; Picha 739,345; Pilaar 3,421,417; Pincon et al 3,171,335; Porter 2,060,746; Porter C. K. 681,834; Plotner  2,919,634; Prestele 5,713,155 (T); Prindle 150,710X; Promoli 345,726; Prouty 1877,481X; Puccini et al 4,135,840 (T); D257,824 (T); D257,825 (T), D272,037 (T); Pugh 888,530; Purdy 1,884,216


Ramoneda 3,304,673X; Reinhard 113,929; Reinschütz 4,125,341, 4,313,689 (T); Renkert 2,095,012; Repassky D283,551; Riccobene (9) RE027,694, D471,990, D536,058, D537,501, 6,668,484, 6,881,463, D586,925, 7,393,155; 2008/0209828; Rhodes 1,838,108; Rinninger 4,572,699; 4,792,257; Roberts 3,214,874; Romanoff 801,108; Roming 4,231,677 (T); Rosenberger 3,947,192; 4,349,293 (T); Rothmann 2,215,159; Rotherrham 4,532,748X; Ruckstuhl 4,496,266, 4,907,909; Russell 4,287,141 (T); Ryan D154,156


Saleeba 4,452,419; Salgado 8,490,976; Sampson 1,682,687X; Sammis 1,045,328; Sassenberg D414,281; Scanni et al 3,931,700; Schnaar 3,242,832; Schaeffer 421,618; 2,323,848; Scheiwiller D278,934; D278,935; D317,207, D317,208, 4,524,551X; 4,627,764; 4,922,678; 5,028,167; 5,133,620; 5,348,417, 5,503,498; 5,533,827; 5,560,173, 6,471,440; Scholl 3379104; Schillinger 147,982; Schmitz et al D498,543; Schuring D250,898X, 4,031,678X; Schraudenbach 3,343,468, 4,067,1996; Schutte 768,698; Schwartz, 2,368,330; Sears D248,780; See 479,126;  Seidler 2,013,768; Sellars 250,456; Sheen 1,460,516; Sherman 3,546,792; Shindo 4,781,492; Simpson 472,590; Skaug 5,281,047; Slosberg et al 3,386,001; Smith, E. F 649,963X; Smith 3,270,473; Solon 1,864,153; Steiger 468,840; Stone D184,463; Stoy D197,127X; Stratton 4,593,513X; Streator 1,636,113; Strazinger 3,522,618X; Strobel 3,229,439; Sweeney D393,726X


Talamini 42,404; 42,405; Tau 2493,470; Tavares 5,186,574; Terry D397,802; Thieffry 4,249,358; Thomas 28,670; Thomson 1,660,459; Thomassen et al D550,375; Thorn 400,974; Tilley 2,893,098 (T); Tokunaga 3,759,043; Trieff 2,605,681; Trimmer et al 4,407,480 (T); Triol 1,505,174; Turnquist 2,099,149


Uppström 4,537,001; Urdaneta 4,722,158


Vesterholt D262,742; Von Langsdorff 5,713,173


Wallace 4,350,341; Warden 1,156,117; Weber 6,715,956; Wedberg; Weigand D393,727; Welling Jr 4,997,308; Whitman et al 4,474504X; Widmer 5,249,884; Wilkins 2,915,893X; Wilhelm D365,643; Williams 2,991,213 (T); Wilmot 566,489X; Wong 2,710,335X; Woolford D404,147, 5,884,445; Wormser 2,885,207; Wright 1,778,927, 1,796,973; Wurtz 241,771X; Wilson 4,510,725 (T); Wu 5,409,325; Wyatt 1,794,572;


Yacura 187,739; Yoshida et al 5,051,023 (T), Zahn 1,485,007; Zeigler D431,870


Pentagonal theme:

Calvert 4,357,018; Donchian 110,173; Felt 23,059; Pocklington 5,536,013; VanErmen D549,396; Way 2,901,256

Winfield, 6,527,653


Tilings: (All cross referenced)

Flood 708,470;



Knowlton et al 8,628,087


X – nominally paving related, from a paving reference, seen, but the content is not deemed worthy of saving the PDF, by so denoting I can cross check any references elsewhere

  • same, save for UK and US versions
  • occasional different authors names have the same surname; where this occurs I simply add 1, 2, 3… as appropriate




Cerrone, Kathryn. ‘Tessellations: A Lesson for Every Age’. (23 June 2011)


Chavey, Darrah, P. ‘Periodic Tilings and Tilings by Regular Polygons’. 1984 (18 May 2011)


Harriss, E. O. ‘On Canonical Substitution Tilings’. 2004 (10 June 2008)

Penrose tilings


Kaplan, C. S. ‘Computer Graphics and Geometric Ornament’. 2002 ()



Reinhardt, K. ‘Über die Zerlegung der Ebene in Polygone’. Universität Frankfurt 1918 (24 May 2012)

This is interesting in many ways, and specifically in context with the Cairo tiling, in that it shows what I term as a ‘skew’ Cairo tiling, pages 80-81, and the first instance I have of this. It’s also curiously noticeable for the almost total exclusion of diagrams, despite it being a paper on tiling!




