My Library - Articles

ARTICLES

 

A

 

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. pp. 100-112 (30 April 1994).

Abbas, Masooma. ‘Ornamental Jālīs of the Mughals and Their Precursors’. International Journal of Humanities and Social Science Vol. 6, No. 3, March 2016 (October 2018)

Of recent (October 2018) jali interest. Gives a good, semi-popular account, without too many pages.

 

Abercrombie, M. L. J. ‘Studies, Concepts and Research. The Uses and Abuses of Boundaries - Perception: the Structure of Space and Group Process’. Group Analysis, Vol. 12, 1: pp. 30-40. 1979.

Occasional Escher: p. 32 Hand with Reflecting Sphere p. 38, Three Spheres p. 39. Paper given at the Royal Society of Medicine in the section on psychiatry.

Aboulfotouh, Hossam M. K. and Gamal A. Abdelhameid. ‘Retrieving the Design Method of the Islamic Decagonal Girih Patterns’. Proceedings of the 3rd International Conference of the Faculty of Fine Arts, Alexandria University, at Bibliottheca Alexandrina, 2007 (Date not stated.)
Broadly, a rebuttal of Peter Lu and Paul Steinhardt’s conclusions in their widely accepted controversial paper in Nature as to the design process.

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, pp. 41-51 (16 December 2011)

Academic. Note that I have other articles by Adams, from his ‘Mathematical Bent’ column, but these are somewhat frivolous in nature, and so not included here.

 

Akiyama, Shigeki. ‘A Note on Aperiodic Ammann Tiles’. Discrete & Computational Geometry (2012) 48: pp. 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, pp. 1-10

 

Akelman, Ergun, Vinod Srinivasan, Zeki Melek and Paul Edmundson. ‘Semiregular pentagonal subdivisions’. Journal not stated. (8 April 2011)

3D mesh covering, including a Cairo tiling, albeit of limited value, and indeed interest here, given its academic nature.

 

Alexanderson, Gerald. L. ‘Award for Distinguished Service to Professor Murray Klamkin’. American Mathematical Monthly pp. 3-4, January 1988 (6 March 2013)

Praise for Klamkin. General interest as to Klamkin.

 

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 pp. 192-196

From a reference in Tilings and Patterns. Semi-popular.

 

Alexanderson, Gerald. L. and John E. Wetzel. ‘Simple Partitions of Space’. Mathematics Magazine Vol. 51, No. 4 September 1978 (6 March 2013)

Of an academic nature. Perhaps the title deceived me into thinking it would be simple….

 

————. ‘Divisions of Space by Parallels’. Transactions of the American Mathematical Society, Vol. 291, Number 1, September 1985 (6 March 2013)

Of an academic nature. Perhaps the title deceived me into thinking it would be simple….

 

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-Escher’. Rolling Stone. 52. p. 40, February 21, 1970 (28 March 2011)

An (early US) 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 Liberation, Self Portrait and Waterfall.

 

Aljamili, Ahmad M. and Ebad Banassi. ‘Grid Method Classifciation of Islamic Geometry Patterns’. In M. Sarfraz eds Geometric Modelling: Techniques, Applications, Systems and Tools. Springer Dordecht, 2004. 233-254.

Popular account, although as with much Islamic articles of a similar nature, has not led me to any new insights.

 

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

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

 

Amiraslan Imameddin. Backgoundless Geometrical Calligraphy. Source not stated. (22 October 2010)

Arabic tessellations with words.

 

Alvarez, Josefina and Christina Varsavsky. ‘Tilings’. Function Volume 28 Part 4 pp. 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 and G. C. Shephard. ‘Aperiodic Tiles’. Discrete & Computational Geometry 8: pp. 1-25 1992 (25 May 2012)

Academic, although written in a largely popular manner. I strongly suspect that although Ammann fronts this (having discovered the tile in question), the writing is essentially from Grünbaum and Shephard. Essentially of a ‘stepped’ tile, with an elliptical ‘key’. As such, not of any real interest.

 

Andraos, John. ‘Named Optical Illusions’. Self published 1-25

A listing of named sources and the journal in which they appeared. Useful.

 

André, Jacques and Denis Girou. ‘Father Truchet, the typographic point, the

Romain du roi, and tilings’. TUGboat, Volume 20, 1999, No. 1 pp. 8-14 (25 August 2016)

In short, a brief discussion on Truchet, of typographic point and tilings, all of a popular level, of interest.

 

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

N.B. The chronology discrepancy between date of publication and owning 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. pp. 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.

 

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 pp. 25-31. 2003 Texas Instruments Incorporated (2010)

Brief discussion of the Cairo tiling, p. 30.

 

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

 

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

‘Speaking of Pictures’ is a generic term, Life has other articles of this title.

 

Anonymous. Time. 25 October 1954. ‘The Gamesman’, Vol 65, No. 17. p. 68 (28 March 2011)

 

Anonymous. Time. 2 April 1951. ‘Prying Dutchman’, p. 50 (28 March 2011) Vol. 57 No. 14 (Locher reference)

 

Anonymous. H. S. M. Coxeter. Biography

 

Anonymous. Scholastic Art. ‘Art Meets Math’. February 2010. pp. 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.

 

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

 

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

 

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

 

Anon. ‘The Lure of Puzzle Inventing’. Popular Mechanics, January 1935, pp. 20-23, 128A, 130A. (22 March 2016)

From a reference in Williams. Minor mathematics. Loyd, 14-15, Jigsaws, wedge tromino.

 

Anonymous. ‘Unreal Reality, Real Unreality’. Intellectual digest [sic]. June 1972. Volume II No. 10. pp. 71-73

From a reference in a letter of Cornelius van. Roosevelt in The New York Times. A speculative purchase resulting in a major disappointment! Given the title of the journal, I was under the impression there was going to be an worthy article on Escher, but of which in a sense there is an article, but it cannot be described as worthy - all it does is repeat select text and illustrations from The Graphic Work!. There is nothing in the remotest sense ‘original’! Prints include Hand with Reflecting Globe, Puddle, Reptiles, Metamorphosis Magic Mirror, Relativity, Descending ascending Belvedere, Waterfall.

 

Appel, Kenneth and Wolfgang Haken. ‘Every Planar Map is Four Colorable’. Illinois Journal of Mathematics 21 pp. 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, pp. 10-20 (9 December 2011)

Academic nature throughout.

 

ApSimon H. G. ‘Almost Regular Polyhedra’. The Mathematical Gazette, Vol XL, No. 332 May 1956, pp. 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 pp. 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 pp. 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.

 

Aslet, C. ‘Art is Here: The Islamic Perspective’. Leighton House. Country Life, Vol 16, 1642-1643, 1983. WANTED

From a reference in Abbas.

 

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

Largely popular account.

 

Avital, Doron. ‘Art as a Singular Rule’. Journal of Aesthetic Education, Vol. 41, No. 1 (Spring, 2007), pp. 20-37 (28 February 2013)

Brief Escher discussion and illustration.

 

 

B

 

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 pp. 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, pp. 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., pp. 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): pp. 63-73, 2011.

 

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

The Strand, 1907, pp. 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) pp. 295-307 (30 May 2012)

Academic.

 

Bell, A. W. ‘Tessellations of Polyominoes’. In Mathematical Reflections edited by members of the ATM  (Cambridge University Press, 1970.  WANTED

Quoted by Gardner, ‘More about tiling the plane…’. Mathematical reflections : contributions to mathematical thought and teaching, written in memory of A. G. Sillitto. / Edited by members of the Association of Teachers of Mathematics.

 

Bantegnie, Robert. ‘Sur Quelques points de Geometrie des Nombres’ (31 May 2017)

 

————. Sur les configurations de Hadwiger. Arch. Math 534-538 (31 May 2017)

 

————. ‘Etalements Cristallogrophiques’. Acta Mathematica Academiae Scietiarium Hungaricae Tomus 30 (3-4) 1977, pp. 283-302. (31 May 2017)

From a reference in Tilings and Patterns and Schattsneider, ‘Will it Tile’

.

————. ‘Animaux Plans et Gropoes Cristallographiques’. Theorie des Nombres, 1-16, 1976-1977. (31 May 2017)

 

————. ‘Pavements Equilibres du Plan’. Theorie des Nombres, 1-27, 1978-1979. (31 May 2017)

 

————. ‘Parties fondamentales des groupes  cristallographie plans’. Mimeographed  notes Besaçon, 1978. WANTED

From a reference in Tilings and Patterns.

 

Barcellos, Anthony. ‘A Conservation with Martin Gardner’. The Two-Year College Mathematics Journal. Volume 10 No. 4 Sep 1979 pp. 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 pp. 59-71. (2016)

On polyhedra, with Cairo-like similarities. Has a Cairo tiling p. 65.

 

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

 

Barnette, David W. ‘The Graphs of Polytopes With Involutory Automorphisms’. Israel Journal of Mathematics. Vol. 9, 1971, pp. 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, p. 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, pp. 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 pp. 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 pp. 129-144 (23 September 2013)

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

 

Barrowcliff, Vikki. ‘My experience as an NQT Head of Year’. Management in Education, Vol. 24, 3: pp. 94-95. July 2010.

Non-tessellating article, with a one-line mention of Escher, p. ?, no illustrations.

 

Basin, S. L. ‘The Fibonacci Sequence as it Appears in Nature’. The Fibonacci Quarterly, 1 1963 pp. 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 pp. 318-319 (11 April 2013)

Polyominoes.

 

Bays, Carter. Cellular Automata in the Triangular Tessellation’. Complex Systems Publications, Volume 8, Issue 2, pp. 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 *, pp. 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, pp. 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) pp. 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. pp. 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 pp. 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.

 

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, pp. 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, pp. 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 pp. 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 pp. 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, pp. 21-22 1957, and reproduced from the translation in Sov. Phys Crystall 2, pp 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 pp. 220-221 (13 October 2006)

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

 

Benedikt, M. L. ‘On Mapping the World in a Mirror’. Environment and Planning B: Planning and Design, Vol. 7, 4: pp. 367-378. 1980.

Non-tessellating article, with a one-line mention of Escher, illustrated with Escher’s ‘Hand With Reflecting Globe’, p. 368.

 

Bennett, Curtis D. ‘A Paradoxical View of Escher’s Angels and Devils’. The Mathematical Intelligencer , pp. 39-46 (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.

 

Berglund, John. Is There a 5k (2016)

Includes a freehand drawing of the Stein tiling.

 

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

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

 

Bigo, Didier and R. B. J. Walker. ‘Political Sociology and the Problem of the International Millennium’, vol. 35, 3: pp. 725-739, 2007.

Non-tessellating article, with a brief paragraph mention of Escher, p. 737, Mobius Band illustration p. 739.

 

Bilney, Bruce. ‘Ozzie The Magic Kangaroo’. The Australian mathematics teacher vol. 52 no.4, pp. 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.

 

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.

 

Bollabas, Bela. ‘Filling the Plane with Congruent Convex Hexagons Without Overlapping’. Tudomanyegyetem Annlaes Sectio Mathematica 1963, pp. 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 pp. 339-342 (20 February 2013)

Child mathematics.

 

Bolster, L. Carey and Evan M. Maletsky. ‘Tangram Mathematics’. December 1975, pp. 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)

 

Boselie, Frans and Annalisa Cesàro. ‘Disjunctive Ambiguity as a Determinant of the Aesthetic Attractivity of Visual Patterns’. Empirical Studies of the Arts, vol. 12(1) pp. 85-94, 1994

Non-tessellating article, with a one-line mention of Escher, p. 85, no illustrations, and bibliography.

 

Bossert, Philip J. ‘Horseless Classrooms and Virtual Learning: Reshaping Our Environments’. NASSP Bulletin, Vol. 81, 592: pp. 3-15. November 1997.

Non-tessellating article, with a one-line mention of Escher, p. 14, no illustrations.

 

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!

 

Bovill, Carl. ‘Using Christopher Alexander’s Fifteen Properties of Art and Nature to Visually Compare and Contrast the Tessellations of Mirza Akbar’. Nexus Network Journal, Volume 14, No. 2, 2012, pp. 333-343 (13 September 2018)

Found upon the contemporary search on Christopher Alexander. Not sure of the legitamacy of Bovill’s (an architect) premise, in which two tessellations of Akbar are ‘assessed’. Whatever, of little direct interest.

 

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, pp. 58–86, 2010 (14 November 1997) Eh? Dates contradict

Somewhat advanced, although there is the occasional diagram of interest

 

Boyd, David. W. ‘The disk-packing constant’. Aequationes Mathematicae 7 1971, pp. 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), pp. 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, pp. 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 pp. 186-187 1921

 

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

(Not in Frederickson).

 

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

Form a 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, pp. 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, pp. 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, pp. 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.

 

Brêchet, Michel. ‘Le Coin Des Pavages’, 1-4. Math Ecole pp. 207-210, 2004. (In French)

Tessellations, with illustrations by schoolchildren.

 

Bricard, R. ‘Sur une question de géométrie relative aux polyhèdres’. Nouvelles Annales de Mathématique 15, pp. 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, pp. 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) pp. 305-317

 

Broos, C. H. A. ‘Escher: Science and Fiction’. In The World of M. C. Escher. Abradale Press 1988 pp. 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 Geometric Design’. pp. 20-27 (19 October 2015)

From catalogue of 2012 exhibit. Makes use of Escher’s works. Sketches from the Alhambra, PD, Order and Chaos II prints.

 

Brown, Harold I. ‘Self-Reference in Logic and Mulligan Stew’. Diogenes, vol. 30, 118: pp. 121-142. June, 1982.

Non-tessellating article, albeit with a slant towards Gödel, Escher, Bach by Douglas Hofstadter, with many references, pp. 122,129, 131-132 no illustrations. However, the text is largely unreadable, in the Hofstadter vein.

 

Browne, Cameron. ‘Duotone Truchet-like tilings’. Journal of Mathematics and the Arts. Vol. 2, No. 4, December 2008, pp. 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), pp. 201-220, June 18 1990 (27 September 2013)

Largely academic, of no use.

Also see interview with Jeane Daems the 1954 exhibit at the Stedelijk Museum, of two interviews, separately, with N. G. de Bruijn and Hendrik Lenstra.

 

————. ‘Penrose patterns are almost entirely determined by two points’. Discrete Mathematics 106/107 1992, pp. 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, pp. 748-753

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

 

Bunch, Phillipa. Logical Challenge. Future Publishing Limited. Issue 10, 2006 (14 May 2016)

Various popular logical puzzles of no real interest. However, included here as it has Escher’s Concave and Convex on the front cover, which is its sole attraction. Note that there is no discussion of Escher or the artwork inside.

 

Burgiel, H and M. Salomone. ‘Logarithmic spirals and projective geometry in M.C. Escher’s Path of Life III’. Journal of Humanistic Mathematics, Volume 2 Number 1 (January 2012), pp. 22-35.

 

————. ‘How to lose at Tetris’. The Mathematical Gazette. Vol. 81, 491 July 1997, pp. 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 pp. 372-373 (23 February 2013)

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

 

Burt, M. ‘The wandering vertex method’. Structural Topology 6 1982, pp. 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.