All these maths calendars are from John Bibby. Various oddments of mathematics, still relevant


The Mathematics Calendar 1992


The Fun Maths Calendar 1993 (9 April 1993)


Fun Maths [calendar] 1994 (30 April 1994)


Mathematics Education Calendar 1994 (30 April 1994). Historical figures for each month




Emboden, William A., Jr. ‘To Cast a Lovely Dream: Natural History and the Art of M. C. Escher’. Terra (Natural History Museum of Los Angeles County) 11, no. 2 (Fall 1972): 3-10.

(c. 11 March 2014)

‘Snakes’ on front cover, ‘Three Worlds’ back cover. A slim booklet, of just twelve pages, with numerous prints in the catalogue to the exclusion of tessellations. This is essentially an essay by Emboden, albeit of no particular insight by way of anything new. However, Emboden does indeed point out a recurring church theme in Escher’s work. Also see an earlier 1984 catalogue of Embdoen.


Escher: Pattern & Paradox. Catalogue of the exhibition held at Lowe Museum, Miami, USA, October to November 1984. (17 March 2014)

Essentially an brief essay (more or less of just two solid pages of text) by Emboden, with a premise of natural history, illustrated by 16 of Escher's prints; oddly not all are discussed, for instance Sky and Water. Nothing of any real insight is shown, although Embolden makes a fair attempt, and is not simply repeating well-worn text.


Escher: Pattern & Paradox. Catalogue of the exhibition held at Lowe Museum, Miami, USA, October to November 1984 (28 March 2011)


Catalogue of the exhibition held at Artists’ Market, Norwalk, Connecticut, USA, March-April 1987 (28 March 2011)


Structural Topology 15 1988 (Escher Special, articles by M. Emmer, D. Schattschneider, H. S. M. Coxeter, M. Senechal, G. C. Shephard)


Structural Topology 17 1991 (‘continuation’ of Escher Special: articles by M. Emmer, Roger Penrose, J. F. Rigby, A. Dress and D. H. Huson)


O Mundo Mágico de Escher Brazil exhibit catalogue, in Brasilia 12 October-26 December 2010, Rio de Janeiro 17 January-27 March 2011, Sao Paulo 18 April-17 July 2011. Curated by Pieter Tjabbes (10 October 2012)




Anon. ‘Ever so plane and beautiful’. Times Higher Education 14 January 2005 (12 August 2014)


Brooks, David. ‘Escher: Unusual marriage of art and mathematics’. The Telegraph. 2 December 1990 (12 August 2014)

Review of Visions of Symmetry


Cain, Barbara A. Tessellation Winners: The First Contest and The Second Contest. Mathematics Teaching in the Middle School (25 February 2013)


Cooper, Donna. ‘Three-Dimensional Symmetry’. Ann E. Fetter et al. Vol. 89, No. 6 September 1996.


Coxeter, H. S. M. ‘Islamic patterns: An Analytical and Cosmological Approach’ by Keith Critchlow. American Scientist  504 Volume 65 (20 February 2013)


Cromwell, Peter. ‘Dissections: plane and fancy’, by Greg N. Frederickson 359-360. The Mathematical Gazette (21 March 2013)


Cundy, H. Martyn. ‘Topics in Recreational Mathematics’ by J. H. Cadwell. The Mathematical Gazette p.186 (1 March 2013)


Daniel, Patricia. ‘Escher Interactive: Exploring the Art of the Infinitive’. The Mathematics Teacher 604 (23 February 2013)


Dembert, Lee. Book Review: ‘Art Meets Math in Kaleidocycles’. Los Angeles Times 27 May 1988. (12 August 2014)


Dunham, Douglas. ‘Visions of Symmetry’ by Doris Schattschneider. 78-81 (27 February 2013)


Eisenberg, Mike. ‘King of Swing’ Hinged Dissections by Greg N. Frederickson. American Scientist. Vol. 91 269-270 (21 March 2013)


Emmer, M. ‘Visions of Symmetry’. 389 (17 February 2013)


Folletta, Gina. ‘Teaching Tessellating Art: Activities and Transparencies Masters’ (28 march 2013)


Grünbaum, Branko and Marjorie Senechal.  Review of Symmetrie-Gruppe-Dualität by Erhard Scholz 1989 406 377-380. (9 December 2014)


Jenson, John M. ‘M. C. Escher Kaleidocycles’ by Doris Schattschneider and Wallace Walker