 

C

 

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 pp. 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, pp. 38-40 (29 July 2015)

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

 

Carrasco, Marisa, Svetlana M. Katz, Julia Winter. ‘Multidimensional scaling and experimental aesthetics: Escher’s prints as a case study’. Empirical studies of the arts, vol. 11(1) pp. 1-23, 1993 (17 October 2016)

 

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.

 

Chapelot, Pierre. 'Une Decouverte: le visionnaire Escher.' Planete, no. 8 (1963), p. 60. (28 August 2016)

First saw Sunday 28 August 2016 on Scoplato site as read only, it is not available as a pdf.

 

Chassagnoux, Alain, Michel Dudon, Didier Aubry, J. Chassagnoux, A. Chomarat, C. Diacon, F. Miguet, J. Savel. ‘Teaching of Morphology’. International Journal of Space Structures, Vol. 17, 2-3: pp. 197-204.  2002.

 

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

 

Chen, Elizabeth R. ‘A Dense Packing of Regular Tetrahedra’. Discrete & Computational Geometry (2008) 407: pp. 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, pp. 17-32, 1989 (27 September 2013)

Academic.

 

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 much of interest in a generalised way. Has interesting Cairo tiling references, pp. 783-784, derived from James Dunn’s 1971 article and references this in the bibliography

Of note is the reference to the well-known eight-pointed star and pointed cross tiling, p. 759, in which this is described by Bakhtiar mystically, as ‘The Breath of The Compassionate’, and of which the term seems to have spread. Chorbachi takes him, and other mystics, to task. This reference only came to light in September 2017, upon research on the star and cross tiling in connection with a variation by Muriel Higgins; I had forgotten the reference!

 

Choi, John and Nicholas Pippenger. ‘Counting the Angels and Devils in Escher's Circle

Limit IV’. Journal of Humanistic Mathematics, Volume 5 Issue 2 July 2015 (3 August 2016)

Academic throughout, of no use.

 

Chow, William W. ‘Automatic Generation of Interlocking Shapes’. Computer Graphics and Image Processing 9 (1979), pp. 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.

 

————. ‘Interlocking shapes in art and engineering’. Computer Aided Design, Vol. 12, No.1, Janaury 1980, 29-34 (22 October 2018)

Illustrated with works of Escher-like tessellations from his (anonymous) engineering students, of which they are a few novel instances, with forks and  keys, with a nod to the needs of industry. Heesch nomenclature is used once again, seemingly in the same vein in his earlier paper.

 

Christensen, A. H. J. ‘Recursive Patterns or the Garden of Forking Paths’. Leonardo 15 1982 pp. 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, then becomes academic.

 

Chung, Ping Ngai, Miguel A. Fernandez, Niralee Shah, Luis Sordo Viers and Elena Wilker. ‘Perimeter-minimizing pentagonal tilings’. Involve Vol. 7, No. 4 pp. 453-488, 2014 (6 March 2019, although seen much earlier? Or perhaps I am confusing a like paper by the many authors here.)

Makes much use of the Cairo tiling and prismatic tiling in conjunction. Although the tenure of the paper is of an advanced nature, way beyond my understanding, it is still broadly readable, at least of the initial pages. Input is provided by Frank Williams. Also see ‘Isoperimetric Pentagonal Tiling paper’ by Chung et al.


Ping Ngai Chung, Miguel A. Fernandez, Yifei Li, Michael Mara, Frank Morgan, Isamar Rosa Plata, Niralee Shah, Luis Sordo Vieira, Elena Wikner. ‘Isoperimetric Pentagonal Tilings’. Notices of the American Mathematical Society 59:5, 2012, pp. 632-640. (First saw 9 December 2011, preprint)

Makes much use of the Cairo tiling and prismatic tiling in conjunction. Although the tenure of the paper is of an advanced nature, way beyond my understanding, it is still broadly readable, at least of the initial pages.

 

Chung, Priscilla. ‘Imprint (NYC): The evolution of motifs in fashion Houndstooth’. (18 October 2018, although seen much earlier, but not sure when,  as a guess, in 2015?)

Available on online catalogue site ‘Yumpu’, as a flipbook and PDF. Quite how best to categorise this, as a book or article is not clear. A very nice history on houndstooth indeed, with much that is new.

 

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.

 

Clarke, Eric. ‘Meaning and the Specification of Motion in Music’. Musicae Scientiae, vol. 5, 2: pp. 213-234. Fall 2001.

Non-tessellating article, with a one-line mention of Escher, p. 217, no illustrations.

 

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

Many interesting ‘simple’ diagrams.

 

Clauss, Judith Enz. ‘Pentagonal Tessellations’. The Arithmetic Teacher (NCTM), 38(5): pp. 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 then known.

 

Clemens, Stanley R. ‘Tessellations of Pentagons’. Mathematics Teaching, No. 67 (June), pp. 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 pp. 169-173 (9 April 2013)

Packing rectangular trays, non orthogonally.

 

Collatz, Lothar. Vortrag und Ansprachen. Augsburger Universitätsreden 8. 1986. Geometriche Ornamente (in German) 1-56. (3 September 2018)

From a reference in Trinajstić. Replete with tiling diagrams, albeit effectively with my having only the most minimal German is unreadable. To what extent this is original is unclear. Of most intest is a Cairo tiling diagam, p. 28, and again, p. 45. Pp. 2-13 are all text, and can be disregreded. Attempts are made occasionally with Escher-like tessellation pp. 25, 27, with minimal detail, of a eye. However, these are mostly in effect designed as ‘overlaps’. One exception, of a polyomino, is that of a dog, p. 29. which is quite reasonable. P. 27 shows the well-known German paving tile. of note is the bibligaphy for an intersting reference of a listing of 20 entires, all academic save for one, of which this is of a recent correspondent, namely Muriel Higgins and her patchwork book!

 

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

General interest, academic.

 

Conlan, James P. ‘Derived Tilings’. Journal of Combinatorial Theory, Series A 20, pp. 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 pp. 903-904 (7 March 2013)

 

Cousineau, Guy. ‘Tilings as Programming Exercise’. Journal not given (5 April 2012)

Mostly on computer programming, of an advanced level. occasional Escher, and use of Raoul Raba Kangaroos.

 

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

Not mentioned in any of Frederickson’s three books

 

Cotter, J. R.  ‘Pythagoras’s Theorem as a Repeating Pattern’. Nature, May 6, 1922, p. 579. (31 May 2017)

 

Coxeter, H. S. M. ‘The Polytopes with Regular-Prismatic Vertex Figures’. Philosophical Transactions of the Royal Society of London. Series  A  Vol. 229, 1930, pp. 329-425 (24 October 2018)

First, as a general statement, typically, Coxter’s works are of a too advanced nature, way beyond my understanding. However, as he has an interest in tiling and Escher, I do indeed see the odd diagram at least of interest. Further, the Cairo tiling appears among his works (book cover and *) and so it is possible that it has appeared in his publications elsehere. Therefore, I am more inclined to peruse such material than perhaps otherwise.

Academic, of no real interest; far too advanced for me. Replete with text and equations, with only occasional diagram.

 

————. ‘Crystal Symmetry and Its Generalizations’.  in A Symposium on Symmetry, Trans Royal Society, Canada 51 series 3 sec 3 1957, pp. 1-13. WANTED

 

————. ‘Twelve points in PG(5,3) with 95040 self-tranformations’. Proceedings of the Royal Society A, 1958

Academic, of no real interest; far too advanced for me. Replete with text and equations, with only occasional diagram.

 

————. ‘The four-color map problem, 1840-1890’. The Mathematics Teacher April 1959. pp. 283-289. (28 February 2013)

General historic interest

 

————. ‘Regular compound tessellations of the hyperbolic plane’. Proceedings of the Royal Society A, 1964 pp. 147-167 (26 October 2018)

Has many circle limit type diagrams, of various forms. Academic, of no real interest; far too advanced for me.

 

————. Review of Regular Figures by L. Fejes Toth, Science, Vol. 146 4 December 1964 pp.1288

 

————. ‘The Problem of Apollonius’. The American Mathematical Monthly, Vol. 75, No. 1 Jan, 1968, pp. 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 pp. 297-310. 1971. (30 January 2012)

‘Typical Coxeter’, too advanced.

 

———— ‘Kepler and Mathematics’. In Vistas in Astronomy. Four Hundred Years Proceedings of Conferences held in honour of Johannes Kepler. Vol.18 pp. 661-670. Pergamon Press. 1975. Beer, Arthur and Peter Beer (editors). (c. 2001)

A major collection of articles (of 1034 pages!) arising from the conference. Perhaps somewhat surprisingly tessellations, and to an extent polyhedra, are not really discussed. Instead, this is really more of his astronomical work. Chapter 11 is described as ‘Kepler as Mathematician and Physicist’. Of most interest here is Coxeter’s essay ‘Kepler and Mathematics’ pp. 661-670. N.B. Also discussed under Beer et al.

 

————. ‘The Trigonometry of Escher’s Woodcut Circle Limit III’. In M.C. Escher’s Legacy A Centennial Celebration. Doris Schattschneider and Michele Emmer, Editors First edition 2003, and second 2005. pp. 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. pp. 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, pp. 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 pp. 327- (January 2016)

‘Typical Coxeter’, too advanced.

 

————. ‘Escher’s Lizards’. In: Structural Topology No.15 1988. pp. 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), pp. 27-38, March 1993 (27 September 2013)

Academic, of no use.

 

————. ‘Escher’s Fondness for Animals’. In M.C. Escher’s Legacy. A Centennial Celebration. Springer. First edition 2003, and second 2005. pp. 1-4 (31 August 2005)
A footnote of the page states that this is a reprint of the catalogue of the 1985 Escher congress at the University of Rome, with George Escher giving additional comments in an addendum.

 

Coxeter, H. S. M; M. S. Longuet-Higgins and J. C. P. Miller. ‘Uniform Polyhedra’. Philosophical Transactions of the Royal Society of London. Series  A  246, 1953/54, pp. 401-450

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

 

Craig, E. J. ‘Phenomenal Geometry’. The British Journal for the Philosophy of Science, Vol. 20, No. 2 (Aug., 1969), pp. 121-134 (28 February 2013)

Use of Escher’s print Ascending and Descending, pp. 130-131.

 

Crompton, Andrew. ‘Lifelike Tessellations’. In Manchester Architectural Papers 2000, edited by Geoff McKennan, pp. 17-24 (30 May 2006)

Pleasing, although brief. Some hints and tips on drawing lifelike tessellations. Begins by gives a history of lifelike tessellations, pp. 17-18, of the Art Nouveau period.  A listing of the ‘permissible’ isohedral tilings is given, as well as a brief history of lifelike tiling. 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, pp. 36-56, 2009 (25 November 2011)

Popular account. Conclusion drawn is a negative; no such quasi-periodcity is found.

 

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

Popular account. Of peripheral interest only (being on knots).

 

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

General interest. The first of a two-part article.

 

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

General interest. As detailed above.

 

————. ‘Kepler’s Work on Polyhedra’. ‘Years Ago’ column in The Mathematical Intelligencer. Vol. 17, No. 3, 1995 pp. 23-33 (16 December 2011)

Popular account. This article predates his book on Polyhedra and acted as the inspiration.

 

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

Popular account.

 

Crowe, D. W. ‘The Geometry of African Art II. A Catalog of Benin Patterns’. Historia Mathematica 2 1975, pp. 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. pp. 407-411, 1986 (9 September 2010)

 

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

Potted account of tile history.

 

Cummings, Meridith. ‘Alabama’s Houndstooth History’. Crimson Magazine.Net (18 October 2018)

Seen earlier I believe. 54-58. on Bear Bryant. Yumpu publication.

 

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

Polyhedral lampshade.

 

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

 

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

Largely popular account

 

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

On matters of incorrect usage of the terms.

 

Also see letters and reviews.

 

Curl, Robert F and Richard E Smalley. ‘Fullerenes’. Scientific American October 1991. pp. 32-41.

 

D

 

Daems, Jeanine. ‘Escher for the mathematician’ (as in original). NAW 5/9 nr.2 June 2008. (15 December 2009)

Two interviews, separately, with N. G. de Bruijn and Hendrik Lenstra. De Bruijn addresses the 1954 exhibit at the Stedelijk Museum, whilst Lenstra primarily concerning aspects of Escher’s print ‘Print Gallery’. Cross referenced with entries for both.

 

Danzer, Ludwig, Branko Grünbaum and 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.

The 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!

 

Deręgowski, Jan B. ‘Bilateral Symmetry and Perceptual Reversals’. Perception, Vol. 43, 1: pp. 85-89. 2014.

Non-tessellating article, with a one-line mention of Escher, p. 88, no illustrations.

 

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

Composing Islamic patterns by means of lattices of circles. Relatively sparsely illustrate, and I recall this procedure was critiqued, but by who I forget.

 

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.

 

————. ‘Geometry Comes Up to Date’. New Scientist 5 May 1983. WANTED

From a reference in Abbas.

 

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)

Academic.

 

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!

 

Dotto, Edoardo. ‘Drawing Hands. The Themes of Representation in Steinberg and Escher’s Images’. MDPI Proceedings 2017 (14 May 2018)

 

Draper, Stephen W. ‘The Penrose Triangle and a Family of Related Figures’. Perception, Vol. 7, 3: pp. 283-296. June, 1978.

 

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.

 

Druick, Douglas and Elizabeth Driver. ‘The Graphic Work of M.C. Escher’. The National Gallery of Canda Journal, 3, Jnaury 1975, pp 1-8. WANTED

From a reference in E. B. Versluis, M. C. Escher: Art and Science.

 

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’

 

————. ‘Creating Repeating Hyperbolic Patterns—Old and New’. In Notices of the AMS, Volume 50, Number 4 April 2003

 

————. ‘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.

 

E

 

Earle, Robert. ‘On the Campus’. Princeton Alumni Weekly Volume 72 (16 July 2015)

From Google books. Shows Verbum

 

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

 

Eberhart, Mark. ‘Classics’. Resonance, June 2006. (October 2018)

Has cairo tiling diagram, without attribution, found in a excerpt page on C. S. Smith’s book A Search for Structure.

 

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.

 

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. 2nd Printing 2005 (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.

 

————. ‘Ravello: An Escherian Place’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration. 2nd Printing 2005 (31 August 2005)

Some nice stories on Ravello.

 

————. ‘Mathematics and Art: Bill and Escher’. Bridges 2000, pp. 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.

 

————. ‘Escher, Coxeter and Symmetry’. International Journal of  Geometrical Methods in Modern Physics. World Scientific Publishing Company, Vol. 3 Nos. 5 & 6, 2006, pp. 869-879 (27 October 2017)

 

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.

 

Engel, Peter. ‘A Paper Folder’s Finding’. The Sciences 16-22 (19 April 2017)

Minor mention of Escher, p. 18, and an Alhambra sketch of his, p. 19

 

Eodice, Michael T., Larry J. Leifer and Renate Fruchter. ‘Analyzing Requirements - Evolution in Engineering Design Using the Method of Problem-Reduction’. Concurrent Engineering, vol. 8, 2: pp. 104-114. June 2000.

Non-tessellating article, with a one-line mention of Escher, p. 109, no illustrations.

 

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)

 

————. ‘Selection is Distortion’. In Doris Schattschneider and Michele Emmer, Eds. M.C. Escher’s Legacy A Centennial Celebration 2003 (31 August 2005) 5-16

Of note here is that on p. 11 Ernst states that Escher had a copy of Jamnitzer’s book on polyhedra in his possession. In recent times, I have come to realise that Ernst’s main interest in Escher is his spatial work, rather than tessellations, as this article shows. Again, Ernst here largely dismisses Escher’s tessellations as a body of work; the article is mostly on other aspects of his work.

 

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

 

Epstein, David. ‘Geometers unify their ideas of space’. New Scientist 11 April 1985. 29-32 (5 January 2017)

Some advanced concepts of hyperbolic geometry illustrated with Escher’s Circle Limit IV, and a brief discussion, pp 30-31.

 

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)

 

F

 

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 Robert Bosch, Reza Sarhangi & Douglas McKenna (eds) Proceedings of Bridges Towson Conference 2012, Mathematics, Music, Art, Architecture, Culture. 299-306
First, beginning in 2012, and continuing to the present day (2018), Loe Feijs, of TUE university, has written a series of articles on Houndstooth/Pied-de-polue, for successive Bridges conferences. Each article is broadly a combination of popular and scholary accounts, and at times he uses his superior mathematicsal ability to write somewhat techicanlly, far beyond my undertstanding. Be that as it may, these can be consderd as a must read, and  they leave writimgs by others far in the shade.
A most pleasing discussion, although technical in places, but still lagley popuar, with some pleasing innovations, such as the houndstooth appearing in different orientations. Much mention of Heesch and Keinzle.

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 of a fractal premise, although somewhat technical in places.

Feijs, Loe M. G., Marina Toeters, Jun Hu and Jihong Liu. ‘Design of a Nature-like Fractal Celebrating Warp-knitting’. Proceedings of Bridges Conference 2014, Mathematics, Music, Art, Architecture, Culture. 369-371
On warp knitting, rather than pied de poule, which is not mentioned. Of no real interest.

Feijs, Loe M. G. and Marina Toeters. ‘A Novel Line Fractal Pied de Poule (Houndstooth)’
Bridges 2015 Conference Proceedings
In short, houndstooth as a line fractal. Somewhat advanced in nature.

Feijs, Loe M. G. and Marina Toeters. ‘Pied de Pulse: Packing Embrodered Circles and Coil Actuators in Pied de Poule (Houndstooth)’. Bridges Finland Conference Proceedings, 2016 415-418
In short, houndstooth as circles packing. Somewhat advanced in nature.

Feijs, Loe M. G. and Marina Toeters. ‘A Cellular Automaton for Pied-de-Poule (Houndstooth)’. Proceedings of Bridges 2017, Mathematics, Art, Music, Architecture, Education, Culture. pp. 403-406. Tessellations Publishing, Phoenix, Arizona
In short, houndstooth as celluar automontons. Somewhat advanced in nature.

Feijs, Loe. ‘Descending A Staircase’. Aplimat Proceedings 2018. Slovak University of Technology in Bratislava, Faculty of Mechanical Engineering. (October 2018)
Houndstooth in relation to John Horton Conway’s game of life, and Marcel Duchamp.

Férey, Gérard. (2014). ‘Mosaics, Quilts, Science and Crystal Structures – Which one Inspires the Other?’. Zeitschrift für anorganische und allgemeine Chemie. 2014,  (15), 3212-3216. (2 October 2017)
Has two Pavage De Diane references, figures 3 and 7.

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)

Mentioned in Grunbam. 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, of a frieze, although this is more space filling with a few elements, with considerable vacant space, far too much so,.

 

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.

 

Frank, F. C. and J. S. Kasper. ‘Complex Alloy Structures as Sphere Packings. I. Definitions and Basic Principles’. Acta Crysta 1958 184-190 (31 October 2018)

Upon seeing a reference to equilateral pentagon tilings (i.e Cairo tiling) in Frank’s preface to The Kelvin Problem, reference is seemingly made to equilateral pentagon tilings (i.e Cairo tilings) in conjunction with John Kasper’s own work on this. However, the exact publication was left unstated. Upon seeing a list of Frank’s publications in his obituary, this (and the following article) would apper to be what Frank is quoting. However, disappoinringly, of the following there is not any Cairo tile tile diagrams (or indeed, pentagon diagrams per se). As the title suggests, an academic article, mostly beyond my understanding.

 

————. ‘Complex Alloy Structures as Sphere Packings. II. Analysis and Classifcations of Reprsentative Structures’. Acta Crysta 1959 12, 483-499 (31 October 2018)

See above.

 

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

Although not strictly of a mathematical nature, included here as it is I believe quoted in tessellation matters, although I do not recall exactly where. Of note is that I must have purposeful sought this out; I photocopied it in Hull reference library, along with an article by Robinson and Wilson. Notably contains examples of the ‘Fraser spiral’, plates III-VII and other twisted cord type illusions. However, as regards tessellation it is inconsequential; although of interest in ‘optical illusion phenomena’, there is nothing tessellation related here.

 

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.

 

Fujika, Shin. VISION Vol. 20, No. 1, 000–000, 2008 (15 September 2016)

Somewhat surprisingly, this was the first conscious exposure to Shin’s Escher-like tessellation work, found on the Japanese tessellation page. Somehow or other, despite having looked at this, he had previously escaped me. His PDFs are full of interest

 

————. ‘Considerations of Penrose’s nonperiodic patterns and Escher’s patterns’.

Bulletin of JSSD Vol. 47, No. 5, 2001 (15 September 2016)

 

————. ‘Pattern Design research Using the Penrose Pattern (I)’

JSSD Vol.51, No. 5, 2005 (15 September 2016)

 

————. ‘Pattern Design research Using the Penrose Pattern (II)’

JSSD Vol.51, No. 5, 2005 27- (15 September 2016)

 

————. ‘Sampling Survey of ‘17 Kinds of Wallpaper Patterns’ in Marketing’.

Bulletin of JSSD Vol.47, No. 5, 2001 (15 September 2016) 31-

 

————. Figurations of Mine on 17 Kinds of Wallpaper Patterns”

Bulletin of JSSD Vol.53, No. 3, 2006 (15 September 2016) 21-

 

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.

 

Fukuda, Hiroshi, Michio Shimizu, and Gisaku Nakamura. ‘New Gosper Space Filling Curves’. Conference paper 2001 (4 August 2016)

 

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

Academic, of no practical use.

 

G

 

Gailiunas, 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)

Of a broadly popular level.

 

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

Nice treatment 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)

The first of a two-part series on MacMahon, by his champion, concentrating on his recreational interests. Popular account of MacMahon’s slab ‘stacking’ puzzles, and brief background as to MacMahon himself.

 

————. ‘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, although there is no mention made of the Cairo connection. Concentrates on tilings.

 

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, Martin

Martin Gardner, being a significant figure, gets a more thorough treatment than others. However, so numerous is his writings that to discuss everything in depth is simply too much. This being so, I concentrate primarily on his tiling and Escher articles.

 

————. ‘H. S. M Coxeter’. Scientific American

 A discussion of Coxeter and his new book, Introduction to Geometry.

 

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 on Escher. 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.

Popular account of hexagonal and pentagonal tiling (giving the eight types known as of that writing). Contains the second recorded reference to the Cairo pentagon, p. 114, giving an erroneous equilateral pentagon, and its construction p. 117. Much discussion on pentagon and hexagon tiles. Escher illustration of ‘tadpoles’ p. 112.

 

————. ‘More about tiling the plane: the possibilities of polyominoes, polyiamonds, and polyhexes’. Scientific American. August 1975 112-115. (24 February 1987 and 11 February 1988)

Popular account of polyominoes, polyiamonds, and polyhexes. Mentions the Conway criterion p. 112. Penrose loaded wheelbarrow tile mentioned, and illustrated, p. 115. Mention of Percy MacMahon’s book New Mathematical Pastimes, p. 115. Minor Escher reference p. 115.

Of note is the date when this was first seen, or at least first studied, namely 24 February 1987, which is one of the earliest of my tiling studies. However, this was not photocopied, but of necessity copied by hand in the Grimsby Central library, along with ‘On tessellating the plane with convex polygon tiles’.

 

————. ‘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. Escher bird and fish PD * p. 110. Mentions of Voderberg spiral, p. 111, and Robinson’s six tiles, p. 112. Then Penrose tiles in depth, with Conway nomenclature, 112-120, including thin and thick rhombs p. 120.

 

————. 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, and 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.

 

Gethner, Ellen, David G. Kirkpatrick, Nicholas J. Pippenger. ‘Computational Aspects of M.C. Escher’s Ribbon Patterns’. Theory Comput Syst (2014) 54:640–658 (17 October 2016)

Largely of an academic nature, the usefulness or otherwise as above.

 

Gibbons, Brian. ‘The Question of Place’. vol. 50, 1: pp. 33-43.  Oct 1996.

Non-tessellating article, with a one-line mention of Escher, p. 41, no illustrations.

 

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.

 

Ginzburg, Ralph (editor). Avant Garde. Large Charge The Exciting Art of M.C. Escher magazine prototype 1974 Use of Escher’s Dragon’s print, with a heading ‘Up against the Wall! The Avant Garde Poster’, and below, ‘Do Your On Thing, Wood Engraving by M. C. Escher’. Saw on eBay, not in possession. First saw October 2017

From Abe books

About this Item: Avant Garde Media, New York, 1974. Soft cover. Condition: Fine. Carruthers, Roy and M.C. Escher (centerfold) (illustrator). 1st Edition. Tabloid Size Magazine. First Printing. 16 x 11 inches. Printed on newsprint. 24 pages.

 Seemingly no article on Escher, just the centrefold mage and captions, available at ridiculous pieces, from $150

From Wikipedia: Avant Garde was a magazine notable for graphic and logogram design by Herb Lubalin. The magazine had 14 issues and was published from January 1968 to July 1971. The magazine was based in New York City. The editor was Ralph Ginzburg. Avant Garde 3, published in May 1968, lists in the masthead:.

 

Glasser, L. ‘Teaching Symmetry The use of decorations’. Journal of Chemical Education 44, no 9 (September 1967) pp. 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

 

Glassner, Andrew S. ‘Frieze Groups’. In IEEE Computer Graphics and Applications, pp. 78-83, May 1996

Note that from 1998 to 2005 Andrew Glassner, a computer scientist at Microsoft Research, with a strong interest in graphics, had a column in IEEE in which he discussed a variety of subjects from a graphic viewpoint, some of which are direct interest, being of a tiling nature as well as others of a more peripheral nature. These articles I have long known about, possibly from Craig Kaplan’s bibliography in his thesis, of which previously they were unavailable to me (save for purchasing from the IEEE site). Only of today have I found these freely available on his website. As such, in a broad sense there is much of interest here, although for the sake of brevity I list here only those that are of the most significant.

 

————. ‘Origami Platonic Solids’. In IEEE Computer Graphics and Applications, pp. 85-91, July 1996

 

————. ‘More Origami Solids’. In IEEE Computer Graphics and Applications, pp. 81-85, September 1996

 

————. ‘Signs of Significance’. In IEEE Computer Graphics and Applications, pp. 79-82, May/June 1997

 

————. ‘Net Results’. In IEEE Computer Graphics and Applications, pp. 85-89, July/August 1997

Polyhedra nets.

 

————. ‘Aperiodic Tiling’. In IEEE Computer Graphics and Applications, 18 (3) pp. 83-90 May/June 1998 (4 July 2016)

Arguably the article of most of interest. Penrose, Voderberg, Robinson, Ammann. Also see the following article on Penrose tiling

 

————. ‘Penrose Tiling’. In IEEE Computer Graphics and Applications, 18 (4) pp. 78-86. July/August 1998 (4 July 2016)

Penrose, Quaiscrystals, Wang.

 

————. ‘Fourier Polygons’. In IEEE Computer Graphics and Applications, 19 (1) pp. 84-91. 1999 (4 July 2016)

 

————. ‘Celtic Knotwork, Part I’. In IEEE Computer Graphics and Applications, 19 (5) pp. 78-84, 1999. (4 July 2016)

 

————.  ‘Celtic Knots, Part 2’. In IEEE Computer Graphics and Applications, 19 (6) pp. 82-86, 1999. (4 July 2016)

 

————.  ‘Celtic Knots, Part 3’. In IEEE Computer Graphics and Applications,  pp. 70-75, January/February 2000. (4 July 2016)

 

————. ‘An Open and Shut Case’. In IEEE Computer Graphics and Applications, 19 (3) pp. 82-92, 1999. (4 July 2016)

 

————. ‘String Crossings’. In IEEE Computer Graphics and Applications, March/April 1999

String art.

 

————. ‘Hierarchical Textures’. In IEEE Computer Graphics and Applications, July/August 2000

 

————. ‘Crop Art, Part 1’. In IEEE Computer Graphics and Applications, September/October 2004

Some advanced maths in places.

 

————. ‘Crop Art, Part 3’. In IEEE Computer Graphics and Applications, January/February 2005

Some advanced maths in places.

 

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

Of pavement interest. Glickman, Macaulay Corporation Limited, UK,  appears to be an authority on the subject. 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.The first of two papers of his I have. This is the more technical, on 'G-Blocks' (although I fail to see why these are so named). Strictly, there is nothing here of direct interest.

 

————. ‘Pattern, Texture and Geometry in the Paved Surface’. Proc. Third International Conference On Concrete Block Paving, 1988?. 71-74 (29 July 2014)

Despite the title, begins with a brief history of paving, with the Romans. The more interesting of his papers. Shows the 'wavy rectangle'. Only of mild interest.

 

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.


Tetrahedra equivalent to cubes by dissection. Elemente der Mathematik


————. 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.

 

Goodfellow, Caroline. Board Games Studies 1, 1988, 1-11. (27 November 2018)

A follow-up arising from her book.

 

Goodman-Strauss, Chaim. ‘Compass and Straightedge in the Poincaré Disk’. American Mathematical Monthly 108 (2001): 38-49 (25 August 2016)

From a reference in Schattsneider. Of an academic nature. readable to begin with, with a mention of Escher's Circle Limit prints before then being heavily advanced. of no practical use.

 

————. Tessellations.

A survey of tessellations, which appeared in Italian in La Matematica Vol. 3 in 2010. The Tiling Lemma appears for the first time.

 

————. ‘Compass and Straightedge in the Poincaré Disk’, American Mathematical Monthly 108 (2001) 38-49

 

————. ‘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 semantic commentaries way beyond me (Goldstein is a philosopher). Profusely illustrated with Escher's non-tessellation prints, of impossible objects and situations (as described by Goldstein), namely Depth, Print Gallery, Drawing Hands, Hand with Reflecting Sphere, Ascending and Descending, Waterfall and Belvedere.