Vol. 90 No. 5 May 1997 (17 February 2013)


Joël, Nahum. ‘The World of M. C. Escher’. Leonardo Vol. 9 No. 3 Summer 1976 254-255 (17 February 2013)


Jones, Lesley. ‘The Magic of M. C. Escher’. Mathematics in School March 2001 (17 February 2013)


Lawrence, K. W and Anne Lawrence. Tesselmania! Mathematics Teaching in the Middle School, 840 (28 March 2013)


Loeb, A. L. ‘Transpolyhedra: Dual Transportations by Explosion-Implosion’ by Haresh Lalvani  Leonardo, Vol 11, No. 4 (autumn) 1978 342 (4 April 2013)


Mackay, Alan L. ‘Stimulating patterns’. Nature Vol. 349 7 February 1991, 471-472.


————. ‘Visions of Symmetry, Notebooks Periodic Drawings, and Related Works of M.C. Escher’. (15 June 2011)


Marchand, Carolyn. ‘Escher Interactive: Exploring the Art of the Infinitive’ (23 February 2013) Vol 3 No. 7 May 1998


Massey, Ann. ‘Introduction to Tessellation and Tessellation Teaching Masters’, by Dale Seymour and Jill Britton. Mathematics Teacher 482


‘Explorations with TesselMania!’ Activities for Math and Art Classrooms, by Jill Britton. (23 February 2013) Vol 3 No. 7 May 1998


‘Holland’ (pseudo name) ‘Creating Escher–Type Drawings’. E. R. Ranucci and J. L. Teeters October 1978 (16 February 2013)


————. Shmuzzles Puzzle. Mathematics Teacher, 1980 623-624


Orton, Tony. ‘Dissections: Plane and Fancy’, by Greg N. Frederickson. Mathematics in School, May 1999, 46 (21 March 2013)


Pargeter, A. Robert. ‘Dissections: Plane and Fancy’ by Greg N. Frederickson. The Mathematical Gazette 371-373 (21 March 2013)


Pederson, Jean. Geometry Turned On: Dynamic Software in Learning, Teaching and Research, by James King and Doris Schattschneider


Pickover, Cliff. Tribute to a Mathemagician (Martin Gardner tribute). Edited by Barry Cipra, Erik D. Demaine, Martin L. Demaine and Tom Rodgers. Review in The Mathematical Intelligencer Vol. 29, No. 3 2007, 71 (25 November 2011)


Radin, Charles. Quasicrystals and geometry. Review of Marjorie Senechal’s Book in Notices of the AMS Vol. 43, No. 4, April 1996 (12 December 2012)

Broadly accessible, occasional academic aspects


Reynolds, P. The Graphic Work of M.C. Escher. Mathematics in School 34 (17 February 2013)


Ruane, P. R. Hinged Dissections: swinging and twisting by Greg N. Frederickson. The Mathematical Gazette 183-184 (21 March 2013)


Ruane, P. N. M. C. Escher’s Legacy. The Mathematical Gazette 135-136 (17 February 2013)


Satterfield, Melanie. ‘Tessellation Exploration’. 2001 Teacher’s Guide + CD-ROM. Tom Synder Productions. Mathematics Teaching in the Middle School 314 (25 February 2013)

Software review


Schattschneider, Doris. ‘The Mathematical Theory of Chromatic Plane Ornament’. By Thomas W. Wieting. American Scientist, Vol. 71 324


————. ‘Symmetries of Culture Theory and Practice of Plane Pattern Analysis by Dorothy K. Washburn and Donald W. Crowe’. 383-386 (27 February 2013)


————. Dissections: Plane and Fancy, by Greg N. Frederickson 220-223. (21 March 2013)


————. ‘III. Software Reviews’

Reviews of geometric software; Adobe Illustrator, The Geometer’s Sketchpad etc.


Shufelt, Gwen (ed). ‘Shmuzzles’. April 1979 43. (28 March 2013)


Sigmund, Karl. Kepler’s Conjecture: How Some of the Greatest Minds in History Helped Solve One of the Oldest Math Problems in the World by George G. Szpiro. The Mathematical Intelligencer Vol. 26, No. 1 2004, 66-67


Sykes, John. ‘Hinged Dissections: Swinging and Twisting’, by Greg N. Frederickson. Mathematics in Schools, November 2003 (21 March 2013)


Web, Nigel. ‘Tessellations’, by Fred Daly, Bob Burn and Chris Forecast (14 March 2013)


Williams. H. C. Tilings and Patterns by B. Grünbaum and G. C. Shepherd




Bennett, Geoffrey Thomas 1868-1943. 597-615. Obituary notices of Fellows of the Royal Society (27 March 2013)

From references in Frederickson and Garcia



Gardner, Martin (all 13 August 2014)