 

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’. American Mathematical Monthly, 61, No. 10 December 675-682 (9 April 2013)

Semi-academic.

 

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

Academic.

 

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 others 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.

 

Branko Grünbaum only

Grünbaum on his home page has a long and extensive listing of geometry-based articles (212!), with some made freely available, most of which are way beyond me. Rather than have a lengthy, ‘inappropriate’ listing here, I only list those that are of note in some way, those that are broadly understandable, or have been oft quoted that may be of a more academic nature.

 

————. ‘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.

 

————. ‘Levels of Orderliness: Global and Local Symmetry’. In Symmetry 2000. Proceedings of a symposium at the Wenner–Gren Centre, Stockholm. I. Hargitai and T. C. Laurent, eds.

Portland Press, London 2002. Vol. I, pp. 51 – 61.

Mention of Peter Raedschelders.

  

Grünbaum and Shepherd:


Grünbaum, B. and G. C. Shephard. ‘A Variant of Helly’s Theorem’. Proceedings of the American Mathematical SocietyVol. 11, No. 4 (Aug., 1960), pp. 517-522  (26 February 2013)

Of an academic nature throughout; of no use.


Grünbaum, B. and G. C. Shephard ‘A Generalisation of Theorems of Kirszbraun and Minty’. 812-? 1961 (26 February 2013)

Of an academic nature throughout; of no use.


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)

From a reference in Tilings & Patterns. Somewhat of a let down as regards tiling content, the subject matter being of no real interest of the day. There is no reference to tessellation per se. However, subsequently, in more recent times, of 2019, with a recent interest in houndstooth and related fabric matters, this is once more examined in the new context. In short, Grunbaum and Shephard take the weaving community to task for a lack of rigour, and indeed the mathematical community, for a shortsighted lack of interest, noting that only papers by Lucas (1867, 1880, 1911), Shorter (1920) and Woods (1935) pass any degree of muster. As to be expected, they are considerably more rigorous, with clear definitions. Aside from clarifying standard weaving terms, they also introduce some mathematics that quickly loses me, namely of isonemal and monomenal. Of particular note is their Fig. 5, containing the ‘minimum houndstooth’, although I do not fully understand their premise, of in effect ‘failed fabrics’. Titles such as isonemal and monomenal are introduced, likely of originality on their part (Akelman, 2011, asserts so); again, I do not understand these distinctions, and indeed much of the mathematical discussion. Further, even the weave authority and computer scientist Ralph Griswold is lost as to their term ‘hanging together’! The ambiguous use of satin and sateen is also taken to task.There seems to be some fearsome mathematics underlying isonemal matters. For example, see Bohdan Zelinka’s paper ‘Symmetries of Woven Fabrics’.  The various complexities and nuances here, are, at least for now (2019), are put aside. Much time could otherwise be wasted on matters largely inconsequential to my researches and understanding.

 

————. ‘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.


Grünbaum, B.  and G. C. Shephard. ‘An extension to the catalogue of isonemal fabrics’

Discrete Mathematics Volume 60, June-July 1986, pp 155-192 (19 February 2019)

Of recent (2019) fabic interest. Advanced.


Grünbaum, B. ‘How to Cut All Edges of a Polytope? The American Mathematical Monthly Vol. 79, No. 8 (Oct., 1972), pp. 890-895

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)

 433-

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?

37-

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 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 in a mail 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)

Advanced.

 

————. ‘A starshaped [sic] polyhedron with no net’. Geombinatorics 11 (2001) 43-48 (30 January 2012)

Advanced.

 

————. ‘Families of point-touching squares’. Geombinatorics 12 (2003) 167-174 (30 January 2012)

Advanced.

 

————. ‘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, Peter Mani-Levitska and G. C.  Shephard. ‘Tiling three-dimensional space with polyhedral tiles of a given isomorphism type’ Journal of the London Mathematical Society (2) 29,  1984, 181-191

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

 

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)

Academic.

 

 ————. ‘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.

 

————. ‘Idiot-Proof Tiles’. Mathematical Gazette 75, No. 472 (June 1991) 143-147

 

————. ‘Ceva, Menelaus, and the Area Principle’. Mathematics Magazine, Vol. 68, No. 4 Oct. 1995) pp. 254-268 (30 January 2012)

Advanced.

 

————. ‘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.

 

Guy, Richard. ‘The Penrose Pieces’. Bulletin London Mathematical Society 8 1976 pp. 9-10

 

H

 

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). N.B. ‘regelmässigen’ in the title  is lower case as in the original, and is not a typo on my part.

 

————. ‘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)

Academic.

 

————. ‘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.

 

Hannas, Linda. ‘Two centuries of jigsaws’. Illustrated London News, Sunday, November 2 1975, 41-44. (12 April 2017)

This appears to be largely, if not all, derived from her 1972 book. Spilsbury is discussed extensively. 9 illustrations. Be all that as it may, Hanass’s research is most impressive, and even more so considering the limited resources of the day (i.e. no internet).

 

Hann, Michael. The Fundamentals of Pattern Structure: Part I: Woods Revisited. The Journal of The Textile Institute. 2003 94:1-2, 53-65 (17 October 2016)

The first of three sequential papers of a underlying premise of H. J. Woods. As such, I find most of Hann’s work, not just here but elsewhere, obscure, at least to my interests. Finding any hidden gems is disproportionate as to the time required to examine his lengthy writings, not that there is anything wrong with that. Of note is a preoccupation with counterchange, of which he complied a short but significant reference.

 

————. ‘The Fundamentals of Pattern Structure: Part II: The Counter-change Challenge’

The Journal of The Textile Institute. 2003 94:1-2, 66-80 (17 October 2016)

See above. Profusion of diagrams!

 

————.The Fundamentals of Pattern Structure: Part III: The Use of Symmetry Classification as an Analytical Tool’. The Journal of The Textile Institute. 2003 94:1-2, 81-88 (17 October 2016)

See above. No diagrams!


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)

First, as to Hankin’s series of five articles, arising from his visit to India (1892-early 1920s), and specifically of the Fatehpur Sikri and Tomb of Itimad-Ud-Daula. Incidentally, Hankin was a most interesting character, with many scientific interests. These articles differ in their scope and extent. The more substantial articles are ‘The Drawing of…’ and ‘On some Discoveries of the Methods…’. The others, all in the Mathematical Gazette, are merely one or two pages.

This paper was seemingly read to a Journal of the Society of Arts meeting, dated March 17, 1905. This gives a general overview , with geometrical patterns divided into four classes: square, hexagonal, octagonal and arabesques, and of which I have issue with. Nonetheless, it is an impressive piece of writing, liberally illustrated, and largely popular, although still not yet studied as such by myself. Lewis F. Day gets a mention, p. 475.

————. ‘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’,in five broad parts: Hexagonal patterns, Octagonal patterns, Geometrical Arabesques, Floral Arabesques and Decoration of domes. These differ in extent, with Geometrical Arabesques the more lengthy, whereas Floral Arabesque is a mere half page. The articles consists of an essay pp. 1-25, followed by illustrations (thirteen pages, non-paginated) thereof. Again, I have yet to do this justice. Of a broad overview, this discusses Hankin’s ‘polygons in contact’ and 'overlapping' methodology. Of particular interest is that of two diagrams of a line drawing analysis of a fused pentagon, with a Cairo tile premise if suitably joined. Both seemingly give an equilateral pentagon. Specifically, see Figure 24, a screen at the tomb of Itimad-Ud-Daula, Agra, and Figure 25, a vestibule (entrance hall) at the tomb of Akbar, Sikandra.

Subsequently, Craig Kaplan ‘Islamic Star Patterns from Polygons in Contact’ (2005) and B. Lyne Bodner ''Polygons in Contact' Grid Method for Recreating a Decagonal Star Polygon Design’ has followed up on his methodology.

————. ‘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, all in the Gazette. Only much later, after nine years, did Hankin decide to continue, with II in 1934, and later still with III, in 1936.

An analysis of three ‘complex’ Arabic patterns, from a dome in the Alhambra (Figs. 1 and 2) and a copy from Bourgoin.

————. ‘Some Difficult Saracenic Designs II’. A Pattern Containing Seven-Rayed Stars. Math. Gazette 18 (1934), 165-168 (25 February 2013)

An analysis of a ‘complex’ seven-rayed Arabic pattern, albeit not sourced. Also see his 1936 paper, of a fifteen-rayed star.

 

————. ‘Some Difficult Saracenic Designs III’. A Pattern Containing Fifteen-Rayed Stars Math. Gazette 20 (1936), 318-319 (25 February 2013)

An analysis of a complex’ fifteen-rayed Arabic pattern, from Bourgoin. Also see his 1934 paper, of a seven-rayed star.

 

Hansen, Vagn Lundsgaard. ‘From Figure to Form’. Math Horizons November 1999 8-11

Popular account.

 

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.

 

————. ‘Konvexe Fünfecke in ebenen Punktmengen’. (in German) Elemente der Mathematik Band (Jahr): 33 (1978) Heft 5 116-118

Academic, one diagram.

 

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. Begins with discussion on Kepler.

————. ‘Dethronement of the Symmetry Plane’. In Doris Schattschneider and Michele Emmer, Eds. In 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)

 

————. ‘Symmetries in Moscow and Leningrad’. Comput. Math. Applic. Vol. 16, No. 5-8, pp. 663-69, 1988 (25 April 2016)

Somewhat lightweight account of symmetries in Russia. Three pavements are shown, of cobbles. Nothing of any real interest per se.

 

————. ‘Symmetry in Crystallography’. Acta Crystallography 1998 A54 697-706.

Largely popular account. Brief mention of Escher pp. 699 and 705. (19 April 2017)

 

Hargittai, István. ‘John Conway – Mathematician of Symmetry and Everything Else’. The Mathematical Intelligencer Vol. 23 Number 2, 2001, 6-14 (28 November 2011)

Interview.

 

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.

 

Harrison, Wendy Cealey. ‘Madness and historicity: Foucault and Derrida, Artaud and Descartes’. History of the Human Sciences, vol. 20, 4: pp. 79-105, 2007.

Non-tessellating article, with a one-line mention of Escher, p. 80, no illustrations.

 

Hart, George W. ‘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

In short, an article inspired by Escher’s print ‘Planaria’, with the print having common connections to Hart’s interest in sculpture, and in particular here that of octahedra and tetrahedra. Begins with a brief discussion on the print, with references to the above polyhedra, and also of the planaria, and then more extensively Hart’s own work in the field, concentrating on the polyhedral aspect per se. For no particular reason, planaria, and more specifically of the creatures themselves, has attracted my attention at odd intervals, with the latest of May 2019, on the occasion of finding new detail on the print (by Sherry Buchsbaum in a blog reply). Hence my more circumspect appraisal of Hart’s piece, in the search for anything of interest, given that this print is not too frequently commonly discussed in the literature. Hart, in general, is more concerned with his special interest, rather than planaria. However, he does indeed make one interesting unconnected point in the introduction, commenting on none of Escher's’ trademarks’ being here, but is rather a portrayal of a plausible, albeit unfamiliar, scene. Simply stated nothing per se new on planaria.

 

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)

Although not strictly of a mathematical nature, included here as it is often quoted in tessellation matters. As such, I believe this first came to my attention as a result of an article on Escher in Scientific American ‘Sources of Ambiguity in the Prints of Maurits C. Escher’, by Marianne Teuber, of which on p. 94 she quotes this reference. Consequently, I thus apparently followed up on this (ordered from the British library). However, tessellation wise, of figure and ground, this is a relative disappointment. As such, it is inconsequential. Rubin’s face-vase is discussed/illustrated, pp. 410-415, but what it contains is not of practical use.

 

Hayward, Roger. ‘The Jigsaw Puzzle and the Inventive Mind’. The Worm Runner's Digest, 11, 1 (83-84). 1969.
Just of general interest in jigsaw history. Some ruminations on the jigsaw puzzle. Nothing cluster-like, although he has indeed a loose cluster puzzle like diagram to his name.

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)

Parabola is a Australian mathematics magazine, first appearing in July 1964. In 2005 it merged with another mathematical magazine for high school students, Function.

A seminal work. Considerable Cairo-esque type pentagons. Hirschorn also has another two tiling articles of note in Parabola.

 

————. ‘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 Hirschhornn junior!

 

Hofstadter, Douglas R. 'Parquet deformations: patterns of tiles that shift gradually in one dimension’. Metamagical Themas in Scientific American (July 1983): 12-18.

The importance of this article can hardly be overstated. This is the first popular account of parquet deformations, with William Huff’s student-inspired works, of which Hofstadter does it full justice, with 12 stunning examples. And the titles are most amusing too! To pick a favourite is invidious. However, if pressed ‘Fylfot Flipflop’. Of note is that these are all linear. Absolutely delightful!

 

————. ‘Parquet Deformations: A Subtle, Intricate Art Form’. July, 1983 190-199. In Metamagical Themas: Questing for the Essence of Mind and Pattern. Basic Books; New edition 1996, First Printing edition 1985 (21 November 2016) PDF

This essentially repeats Hofstader’s original July 1983 column in Scientific American (his last), with extra, minor text, but also, more importantly, a ‘post scriptum’, in which a parquet deformation of David Olseon’s ‘I at the Center’ is illustrated and discussed, and much praised.

 

————. ‘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.

 

Hogendijk, Jan P. ‘Mathematics and geometric ornamentation in the medieval Islamic world’.

Source not stated (25 May 2016)

 

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)

Upon subsequent reading of an interview on Taylor, his preoccupation with Escher and Roosevelt becomes clearer.

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

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

Huff, William S. ‘Simulacra in non-reorientable surfaces-experienced in timing’. In Chapter 4, ‘Spatial Lines’, Patricia Muñoz, compiler
Minor reference to parquet deformations.

Huff, William. ‘The Landscape Handscroll and the Parquet Deformation’, 307-314. In Katachi U Symmetry. Tohru Ogawa, ‎Koryo Miura, ‎and Takashi Masunari. Tokyo: Springer-Verlag 1996 (23 November 2016, Google book reference).

This has four new parquet deformations by ‘new people’, namely:

Alexander Gelenscer; Swizzle Stick Twirl, 1986

Pamela McCracken; Cloisonné, 1990

Loretta Fontaine; Seven of One Make Three, 1991

Bryce Bixby; They Come, They Go, 1991

 

————. ‘Defining Basic Design as a Discipline’. In Symmetry: Art and Science, Vol 2 (new series) Numbers 1-4, 2012 (12 August 2016; seen earlier?)
Of interest as to ‘all things’ Huff. Parquet deformation is only mentioned in passing, as a reference.

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.

Huson, Daniel H. "The generation and classification of Ar-isohedral tilings of the Euclidean plane, the sphere, and the hyperbolic plane." Geometrica Dedicata, 47 (1993) 269-296.

Huson, Daniel H. "The generation and classification of Ar-isohedral tilings WANTED

 

Huylebrouck, Dirk Antonio Buitrago and Encarnación Reyes Iglesias. ‘Octagonal Geometry of the Cimborio in Burgos Cathedral’. Nexus Network Journal 13 2011, 195-203

Has mention of the Cordovan proportion (no pentagons though).

 

I

 

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)

 

J

 

Jablan,  Slavik Vlado.  Symmetry, Ornament And Modularity.  World Scientifc Publishing Ltd, 2002 (7 May 2018 PDF)

Generally of an advanced nature. I seem to recall many refences to this book, and so this may come in usiful although is so painfully slow in scrolling through so many pages that it is impractical to view all.

 

 

Jacobi, John V. ‘Dangerous Times for Medicaid’. The Journal of Law, Medicine & Ethics, vol. 33, 4: pp. 834-843. Winter, 2005.

Non-tessellating article, with a one-line mention of Escher, p. 837, no illustrations.

 

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, John. Photographs by Trevor White. ‘A Mathematician’s Guide to the Alhambra’. Second revised edition 2006 (25 October 2012)

Whether this is to be regarded as a book or an article is not clear! The text inside states, ‘This pamphlet was first producied in the 1990s as the result of the BBC/Open University television programme ‘Just Seventeen’. Kindle edition is available 2013, but found as a hard copy. 31 pages. On the 17 symmetries, of a popular account. Oddly, the author (a collabortor with Ian Stewart) here is not stated!

 

Jelliss, G. P. ‘Special issue on Chessboard Dissections’. Chessics 28 Winter 1986 137-152

Reference from Golomb.

 

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, pp. 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.

 

Jung, Hwa Yol. ‘Transversality, Harmony, and Humanity between Heaven and Earth’

Diogenes, vol. 60, 1: pp. 97-104. February, 2014.

Non-tessellating article, with a one-line mention of Escher, p. 101, no illustrations.

K

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 ** (29 July 2015)

Sphere Spirals

 

Kaplan, Craig. S. ‘Computer Generated Islamic Star Patterns’. In Bridges 2000 pp. 105-112 (and Visual Mathematics, 2, (3) 2000)

First, Kaplan has an extensive bibliography in regards of his interest in Computer Science, some of which are either out of my direct interest, or are simply way beyond my understanding. Therefore, the listing below is a compilation of the more useful ones involving tessellation. Pleasingly, he makes his papers easily available on his website.

http://www.cgl.uwaterloo.ca/csk/pubs.html

 

————. ‘Escherization’. In Proceedings of Siggraph 2000. Proceedings of the 27th annual conference on Computer Graphics and interactive techniques pp. 499-510 ACM Press/Addison-Wesley Publishing Co. New York, NY, USA 2000

Most informative, with a considered approach to life-like tiling. Some very clear-cut thinking. Overwhelmingly accessible.

 

————. ‘Islamic Star Patterns from Polygons in Contact’. Proceedings of the Graphics Interface 2005 Conference, May 9-11, 2005, Victoria, British Columbia, Canada

Building upon Hankin’s ‘polygons in contact’ method. Included is a brief discussion on Islamic type parquet deformations exploiting Hankin’s method, Chapter 3. 1, pp**.

 

————. ‘A meditation on Kepler's Aa’. In Bridges 2006: Mathematical Connections in Art, Music and Science, pages 465-472, 2006.

 

 

————. ‘The trouble with five’. + Plus Magazine. December 2007.

For the online + Plus Magazine, and written for a less academic audience than with most of his papers. Tiling with a five theme. No Cairo tiling.

https://plus.maths.org/content/trouble-five

 

————. ‘Metamorphosis in Escher’s Art’, Bridges 2008 (Leeuwarden), 39-46

Of particular interest is parquet deformation, pp. 42-45, based on arbitrary isohedral tiles and then later between the Laves tilings. A major paper on the topic, within a general framework of metamorphosis, as he begins with an overview of such concepts in Escher’s prints.

 

————. ‘Curve Evolution Schemes for Parquet Deformations’. Bridges 2009

 

————. ‘Patterns on Surfaces’ Semiregular patterns on surfaces. In NPAR '09: Proceedings of the 7th international symposium on non-photorealistic animation and rendering, pages 35-39, 2009.

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.

On the computer program Najm. A little technical in places. No parquet deformation, unlike other Islamic papers of his.

 

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 (23 March 2013)
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

Kindt, Martin. ‘Wat te bewijzen is’ (in Dutch) (38) (translated ‘What is to be proved’). Nieuwe Wiskrant 27-1 September 2007 (1 April 2016)

Article on Cairo tiling, 35-36, with initial reference to David Wells. The Nieuwe Wiskrant’, a Dutch journal for mathematics and computer science education, provided news of recent developments in these areas, and appeared quarterly between September 1981 and June 2013. Its focus was mainly on secondary education.

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 (and Golomb). 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 (as indeed are all Klarner papers), with occasional diagram ‘understandable! That said, still of no practical use. From a reference in Tilings and Patterns (and Golomb).

————. ‘A Packing Theory’. Journal of Combinatorial Theory, Series A8, 273-278, 1970 (4 August 2016)

Of an largely academic nature throughout.

 

————. ‘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 (and Golomb). Of an academic nature throughout; of no use.

 

Klarner, David A. and Ronald. 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 (and Golomb). Of an academic nature throughout, no diagrams! Of no practical use.

 

Klarner, David A. and Spyros S. Magliveras. ‘The Number of Tilings of a Block with Blocks’. European Journal of Combinatorics (1988) 9, 317-330 (4 August 2016)

Of an largely academic nature throughout.

 

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

Kowsmann, Patricia. ‘In Lisbon, Some Residents Fear City’s Famous Sidewalks’. The Wall Street Journal, June 1, 2014 (pp. unknown).

Of pavement interest. Laments the dangerous nature of the pavings, as well as instances in Brazil, of a history.


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.

 

Krivý, Maroš. ‘Towards a critique of cybernetic urbanism: The smart city and the society of control’. Planning Theory, vol. 0, 0 2016.

Non-tessellating article, with a one-line mention of Escher, p. 10, no illustrations.

 

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

Although not strictly of a mathematical nature, included here as it is occasionally quoted in impossible object matters. From a reference in Adventures with Impossible Figures by Bruno Ernst (and not The Eye Beguiled as previously thought). Although scientifically slanted, still a largely popular account, with discussions on Thiéry figures and Penrose stairs/triangle. However, I see very little practical use of this. There is nothing (unsurprisingly) tessellation related.

 

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.

 

L

 

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.

 

————. ‘Non-periodic Space Structures’ Vol 2, Issue 2, 1987 (17 January 2017)

 

————. ‘Continuous Transformations of Subdivided Periodic Surfaces’

Vol 5, Issue 3-4, 1990 (17 January 2017)

 

Laninger, Jay A. ‘Metaphoric Usage of the Second Law’. Entropy as time’s (double-headed) arrow in Tom Stoppard’s ‘Arcadia’ 31-37 Chemical Intelligencer October 1996 (31 August 2016)

Escher prints Ascending and Descending p. 35, Waterfall p. 37

 

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.

 

————. ‘Note 1464 Uses of a Geometrical Puzzle’. The Mathematical Gazette 24 no.260 July 1940 209-211

From a reference in Golomb. On Rep-tiles.

 

————. ‘Note 2793. A conundrum for Form VI’. The Mathematical Gazette 42 No. 342 December 1958 287

From a reference in Golomb.

 

————. ‘Note 2864. A Chess-board Puzzle’. The Mathematical Gazette 43, No. 345, (October 1959)

From a reference in Golomb.

 

————. ‘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 pp. 105-110 (1 March 2013)

From a reference in Dissections: Plane and Fancy. The title is a little misleading, it consists, or can be interpreted as of dissections of polygons. Of note is that this shows a Moore pentagon in a regular hexagon, with the tiling implied? However, I am not certain; Langford shows other non-tiling polygons, such a regular pentagon. No references are given in the article; it is unclear if Langford believes these are all his own.

 

————. ‘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)

Landwehr, Klaus. ‘Visual Discrimination of the 17 Plane Symmetry Groups’. Symmetry 2011, 3, 207-219 (1 April 2016)

Brief history of plane tiling, Escher first pages.

 

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’. ‘Space filling’ here is not in the context of tessellation.

 

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

Leighton, Tanya. Spike Art Quarterly, Autumn 2011 On Enzo Mari

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 ‘anolomies’.

Lenstra, Hendrik and Gerard van der Geer. ‘The Gauss Statue in Braunschweig (Brunswick), Federal Republic of Germany’. Mathematical Intelligencer ,Vol 9 No. 2 9 1987, 44-45. Springer-Verlag New York

On Lenstra, also see interview with Jeannes Daems.

 

Levine, Dov and Paul Joseph Steinhardt. ‘Quasicrystals: A New Class of Ordered Structures’. Physical Review Letters, Vol. 53, No. 26, 24 December 1984, pp. 2477-2480 (19 June 2018)

Quoted by Grünbaum in Tilings and Patterns and Penrose (in Art and Science), and others in regards of Penrose tilings. Academic, as expected, predominantly text. Of no practical use.

 

Levy, Silvio. ‘Automatic Generation of Hyperbolic Tilings’. Leonardo, Vol. 25, No. 3 (1992) pp. 349-354 (March 2013)

Academic.

 

Lindgren, Harry. ‘Going One Better in Geometric Dissections’. Mathematical Gazette 1961 94-97 (20 February 2013)

From a reference in Dissections: Plane & Fancy. Accessible.

 

————. ‘Dissecting the Decagon’. Mathematical Gazette 46 305-306 1962 (20 February 2013)

From a reference in Dissections: Plane & Fancy. Accessible.

 

Liu, Yang and Godfried Toussaint. ‘Unravelling 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)

 

Loeb, Arthur. L. ‘The Architecture of Crystals’. In Module, Proportion, Symmetry and Rhythm by Gyorgy Kepes, ed. George Brazillier, 1966, 38-63 (6 September 2016)

From a reference in Locher? Features Escher’s tessellation 48-49. I first saw this c. 1990s at the art school library, but stupidly failed to photocopy the article. However, I did photocopy the front cover, or at least of sister publications by Kepes, ‘education of vision’ and ‘module, symmetry and proportion’ (title all lower case) of 20 May 1997, but likely this was first seen many years before. A brief reference to Escher, almost in passing. Of no significance.

 

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)

 

Locher. P and C. Nodine. ‘The Perceptual Value of Symmetry’. In Computer Maths with Applications Vol. 17, No. 4-6, pp. 475-484, 1989 (10 September 2018)

From a reference in Craig Kaplan’s thesis. A piece on symmetry per se, from two psychologists, in a symmetry special edition of the journal. Nothing on tiling or Escher. Skim read; of mild interest from a symmetry per se viewpoint, but nothing more. No plans to re-read.

 

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, pp. 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, pp. 33-48, 1989 (27 September 2013)

Obscure.

 

Lord, Nick. ‘Constructing the 15th pentagon that tiles the plane’. Mathematical Gazette. Volume 100, Issue 547, March 2016, pp. 154-158 WANTED

 

Lord, Eric A. and S. Ranganathan. ‘Truchet Tilings and their Generalisations’. Resonance, June 2006 (28 October 2018)

Note that Lord, a metallurgist, based in India, has published widely (sometimes with others, namely S. Rangahanathan), of both popular (in the Indian Resonance journal) and academic, of no less than 71 artilels and five books! Although most of his papers are freely available on his site, the academic instances are too advanced for me, and so are not listed here, although saved as PDF on the off chance of needing to refer to.

Mention of Cyril Stanley Smith’s part in bring to prominance Truchet tiles.

 

Lord, Eric A. ‘Quasicrystals and Penrose patterns’. Current Science, Vol. 78, pp. 64-72, 2000. (7 November 2018)

For sure, before a late interest in Lord of late October/November 2018, I had never heard of the Indian ‘Current Science’ , described as ‘a Fortnightly Journal of Research), of which I see (Wikpipedia) that it was first published in 1932! All of the journals are available on their website. As such, this appears to be of an academic nature, and putting in obvous search terms such as ‘tessellation/tiling/Escher/pentagons’ shows nothing.

 

Lord, Eric A, S. Ranganathan and U. D. Kulkarni. Tilings, coverings, clusters and quasicrystals. Current Science, Vol. 78, No. 1, pp. 64-72, January 2000. (7 November 2018)

Semi academic/popular.

 

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.

 

Lu, Peter J. and Paul J. Steinhardt. ‘Decagonal and Quasi-Crystalline Tilings in Medievel Islamic Architiecture’. Science 23 February 2007. In possession?

 

Lück, Reinhard. ‘Dürer–Kepler–Penrose, the development of pentagon tilings’. Materials Science & Engineering A, 2000, Volume 294 pp. 263-267 (October 2016)

This quotes Escher’s ‘monstrum’, albeit the attribution is not categorically stated. However, in the bibliography, Kepler’s Harmonice Mundi is quoted, from which one would assume that it was taken from there. However, * asserts that it is not in Harmonice Mundi!

Note that the book Aperiodic Structures in Condensed Matter: Fundamentals and Applications By Enrique Macia Barber also mention ‘monstrum’ in the context of Harmonice Mundi!

 

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.

 

M

 

Maass. John. ‘The Stately Mansions of the Imagination’. In Horizon A Magazine of the Arts September 1963 Volume V Number 7, pp. 10-26 (31 August 2016)

From a reference in Locher and Schattsneider. A major disappointment! I was under the impression that this was an article on Escher, but is rather a discussion on architecture per se, with the only reference to Escher, p. 22 of the print Relativity with a brief comment!

Note the unusual name and spelling of Maass. I have see this spelt incorrectly in many bibliographies, as Mass, suggesting simply copying of references without the article being  to hand.

 

Macaulay, W. H. ‘The Dissection of Rectilinear Figures’. Mathematical Gazette Vol.7 No. 113 pp. 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 pp. 109-115(4 March 2013)

 

————. ‘The Dissection of Rectilinear Figures’. Messenger of Mathematics, Volume 48, 1919a. pp. 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. pp. 111-121 (29 July 2011)

 

————. ‘The Dissection of Rectilinear Figures (continued)’. Messenger of Mathematics Volume 52, 1919b. pp. 53-56

 

Macbeath, A. M. ‘The classification of non-Euclidean plane crystallographic groups’.

Canadian Journal of Mathematics 19 (1967), pp. 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 pp. 251-255, 1979 (10 August 2010)

Highly significant as regards the Cairo tiling discussion, in depth, the fourth discussion (1979), after Dunn (1971), Gardner (1975) and Schattschneider (1978), and the second in situ account. Also of note is the collinearity aspect, first discussed.

 

Mackay, Alan L [15] ‘Extensions of space group theory’ Acta Cryst., 10, 543-8, 1957. (6 September 2016)

As regards Mackay and his articles, most of these are of an academic title, and so are mostly hard, if not impossible to obtain. However, as of 6 September 2016, I found a website making a substantial proportion of his many papers and others available, as a pdf. These are described as: (i) scientific publications, (ii) miscellaneous publications, (iii) Anecdotal Evidence" columns from "The Sciences" , (iv)indirect material, (v) book reviews, (vi) unpublished papers etc

http://met.iisc.ernet.in/~lord/webfiles/Alan/Alanpapers.html

However, despite being warmly welcomed, these are mostly of limited use and interest, as to be expected given the source, as these are highly academic, way beyond my understanding, not to mention of diverging specialties. However, on occasion, there are indeed crossovers with tiling, and of which some, with diagrams, are broadly followable; for instance, Penrose and pentagon tilings feature strongly.