New York Times (23 May 2010, Douglas Martin)

NBC News Times (23 May 2010)

Huffington Post (23 May 2010)

The Times (24 May 2010, Matt Parker)

Mathematical Association of America (24 May 2010)

Washington Post (24 May 2010, Emma Brown)

New Scientist (24 May 2010, Jeff Hecht)

The University of Chicago Press (24 May 2010, Dean)

The Guardian (25 May 2010, Chris French)

LA Times (26 May 2010, Thomas H. Maugh II)

eSkeptic (26 May 2010)

United States Chess Federation (28 May 2010, Tom Braunlich)

The Independent (29 May 2010, Morton Schatzman)

The Economist (3 Jun 2010)

From references on the site


MacMahon, Percy Alexander
Monthly Notices of the Royal Astronomical Society, Vol. 90, p.373-378 (12 November 2014)



Martin Gardner (all 14 August 2014)

A Conversation with Martin Gardner (Feb 1979, Anthony Barcellos and multiple participants, in The Two-Year College Mathematics Journal, Sep 1979)
Conversation with Martin Gardner, Annotator of Wonderland (1979, Jan Susina, The Five Owls, Jan/Feb 2000)
‘The Annotated Gardner’ (Michael Shermer, Skeptic magazine, vol 5 no 2, 1997), republished with a new introduction as ‘Martin Gardner 1914–2010, Founder of the Modern Skeptical Movement’ (26 May 2010, eSkeptic),
A Mind at Play: An Interview with Martin Gardner (Kendrick Frazier, Skeptical Inquirer, Mar/Apr 1998)
Interview with Martin Gardner (2004, Allyn Jackson, AMS Notices, Jun–Jul 2005)
Martin Gardner's Magic Spells (Mar 2006, MAA Online, Colm Mulcahy, Oct 2006)
The Martin Gardner Interview (2005, Don Albers, in 5 parts online)
Conversations with Martin Gardner (May 2007, Donald E. Simanek)

From references on the site





‘M.C. Escher shape puzzle’, Mosaic II (22 July 2012) I project


‘Perpetual Puzzles’. Makoto Nakamura’s boxed puzzle, ‘Birds of Paradise’ (5 May 2013)


‘Sardines Puzzle’. Bear, Bear & Bear Ltd, Greetham, Rutland, England (26 May 2014)

20 fish to arrange as in a tiling, in layers. I’m not particularly impressed with the quality of the fish design, and indeed of the puzzle concept itself


‘Tesselz’ The Tesselation Jigsaw Puzzle  iproject. (25 October 2014) 507 pieces. Calla lilies by David Muench

A square and rhomb puzzles, as designed by Haresh Lalvani. 1999

see patent 5,007,220




Inspirations. A short video on Mathematics, many references to M.C. Escher 3 minutes 41 seconds by Cristóbal Vila 2010 (15 April 2015)


Nature by Numbers. By Cristóbal Vila 2010. Good slick 3.43 Fibonacci (15 April 2015)


Infinite Escher. John Sanborn, Mary Perillo and Dean Winkler (with Sean Ono Lennon). New York: Sony Corp, 1990 (26 February 2016)




Kluger, David and Jane Richtmyer. ‘A Pentagon Paper’. Not dated

A short compilation in note form on tiling, mostly with pentagons




Crannell, Annalisa. Report: ‘The 2012 Joint Mathematical Meetings Exhibition on Mathematical Art’. Journal of Mathematics and the Arts. Vol. 6, No. 4, December 2012, 211-217 (24 April 2013)


Hall, Rachel Wells. Report: ‘The mathematical art exhibition at BRIDGES: mathematical connections in art, music and science, Leeuwarden, the Netherlands, July 2008’

Journal of Mathematics and the Arts. Vol. 2, No. 4, December 2008, 47-51 (29 April 2013)


Hart, George W. and Natasha Jonoska. Report:  ‘Knotting Mathematics and Art’. Journal of Mathematics and the Arts. Vol. 2, No. 1, March 2008, 47-51 (29 April 2013)

A report: Knotting mathematics and art: Conference in Low Dimensional Topology and Mathematical Art University of South Florida, Tampa FL, 1-4 November 2007. Of general interest.


Knoll, Eva. Report: ‘The 2009 mathematical art exhibition at BRIDGES: Renaissance Banff II Conference’. Journal of Mathematics and the Arts. Vol. 4, No. 2, June 2010, 101-108 (26 April 2013)




Tony Lee’s Islamic patterns notebooks, handwritten, in three parts.(10 May 2013)

No Cairo-like pentagons


The New Yorker. July 5 2010. Escher's Sky and Water II appropriated/adapted for cover design by Bob Staake ‘Gulf Sky and Water’.