To add all these available listing here, of about 250 available papers seems somewhat over officious, given the lack of need. Therefore, I here thus list just those of interest, and where quoted in tiling matters, such as with Grünbaum. Incidentally, Grünbaum lists seven articles of his, all of which I now have.

Also see his review of Visions of Symmetry.

 

————. [42] ‘The structure of structure: some problems in solid state chemistry’,

Chimia, 23, pp. 433-437, 1969 (6 September 2016)

 

————. ‘Crystal Symmetry’. Physics Bulletin, pp. 495-497 November. 1976. (6 September 2016)

 

————. ‘Bending the Rules. Crystallography, Art and Design’ (lecture 1997/98) (2010)

Also see his review of Visions of Symmetry.

 

————. ‘De Nive Quinquangula: on the pentagonal snowflake’. [in Russian] Kristallografiya, pp. 909-918. English version: Soviet Physics–Crystallography 26 (1981) 517-522 (31 January 2011)

Mentioned in Grünbaum bibliography.

 

————. ‘But What is Symmetry?’ Computers and Mathematics with Applications. Vol. 12B, Nos 1/2, pp. 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, pp. 21-37, 1986 (2 October 2013)

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

 

 

MacMahon P. A. ‘On Play “a outrance”’. Proceedings of the London Mathematical Society. 1889 XX pp. 195-198 (19 April 2017)

From a Garcia reference in NMP, which implies on the face of it a possible recreational rendering, of a popular account, but this is not so!

 

————. ‘On the Thirty Cubes that can be constructed with Six differently Coloured Squares’. Proceedings of the London Mathematical Society 1893, 24, pp. 145-155 (19 April 2017)

From a Garcia reference in NMP, which implies on the face of it a possible recreational rendering, of a popular account, but this is not so!

 

————. ‘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)

From a Garcia reference in NMP. 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)

From a Garcia reference in NMP . 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’. Proceedings of the London Mathematical Society (2) 23 (1925) 75-93 (30 April 2012)

From a Garcia reference in NMP. 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 many other MacMahon papers from Garcia’s bibliography, but these are of an academic nature of no practical use, and so rather than a pointless listing are not shown here.  He particularly favoured Proceedings of the London Mathematical Society. Also see others from Garcia reference.

 

‘MacMahon mentions’ in: Nature: October 18, 1906. Nature B, 1906, 1908, September 10, 1908? (April 2012)

 

Macnab, Maggie. ‘Decoding Design. Understanding and Using Symbols In Visual Communication’ (1 June 2011)

On logos, with a leaning towards symmetry.

 

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.

 

Macknik, Stephen L. and Susana Martinez-Conde. ‘Sculpting the Impossible: Solid Renditions of Visual Illusions’. Scientific American Mind. November/December 2011, 22-24 (11 April 2016)

Popular account; use is made of two of Escher’s prints, Waterfall and Belvedere.

 

————. ‘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.

 

Madachy, Joseph S. ‘Recreational Mathematics’. Fibonacci Quarterly. 1968 162-166 (5 August 2016)

On polyominoes.

 

Majewski, Miroslaw and Jiwan Yang. ‘A Journey through Chinese Lattice Designs an introduction to Chinese’. Journal source not given (22 March 2012)

 

Makovicky, Emil. ‘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)

 

————. ‘800-year-old pentagonal tiling from Maragha, Iran, and the new varieties of aperiodic tiling it inspired’. 67-86. In István Hargittai, Five Fold Symmetry, World Scientific (22 April 2016)

On a (far fetched) premise of Penrose aperiodic tiling.

 

Makovicky, Emil and M. Mackovicky. Arabic Geometrical Patterns – a treasury for crystallographic teaching. t Jahrbook fur mineralogy Monatshefte, 2 58-68, 1977. WANTED

From a reference in Abbas.

 

Malcolm, Paul S. ‘Braided Polyhedra’. The Arithmetic Teacher. 386-388 (28 March 2013)

Simple braiding.

 

Malek, Samar and Chris Williams. ‘Structural Implications of using Cairo Tiling and Hexagons in Gridshells’. Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2013 BEYOND THE LIMITS OF MAN 23-27 September, Wroclaw University of Technology, Poland J. B. Obrębski and R. Tarczewski (eds.) 1-4 (25 April 2016, but seen earlier)

Academic in tone. Cairo tiling in the context of ‘gridshells’. As such, this appears to have been an academic study rather than its implementation into actual physical models/structures.

 

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 May 1988, hard copy of magazine of 18 June 2011, and individual articles 18 February 2013)

Note that the article featured in a ‘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’.

The article is most brief indeed, of just four pages padded by large scale diagrams and shows a typical teacher attempt at the Escher-like aspect, with two tessellations showing no understanding of the matter, with little more than detail added to a tile. This also shows an analysis of Escher's  flying fish tessellation.  Admittedly, the article appears slanted to children, but it should have been better. This appears to be his only attempt at Escher-like tessellations, as upon a web search (October 2017), there is no other apparent works by him in this field. Maletsky at least speaks with authority, a maths (and physics) teacher, with a NCTA lifetime achievement award of 2009.

 

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!

http://www.goldennumber.net/un-secretariat-building-golden-ratio-architecture/

 

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)

 

May, Kenneth O: 1965: ‘Origin of the Four Color Conjecture’. Isis vol.56: pp.346-348

Mey, Jos 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)

Marcotte, James and Matthew Salomone. ‘Loxodromic Spirals in M. C. Escher's Sphere Surface’. Journal of Humanistic Mathematics Volume 4 Issue 2 July 2014 (3 August 2016)
Begins at a popular account, and then becomes academic. Even so, much of a popular account remains throughout.

Maynard, Phillip M. ‘Isohedrally compatible tilings’. http://symmerty-us.com (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.

 

McAlhany, Joe. ‘A Brief History of Dissected Maps, the Earliest Jigsaw Puzzles’. Old World Auctions


McConnell, James V. ‘Confessions of a scientific humorist’. impact of science on society, Vol. XIX, No. 3 July-September 1969, 241-252.

Of James McConnell interest, re Escher-flatworms, albeit there is nothing here on Escher, but rather of his (admirable) humour. McConnell’s piece was part of a seemingly special edition on humour and science, from Unesco. (May 2019)

 

————. Article Title Unknown. Worm Runner’s Digest Vol. XVI No. 2, December 1974, pages unknown. WANTED

Of Escher reference, at least of the cover, of which after this  there are many uncertainties here. I do not have the journal in my possession, and quite where I got this reference from is unclear; I may have found it independently, although I doubt it. Be all as it may, an article in The Unesco Courier of April 1976, shows the cover of the WRD, illustrated with Escher’s Flatworm print (a topic of recent (May 2019) interest). Quite what, if indeed there is an Escher related article here  is unclear.

 

————. ‘Worm-Breeding With Tongue in Cheek or the confessions of a scientist hoisted by his own petard’. The Unesco Courier, April 1976, pp. 12-15, 32

As such, the Escher aspect here is only of illustrations; there is not any reference in the text. More exactly, this shows shows the cover of the WRD of 1974, illustrated with Escher’s Flatworm print (a topic of recent (May 2019) interest). The Courier piece is an interesting read in many ways. There is no Escher discussion as such in it, although the Flatworms print is shown on p. 13, with the premise on flatworms (a most interesting creature, I might add. I had no idea of the fascinating science on it). As an aside, I very much enjoy McConnell’s humour.


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.

 

Merow, Katherine. ‘A Toast! To Type 15!’ Math Horizons. November 2015 10-11 (29 March 2016)

Popular account of the Type 15 pentagon discovery by Mann et al. Gives a history, with a small section on the discovery itself.

 

Meurant, Robert C. ‘A New Order in Space - Platonic and Archimedian (sic] Polyhedra and Tilings’ (17 January 2017)

 

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.

 

Mikkonen, Yrjö. ‘Ontology intermingling with onticity and vice versa in M.C. Escher's Reptiles’ International Journal of General Systems. 34:5, 595-601 (2004) (17 October 2016)

Discussion on Escher's Reptiles print, as regard ontology and onticity. Both popular and academic in tone.

 

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 great significance, and is academic in tenure, with not a single diagram!

 

Mueller, Conrad George and Mae Rudolph (consulting editors René Dubos, Henry Margenau and C. P. Snow). Light and Vision (Life science library) 1966 and 1969

Eight chapters, all full of general interest. Escher’s House of Stairs p. 162, and a Fish tessellation, p. 163, apparently by Otto van Eersel, an illustrator, who appears throughout the book.

 

Muscat, Jean-Paul, ‘Polygons & Stars’. Mathematics in School, March 1992 pp. 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?

 

N

 

Naylor, Michael. ‘Nonperiodic tiling: the Irrational Numbers of the Tiling World’. The Mathematics Teacher (5 March 2013)

Popular.

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

 

Neuhaeuser, Stefan, Fritz Mielert, Matthias Rippmann and Werner Sobek. ‘Architectural and Structural Investigation of Complex Grid Systems’. Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2010, Shanghai Spatial Structures – Permanent and Temporary November 8-12 2010, Shanghai, China (29 September 2017)

Features use of the Cairo tiling in ‘grid systems’, although mostly of the terminology used is beyond me. Cairo tiling references on p.7, and liberally illustrated throughout. The paper arose of the work with ILEK

 

Nicki, Richard M. et al. ‘Uncertainty and preference for ambiguous figures, impossible figures and the drawings of M. C. Escher’. Scandinavian Journal of Psychology, Vol. 20, 1979, 277-281

 

Niman, John and Jane Norman. ‘Mathematics and Islamic Art’. American Math Monthly, Vol. 85, 489-490,1978 (18 February 2013)

References of semi-regular tilings in Islamic art. Widely quoted. 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.

 

Norcia, Megan A. ‘Puzzling Empire: Early Puzzles and Dissected Maps as Imperial Heuristics’.

Children's Literature Annual  Volume 37, June 2009 1-32.  WANTED

 

Norgate, M. ‘Non-Convex Pentahedra’. The Mathematical Gazette (18 February 2013) 115-124

Academic nature throughout.

 

Norgate, Martin. ‘Cutting Borders: Dissected Maps and the Origins of the Jigsaw Puzzle’. The Cartographic Journal The World of Mapping Volume 44, 2007 - Issue 4 WANTED

Jigsaw reference.

 

O

 

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. A very nice paper indeed in a generalized sense, although of course, given the academic nature, much is beyond my understanding. Of particular note is two instances of the Cairo tiling, although not stated as such:

P. 557, in relation to use in Mathematical Models by Cundy and Rollett and New Mathematical Pastimes by MacMahon.

P. 567, a diagram, where O’Keefe and Hyde specifically name it after MacMahon, with ‘MacMahon’s net’.

As such this paper seemingly marks the introduction of the term ‘MacMahon’s Net’ for the Cairo tiling, and was used again by them in their 1996 paper, but this time in addition with the Cairo association. However, this is very much an ‘unofficial’ description. Upon correspondence (2012) with him:

I suspect I got ‘Cairo tiling’ from Martin Gardner who wrote several articles on pentagon tilings. He is very reliable. As to ‘MacMahon's net’, I got the MacMahon reference from Cundy & Rollet….We are mainly interested in tilings on account of the nets (graphs) they carry.

Possibly, and plausibly, this by MacMahon, of 1921, was the earliest known representation, and so in a sense, it was indeed broadly justified, even though by 1980 the ‘Cairo tiling’ term was coming into popular use, although if so, it is now been left behind by my subsequent researches. Curiously, the term is used on the Cairo pentagonal tiling Wikipedia page. However, the page leaves much to be desired, including this designation. Toshikazu Sunada has also used this term. However, I do not like this at all; it seems a somewhat artificial, additional naming, and so is unnecessary. Better would simply to have credited MacMahon as the first known instance (at the time) but without naming it after him. Also see a later paper, of 1996.

Biographical details: Michael O’Keefe (1942-) and Bruce Godfrey Hyde (1925-2014) are two prominent people in science, and leading lights in their fields, namely physics and solid state chemistry respectively. In this regard, both have an interest in tilings, of which the paper addresses. Although of an intended academic audience, the paper is still nonetheless has much popular interest. Aside from the Cairo tiling, there are some most interesting simple, but aesthetic tilings, deserving of study.

 

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 pretentiousness. 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 ‘Mathematical Tools for Computer-Generated Ornamental Patterns’

In Electronic Publishing, Artistic Imaging and Digital Typography, Lecture Notes in Computer Science 1375, Springer Verlag, 193-223, 1998. Patterns’ (16 February 2011)

Somewhat advanced. Of limited practical value.

 

————. Ostromoukhov, Victor. ‘Multi-Color and Artistic Dithering’. Siggraph 1999. (14 November 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’.

 

Ostromoukhov, Victor and Roger D. Hersch. ‘Artistic Screening’. Siggraph ’95 Computer Graphics Proceedings 219-228 (15 July 2011)

On a premise of screens, or half toning, with use made of a variety of art works; Escher (Sky and Water I), Islamic design. A little technical in places, although obviously mighty clever.

 

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)

 

Özgan, Sibel Yasemin and Mine Ökar. ‘Playing by the Rules. Design reasoning in Escher’s creativity’ (25 August 2016)

In N. Gu, S. Watanabe, H. Erhan, M. Hank Haeusler, W. Huang, R. Sosa (eds.), Rethinking Comprehensive Design: Speculative Counterculture, Proceedings of the 19th International Conference on Computer-Aided Architectural Design Research in Asia CAADRIA 2014, 23–32.

 

 

P

 

Palmer, Chris K. ‘Spiral Tilings with C-curves Using Combinatorics to Augment Tradition’. In Bridges Renaissance Banff 2005, pp. 37-46

 

Palmer, Kelvin. ‘Jumble-Fits, Cluster Puzzles and Alec Zandimer Plerp’. Game & Puzzle Collectors Quarterly, September 2003 Vol. 4 No. 3, p. 9 (19 March 2015)

Essentially an announcement of the forthcoming publication of his book, rather than a discussion of cluster puzzle in general.

 

Paranandi, Murali. ‘aking Ripples: Rethnking pedagogy in the digital age’ International journal of  structural computing. Issue 4, volume 1 415, 422-423

Of note is my name checks as regards my grid filling method, as well as illustrations.

 

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 pp. 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 pp. 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.

 

Paulino, Glaucio H. Arun L. Gain. ‘Bridging art and engineering using Escher-based virtual

elements’. Struct Multidisc Optim (2015) 51:867–883 (17 October 2016)

Some fearsome mathematics to explain Escher’s ‘simple’ periodic tessellations!

 

Pasko, Galina, Alexander Pasko, Turlif Vilbrandt, Arnaldo Luis Lixandrão, Filho and Jorge Vicente, Lopes da Silva. ‘Ascending in Space Dimensions: Digital Crafting of M.C. Escher’s. Graphic Art’. Leonardo Vol. 44, No. 5 411-416 (12 September 2016)

Only of minor interest, essentially on 3D fabrication matters, the nuances of which are beyond me.


Penrose, L. S.  and R. Penrose. ‘Puzzles for Christmas’. New Scientist, Volume 4, Number 110, 1580-1581, 25 December 1958. (30 May 2019)

Puzzle 2 is a variant of the Penrose staircase. Puzzle 6 is on seven simple (if not more involved), geometric tilings. Solutions are on p. 1597.

 

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, pp. 72-73).

 

Penrose, R. ‘On the Cohomology of Impossible Figures’. Structural Topology 17, 1991 11-16. (c. 2008?)

Largely academic. Impossible tribars, p. 12.

N.B. On Penrose per se, also see interview with Omni Magazine, with the Penrose tiling discussed

 

————. ‘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.

(From a reference in Frederickson).

Ivars Peterson (1948-) is an award-winning mathematics writer. He is Director of Publications for Journals and Communications at the Mathematical Association of AmericaHe worked for 25 years as a columnist and online editor at Science News and continues as a longstanding columnist for the children's magazine Muse. He wrote the weekly online column Ivars Peterson’s MathTrek. He is the author of a number of popular mathematics and related books.


————. 'Tiling to Infinity' Science News Vol. 134  July 16 1988 (2 April 2013)


————. ‘Shadows and Symmetries’. Science News. Vol. 140 408-410. (13 March 2013)

Subtitled ' Quasicrystal geometry brings a new dimension to art and design.


————. Clusters and Decagons. Science News Vol. 155 January 23, 1996 232-233 (2 April 2013)

Subtitled 'picturing complex alloy structures as overlapping atomic structure'. Popular.


————. 'A Quasicrystal Construction Kit'. Science News Vol. 150 January 23, 1999 60-61 (2 April 2013)

Subtitled 'New rules for constructing a quasicrystal'. Popular.


————. ‘The Honeycomb Conjecture’. Science News. Vol. 156 60-61. July 24 1999 (13 March 2013)

Subtitled 'Proving mathematically that honey bee constructors are on the right track'.


————. ‘Pieces of a polyomino puzzle’. Science News. Vol. 132 310. (9 April 2013)

On Karl A. Dahlke.

 

Pickover, Clifford. ‘Mathematics and Beauty: A Sampling of Spirals and Strange Spirals in Science, Nature and Art’. Leonardo Vol. 21, 1988, No. 2, pp.173-181

Typical Pickover. Somewhat advanced.

 

————. ‘How to Design Textures Using Recursive Composite Functions’. Leonardo Vol. 22, 1989, No. 2, pp.219-222

Somewhat advanced computer graphic art, of an equation nature.

 

Pill, Steve. ‘Master Techniques MC Escher’. Artists & Illustrators, November 2015 (24 May 2017)

A piece in conjunction with the contemporary Dulwich exhibit, illustrated with the prints Bond of Union, Castrovalva, Relativity and Reptiles. Nothing new, with the title insinuating technique, which is nothing of the sort.

 

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.

 

Popkin, Gabriel. ‘The Hidden Pattern’. New Scientist, 18 February 2017, pp. 28-31 (24 May 2017)

An article on cosmological matter, illustrated with Escher’s Print Gallery, albeit essentlaly no other mention of him in the text. First saw of the day of publication, of which I noticed a tilling pattern on the cover, and so hence investigated, otherwise the reference would not have come to my attention.

 

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.

Preston, Geoff. ‘Escher’s Delight’. Acorn User. October 1995, p. 69 (21 September 2016)
Essentially on computer graphics software, namely Dial Solutions’ VersaTile rather than on Escher himself. A single page on the merits or otherwise.

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.

 

Journal of European Studies

 

R

 

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. ‘His Designs Come From Math Books’. Popular Science 168 May 1956 205-208 (10 May 2016)

Nothing of much significance. Available from the Popular Science website, although not as a print-out.

 

————. ‘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.

 

————. ‘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.

 

————. ‘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 article is part of a ‘special edition’ on Escher-like tessellations, by Ranucci, Teeters, and Maletsky

 

————. ‘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. Arithmetic Teacher April 1966 . (In Farrell)

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. In Farrell.

 

————. ‘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 pp. 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.

 

————. ‘The Cordovan Proportion: Geometry, Art and Paper Folding’. Hyperseeing May-June 2008 107:114

 

Reeve, J. E. and J. A. Tyrrell. ‘Maestro Puzzles’. The Mathematical Gazette 45 (October 1961) pp. 97-99 (20 March 2013)

Polyominoes, polyiamonds.

 

Reichert, Muchael and Franz Gahler. ‘Cluster model of decagonal tilings’. Source? (Date?)

Relating to Gummelt’s covering rule. Of an academic nature, leaning towards physics, albeit broadly understandable, at least of the first few pages, whereupon it rapidly becomes abstruse, and is not of any practical use.

 

Reid, Michael. ‘Tiling with Similar Polyominoes’. Journal of Recreational Mathematics. 31 (2002-2003) No 1 pp. 15-24 (8 November 2102)

Very accessible.

 

Reinhardt, Karl ‘Über die Zerlegung der hyperbolischen Ebene in konvexe Polygone’ Jahresbericht der Deutschen Mathematiker-Vereinigung. 37 pp. 330-332 1928 (31 December 2014)

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

 

Renz, Peter L. ‘Martin Gardner and Scientific American: The Magazine, Columns and the Legacy’. pp. 1-4

Reminiscences of Martin Gardner by Renz, an editor of Scientific  American. Much new detail, of general interest.

 

Rhoads, Glenn C. ‘Planar tilings by polyominoes, polyhexes, and polyiamonds’. Journal of Computational and Applied Mathematics. 174 (2005) pp. 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) pp. 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, pp. 16-17 (2 April 2013)

Quadrilateral tiling leading to Penrose tiles.

 

Richardson, Martin.Mixed Media: Holography Within Art’. Leonardo Vol. 20, No. 3, pp. 251 -255, 1987 (17 November 2016)

Three brief references to Escher re ‘Cubic Space Division’, no illustrations, nothing of any great importance.

 

Richmond, C. A. ‘Repeating Designs in Surfaces of Negative Curvature’. American Mathematical Monthly, 44, 1937, pp. 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

 

Some new regular compound tessellations. Proceedings of the Royal Society London A 422, pp. 311-318, 1989 (22 October 2018)

Largely academic. Of circle limit type diagrams. Nothing of direct interest.

 

Rigby, John F. ‘Napoleon, Escher, and Tessellations’. Structural Topology. 17, 1991, 17-23. (C. 2008?) Also appeared in Mathematics Magazine 64 (1991) pp. 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) pp. 420-426

————. ‘Precise Colourings of Regular Triangular Tilings’. The Mathematical Intelligencer. Vol 20, Number 1, 1998, pp. 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 pp. 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 pp. 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), pp. 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, pp. 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’. British. Journal of Psychology 64, 3 (1973) pp. 363-365. (10 August 1993). Hard Copy

Although not strictly of a mathematical nature, included here as it is occasionally quoted in impossible object matters. Of note is that I must have purposeful sought this out; I photocopied it in Hull reference library, along with an article by James Fraser. A short article of just three pages, of a popular account, with discussions on the ‘three-stick clevis’. There is nothing (unsurprisingly) tessellation related.

 

Robinson, Raphael M. ‘Undecidability and Nonperiodicity for Tilings of the Plane’. Inventiones Mathematicae. 12, pp. 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, pp. 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, pp. 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, pp. 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.

 

Rodler, Hieronymus. Eyn schön nützlich büchlin und underweisung der kunst des Messens (A Fine, Useful Booklet and Instruction in the Art of Measurement). (9 August 2017)

On perspective, first referenced on John Coulthart’s site.

 

Roelofs, Rinus. ‘Tegels kleuren’ (tile colours). In Pythagoras 4 April 1998, pp. 22-23. (22 April 2016)

An article ostensibly on the Cairo tiling (within the ‘Escher special’ edition), although it begins with Escher’s periodic drawing 3 and a sketch, from Schattsneider, p.102!

 ————. ‘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
Of note is that among the interlinking constructions is an interlocking Cairo tiling (p.254), based on Escher’s PD 133.

Rogers, C. A. ‘The packing of equilateral spheres’.  Proceedings of the London Mathematical Society 1958 pp. 606-620 WANTED

 

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.

 

Rosen, J. ‘Symmetry at the Foundation of Science’. Computers and Mathematics Vol. 17, Nos 1-3, 1989. WANTED

From a reference in Abbas.

                                                         

Rosenbaum, Joseph. Problem E721?

(From Dissections Plane and Fancy).

 

Rosenqvist, I. T. ‘The Influence of Physico-Chemical Factors upon the Mechanical Properties of Clays’. Norwegian Geotechnical Institute (Oslo) Publication 54 (1963): 1-10. (5 September 2016

From a reference in Locher and Schattschneider. An academic article. Some minor use of two of Escher prints, Cubic Space Division and Sky and Water I to illustrate Rosenqvist’s points.

Available from: http://www.clays.org/journal/archive/volume%209/9-1-12.pdf

 

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, pp. 725-750, 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).

An essay by an art teacher, in which she essentially speculates on the appeal of Escher’s works. Also see another letters between Rush and Michele arising from this, referenced below.

 

————. ‘On the Appeal of M. C. Escher’s Pictures (Continued)’. Leonardo p. Vol 14, No. 2, Spring 1981, p. 174 (and reply by Michele Emmer below) (17 February 2013)

Letters following Rush’s article above. Debating (among other matters) on who devised the impossible triangle; Escher or Penrose.

 

S

 

Sachse, Dieter. M.C. Escher. In Munich Round Up 108 (in German, fanzine) No. 108 September 1969, 11-18

Discussion of Escher and his prints Dragon (cover), House of Stairs, Another World, Day and Night, Balcony, Drawing Hands, Relativity, Sky and Water I, Stars (9 August 2016).

 

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.

 

Sakkal, Mamoun. ‘Intersecting squares: applied geometry in the architecture of Timurid Samarkind’. Journal of Mathematics and the Arts, 2018, Vol 12, Nos. 2-3, pp 65-95 (July 2018)

An extensive treatment on the subject. Although much of this is beyond my immediate interests, of note is that of one of Sakkal’s procedures, of double squares, p. 87, where he composes what is in effect a parquet deformation. In his notes, he mentions Makovicky, referring to Shipibo geometric designs Symmetry: Culture Sci 22 (2011) pp. 373-389.

From his web page: Born in Damascus, Dr. Mamoun Sakkal is a native of Aleppo, Syria, who immigrated to the United States in 1978. He practices Arabic type design, graphic design, and calligraphy as principal and founder of Sakkal Design in Bothell, WA.

 

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, Reza. ‘The Sky Within: Mathematical Aesthetics of Persian Dome Interiors’ pp.145-156. In Bridges 1999 (7 March 2006)

 

————. Interlocking Star Polygons in Persian Architecture: The Speclial Case of the Decagram in Mosiac Designs’. Nexus Network Journal Vol. 14, No. 2, 2012, 345-372 (13 September 2018)

Premise is of a historical account. Gives a conveneient time-line of surviving histrocal documents on Islamic designs (pp. 348-350). Various constructions.

 

————. 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, 1-30. ‘ (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 (Escher) concerned. a related piece by George Escher Essentially an update of ‘Potato printing a game for winter Evenings  M.C. Escher Art and Science 9i11’ as an addendum to his article of Escher by the same author.

 

————. ‘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 .

 

————. ‘M.C. Escher and the Crystallographers’. Acta Crystallographica. Section A, Foundations of crystallography, 2005 Volume 61. WANTED

 

————. ‘Lessons in Duality and Symmetry from M. C. Escher’. Bridges Leeuwarden, 2008 1-8 (August 2008)

Likely one of the lesser articles of interest from Schattsneider; there is nothing really new or insightful here. Of interest is a reference, pp. 7-8 to Dylan Thomas and his ‘Coast Salish’ tessellation artwork, although quite why Schattsneider chose to promote this is unclear; his work is undeserving, in both extent and quality. However, they seem to have some kind of association or friendship; Schattsneider comments again on his work in a 2011 paper by him. ‘Artist profile Dylan Thomas: Coast Salish artist’. Journal of the Mathematics and the Arts. Vol. 5, No. 4, December 2011, 199-211.

 

N.B.  For other papers where Schattsneider is listed other than the main author see : (1) 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. (2) Dylan Thomas 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

Also see Review section

Note that although Schattsneider  was one of two of the lead authors of M. C. Escher’s Legacy, with no less than 41 paper by different authors, her contribution was limited to writing the preface.

 

————. ‘Marjorie Rice and the MAA tiling’. In Journal of Mathematics and the Arts, 2018, Vol. 12, Nos 2-3, 114-117 (25 July 2018)

 

Schenk, Robert. ‘Hexagonal Jigsaw Puzzle Pieces’. 1-9. April 2016 (April 2016)

Self published.

 

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.

 

————. ‘Colour Symmetry’. Bulletin of the London Mathematical Society 16 1984: 209-240. (28 March 2017)

From a reference in Tilings and Patterns and others. In three parts: Part One: Symmetry Groups, Part Two: Colour Groups, Part Three: Colouring Patterns. Of an academic nature throughout, 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)

Academic.

 

————. ‘Orderly Dispositions in Space’. March 1988. A report on a workshop meeting. (13 December 2012)

 

————. ‘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.

 

Serlio, Sebastiano. Five bookes [sic] of architecture: translated out of Italian into Dutch and out of Dutch into English. Publisher Robert Peake Year 1611

From a reference in Frederickson.

 

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’.

 

Shechtman, D. et al. ‘Metallic Phases with Long Range Orientational Order and No Translational Symmetry’. Physical Review Letters, Vol. 53, No. 20, 12 November 1984, pp. 1951-1954 (19 June 2018)

Quoted by Grünbaum in Tilings and Patterns and Penrose (in Art and Science), and others in regards of Penrose tilings. Academic, as expected, predominantly text. Of no practical use.

 

Shefrin, Jill. ‘Make it a Pleasure and Not a Task’: Educational Games for Children in Georgian England. In Princeton Catalogue, 251-275 (24 May 2017)

Jigsaw puzzle interest.

 

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 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. ‘Note 1485. Comments on Note 1464’ (C. Dudley Langford proposition). Mathematical Gazette, 24 No. 262 (December 1940 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.

 

Singmaster, David. 82.42 ‘According to Cocker’ 302-303. The Mathematical Gazette 48 (5 March 2013)

General historical interest.

 

————. ‘Covering Deleted Chessboards With Dominoes’. Mathematics Magazine 1975, 59-66 (5 March 2013)

Academic.

 

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.

 

Smit, Bart de and H. W. Lenstra, Jr., ‘The Mathematical Structure of Escher’s Print Gallery’, Notices of the AMS, April 2003. (1 April 2011)

 

Smith, Cyril Stanley. ‘The Shape of Things’. Subtitled. The comparison of crystals, soap bubbles, crazed ash trays, insect wings, living cells and other objects demsonstates the rigorous relationship between natural forces and forms. Scientific American. 1954, pp. 58-65.

A recent  (October 2018) revival in Smith (as regards his interest in the Cairo tiling) has led to a more intensive study of his work. However, much of extensive and lengthy writings when available are mostly outside of my mainstream interest. This being so, I will not formally list these, contenting myself with a title only for the sake of having ‘seen and noted’: ‘On Material Structure and Human History’

Cairo tiling p.

 

————. ‘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 is considered as the person who brought to prominance Truchet’s seemingly forgotten work. Note a special feature of Smith in the Indian journal Resonance, June 2006.

 

Smith, Cyril Stanley. ‘Structure Substructure Superstructure’. In Kepes, G. Structure in Art and Science Braziller, 1965 pp. 29-41. (28 October 2018)

Pages 36-37 mentions his interest in pentagons.

 

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, of a academic nature. Of next to no practical use.

 

Sohncke, Leonard. ‘Die regelmässigen ebenen Punksysteme von umbegrenzter Ausdehung’. Crelle’s Journal für die reine und angewandte Mathematik. Berlin, 1874 (24 December 2014)

I’m not sure why I have this; there is not a single tiling diagram in the article! Perhaps it’s a ‘seen and noted’ reference, given Sohncke’s fame.

 

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)

Although non mathematical, included here as it contains Escher references. 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 277-321(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.

 

————. Cutting a Polygon into Triangles of Equal Area Mathematical Entertainments in Mathematical Intelligencer.

In contrast to the above, largely of a popular level, although indeed academic at times.

 

Steinbach, Peter. ‘Golden Fields: A Case for the Heptagon’. Mathematics Magazine Vol.70, No. 1 February 1997. 22-31 (13 March 2013)

Golden ratio in the heptagon; largely academic.

 

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.

 

Steward, D. ‘Trisides’. Mathematics in Schools Vol. 14, No. 3 (May, 1985), pp. 8-9 (First saw 1991?)

This is worth another look; I seem to recall I couldn’t understand the premise back in 1991

 

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.

 

————. ‘Polyominoes of order 3 do not exist’. Journal of Combinatorial Theory Series A61, No. 1 1992 130-136

From a reference in Golomb.

 

————. ‘The Ultimate Jigsaw Puzzle’. New Scientist 1764  13 April 1991

 

Stillwell, John C. ‘The Tessellating Art of M.C. Escher’. Function Volume 3, 1979, pp. 13-20 (16 January 2015)

‘Function’ was a mathematics magazine published by Monash University mathematics department from 1977-2004. All 140 issues are available online.

Somewhat of a curious article. This begins ‘simply’ by showing ‘absurd’ overlays of grids onto Escher tilings drawn as wireframes, before then moving onto decidedly advanced 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

————. ‘Spatial Realization of Escher’s Impossible World’. Asia Pacific Mathematics Newsletter, January 2011, No. 1 (16 June 2011)
More specifically on the Penrose impossible staircase, begins popular, ends academic.

 

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.

 

T

 

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, pp. 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.

 

Tarnai, Tibor, Zsolt Gaspar, and Lidia Szalait. ‘Pentagon Packing Models for "All-Pentamer" Virus Structures’. Biophysical Journal Volume 69 August 1995 612-618 (5 April 2012)

Of interest.

 

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 Holiday Inn Stratford, 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) First saw 14 May 1988

Note that the article featured in a ‘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.

Note that this article was studied extensively of the day. In short, Teeters purports to set out one of his procedures for drawing Escher-like tessellations, with the subheading ‘a short, clear discussion showing how you or your students can create tessellation art’. If only! It’s not short, it’s not clear, and at the procedure is essentially unfathomable, or at least as given! The time I spent unravelling this unholy tangle was wholly disproportionate as to worth. As such, it is next to useless as a general procedure, as it applies to a specific tiling, of a staggered isosceles triangle, of which Teeters would better have presented as such without such convolutions.

Teeters illustrates the article with three of his works, all oddly not captioned: Unicyclist, St. Bernard Dog and a Red Indian, of which only one (St Bernard) is referenced with a figure number, albeit discussed only in passing. Admittedly, all three are of interest as rare examples of such motifs. All three are of the same broad level of quality, neither particularly good nor bad. I believe that at the time of seeing they were one of the few non Escher-like instances available, and so assumed a position of greater status by default than they deserved.

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. (First saw, or at least recorded, 22 September 1987)

This article has generated considerable discussion, and of which in particular George Escher rebuts. Be that as it may, from Teuber’s premise, she quotes some psychology articles that supposedly Influenced Escher of which I now examine. She quotes, pp. 94 and 98:

During the same period (1936-1938) Escher also became aware of an experimental study by Harrower, who in April 1936, published an article ‘Some Factors determining Figure Ground Articulation in the British Journal of Psychology’.

and

Escher must have known von Fieandt’s experiments.

Both assertions, as far as I am aware, are unproven.

 

Tennant, Raymond. ‘Islamic Constructions: The Geometry Needed by Craftsmen’. International Joint Conference of ISAMA, the International Society of the Arts, Mathematics, and Architecture, and BRIDGES, Mathematical Connections in Art Music, and Science, University of Granada, Spain, July, 2003 (5 November 2010)

 

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)

Of interest mainly due to the Schattsneider collaboration, although this is limited to comments on the contents of the paper only. Quite why Schattsneider, with a primarily tiling and Escher interest chose to ‘promote’ this artist and this piece in particular is unclear; his work is undeserving, in both extent and quality. Although of a geometric nature, these are not inherently of tiling or Escher-like in the true sense. His one loosely defined tessellation artwork ‘Salmon Spirits’, is nothing special, despite a first mention in her 2008 Leeuwarden paper. Further, upon looking for him on the web, on a biography he lists the influences of three artists without even mentioning Escher!

 

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 p. 746. Tuning fork, Necker cubes.

 

Thompson, William. ‘On the Division of Space with Minimum Partitional Area’. Acta Mathematica, Vol. 11, 1887, pp. 121-134 (29 October 2018)

 

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.

 

Trinajstić, Nenad. ‘The Magic of the Number Five’. Croatica Chemica Acta CCACAA 66 (1) pp. 227-254 1993 (3 September 2018)

Seemingly a conference paper, on aspects of symmetry concerning the number five. Of note is p. 234, of the Cairo tiling, of a relatively considered discussion with references, albeit with errors in fact, crediting Blackwell with Martin Gardner’s original line. Perhaps of most note is a reference to Lothar Collatz’s‘ use, of which I was unfamiliar with, and followed up. In general, a pleasing read. Has an extensive bibliography, of 151 references! Trinajstić, was a new name to me.

 

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.

 

Tyler, Tom. ‘Benevolent Confraternity of Dissectologists’. In Game & Puzzle Collectors Quarterly. Volume 1, Number 1, March 2000, 10-11 (10 March 2015)

On jigsaws. On the establishing of the BCD.

 

————. ‘1760s Jigsaw Puzzle Maps of Lady Charlotte Finch’. In Game & Puzzle Collectors Quarterly. Volume 1, Number 3, September 2000, 11-12 (10 March 2015)

Analysis and speculation on the recent discovery of the Lady Charlotte Finch cabinet.

 

U

 

Usiskin, Zalman. ‘Enrichment Activities for Geometry’. Mathematics Teacher 264-266 (5 March 2013)

Minor reference to convex pentagon problem; Marjorie Rice.

 

V

 

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.

 

Vighi, Paola. ‘L’uso di mediatori artistici e informatici per l’insegnamento della Geometria’ Riv. Mat Univ. Parma 6 3 (2000) 183-197. (In Italian) (29 March 2016)

An abstract in English which gives a meaning to this states:

This work was inspired by a periodic drawing of M. C. Escher, based on a pentagonal tessellation. First we have reconstructed the basic pentagon by means of the CABRI software, then we have identified the basic grid and link from which the drawing derives. It

is recognized that it falls under the C4 type in the 17-group classification of patterns design. This allows to treat the issue of geometry and its teaching, providing suggestions of pedagogical relevance. Finally we deal with the theme of the plane covering by pentagons

and its fundamental points are illustrated.

Ostensibly on pentagons. Has many instances of the Cairo tiling (although not stated as such) pp. 185-187, 193. Has an interesting pentagonal paving photo on p. 194.

 

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 2015) that Escher did indeed use this to at least E21 (Running Man), of which De Villiers himself, 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

 

W

 

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’.

In 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. I tried to follow up on this article with both people here. Thurston told me that there might many more pictures of Escher in the Telegraph archive, or they may just have been thrown away! Walker did not reply. Tung Ken Lam (7 November 2017) has pointed out to me two errors of fact in the article that I had missed; a 1932 Alhambra visit, and printing all his own work.

 

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.

 

Weaire, Denis. ‘C S Smith’s Development of a Viewpoint’. Complex Ideas and their Demonstration in the 2D Soap Froth. Resonance June 2006, pp. 31-40 (29 October 2018)

One of a series of writings of the June 2006 dedicated issue paying tribute to C. S. Smith’s work. Of general interest regarding Smith and Weaire’s froth, but nothing specifically of tiling.

 

————. ‘A Philomorph looks at Foam’. Proceedings of the American Philosophical Society, Vol. 145, p. 564, 2001 (5 November 2018)

A reference in the  article above. Of mild interest. Smith discussion.

 

Wells, A. F. ‘The Geometrical Basis of Crystal Chemistry IX  Some properis of Plane Nets’. Acta Chry v. B24, pp. 50-57. WANTED


Welsh, T. ‘Designing and Coloring of Scotch Tweeds’, Posselt’s Textile Journal, October 12, 1912, pp. 90-91 (29 April 2019)

Of houndstooth interest. Reference to Sir Walter Scott and black and white checked trousers, with reference is made to ‘shepherd’s plaid’, presumably equating to shepherd’s check.

From Wikipedia: Emanuel Anthony Posselt (1858–1921) was an authority on Jacquard looms and weaving. His book on the Jacquard machine is considered to be a classic…. Posselt's Textile Journal that ran between 1907 and 1923.


Wenninger, Magnus J. ‘Artistic Tessellation Patterns the Spherical Surface’. International Journal of Space Structures Vol. 5 Nos. 3&4 1990 247-253 (17 January 2017)

No Escher-like tessellation used; the article is more on polyhedra rather than tessellations.

 

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.

 

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.

 

Williams, Anne D. ‘Perplexity Puzzles’. Game Researchers’ Notes 24 5564-5568 October 1996 (11 December 2016)

On Margaret Richardson’s ‘Perplexity’ Puzzles. Contains some new detail.

 

————. ‘Not Spilsbury, Not Finch, But Who? Jill Shefrin’s Discoveries’. Game & Puzzle Collectors Quarterly, September 2003 Vol. 4 No. 3, p. 12 (19 March 2015)

On Shefrin’s investigations as to the originator of the jigsaw puzzle.

 

Anon. ‘The Perplexity Puzzle: The Fad of the Year’, brochure, circa 1908

 

Pauline Wixon Derick Library Dennis, Massachusetts “Emily Evans” Margaret Richardson” NOT SEEN

 

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 pp. 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, pp. 20-43 (28 March 2011)

Delightful! 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. pp. 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.

 

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).

Cairo tiling interest, from a footnote in The Geometrical Foundation of Natural Structures by Robert Williams, p. 43. 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. February 1988, pp. 138-141. (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.

 

Woods, H. J. ‘The Geometrical Basis of Pattern Design. Part I: Point and Line Symmetry in Simple Figures and Borders’. Journal of the Textile Institute (Manchester) Transactions T197-T210 26 1935. (25 April 2016) Published online: 11 Dec 2008 in Journal of Mathematics and the Arts.

From a reference in Grünbaum, and elsewhere. A four-part work: (i) Point and Line Symmetry in Simple Figures and Borders (ii) Nets and Sateens (iii) Geometrical Symmetry in Plane Patterns (iv) Counterchange Symmetry in Plane Patterns.

Of no practical use. Of a crystallographic viewpoint. Of interest in the early usage of the term ‘counterchange’.

 

————.‘The Geometrical Basis of pattern Design Part IV: Counterchange Symmetry in Plane Patterns’. Journal of the Textile Institute (Manchester) Transactions T305-T320 27 1936. (25 April 2016)

 

Wollny, Wolfgang. ‘Contributions to Hilbert’s Eighteenth Problem’. Pacific Journal of Mathematics Vol. 112, No 2, 1984 pp. 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.


Wright, Aaron Sidney. ‘The origins of Penrose diagrams in Physics, Art, and the Psychology of perception’, 1958–62. Endeavour, Volume 37, Issue 3, September 2013, pp. 133-139. WANTED

Wunderlich, Walter. ‘Starre, kippende, wackelige und begwegliche Achtfläche’. Elemente der Mathematik 20, pp. 25-32, 1965 (26 January 2015)

From a reference in Hinged Dissections: Swinging & Twisting. Academic throughout. Of no practical use.

 

X

Jie, Xu and Craig S. Kaplan. ‘Calligraphic Packing’. In GI '07: Proceedings of the 2007 conference on Graphics Interface, pp. 43-50, 2007. (30 September 2009)

Although not strictly mathematics, included for the sake of general interest. All most clever, as is of all of Kaplan’s work. Broadly, taking words and fitting them into a connected object.

 

Y

 

Yazar, Tuğral. ‘Revisting Parquet Deformations from a computational perspective: A novel method for design and analysis’. International Journal of Architectural Computing 2017, Vol. 15 (4) 250-267 (7 December 2018)

Broadly, written from an architecture viewpoint. Tuğral, who I have corresponded with previously on parquet deformations, and has studied these extensively, and here presents his (first?) paper on them. However, I don’t quite know what to make of this, as the writing and explanations are somewhat technical. It is essentially a study of Huff student-inspired works and in particular ‘Trifoliate’ by Glen Paris. Other Huff-related works include ‘Crossover’, by Richard Long and ‘I at the Center’ by David Oleson. Further, his own students works are included as well. Extensive use is made of Grasshopper, much beyond my understanding, or indeed interest, as good as it may be in the right hands. Mentions of myself p. 254 and in the acknowledgements, p. 265. Gives a good bibliography, although many of the references are peripheral.

 

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)

Partly on free-forming Escher's lizards; a little technical in places. Yet another Beyer reference in the paper…

 

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.

 

Z

 

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 pp. 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.

 

Zongker, Douglas. ‘Creation of Overlay Tilings Through Dualization of Regular Networks’. ISAMA 99 Procedings pp. 495-502 (11 September 2018)

From a reference in Craig Kaplan’s thesis. As an aside, the first ISAMA conference. Somewhat advanced as to concept in places. The premise, although simple to state, is not fully understood. Use is made of the Laves tilings. Be that as it may, the resulting four tilings so produced are impressive. However, I have  no plans to re-read or adopt the methodology.

 

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!


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