ARTICLES

A

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 3^rd International Conference of the Faculty of Fine Arts, Alexandria University, at Bibliotheca 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)

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 Classification 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.

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!

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.

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.

Andrews, Noam. ‘Albrecht Dürer’s personal Underweysung der Messung’. Word & Image, Vol. 32, No. 4 October-December 2016, 409-429 (24 June 2019)

The premise is that of Durer’s series of ‘...handwritten revisions and drawn additions by Albrecht Durer in his own copy of the treatise...’. Notable is that of further tilings, p 416. I might just add that I was wholly unaware of these additional tilings until I had found this reference! Further, the article gives a link to a scanned copy of the corrections, which is mighty convenient! Minor spiral references of 2020 interest include pp. 412 and 414. Overall, I must say I am most impressed with Andrews’ scholarship here, and who incidentally is a new name to me. Taylor & Francis online gives:

Noam Andrews holds a PhD in History of Science from Harvard University, and is currently a Jane and Morgan Whitney Fellow in the Department of European Sculpture and Decorative Arts, Metropolitan Museum of Art. He is also a trained architect and has held fellowships in Villa I Tatti and the Max Planck Institute for the History of Science, Berlin. His forthcoming themed exhibition on the relationship between mathematics and aesthetics (co-curated with Jennifer Farrell, Associate Curator, Department of Drawings and Prints) will be exhibited at the Robert Wood Johnson, Jr. Gallery at the Metropolitan Museum in spring 2017.

Angrist, S. W. ‘Perpetual Motion Machines'. Scientific American, Vol. 218, no. 1 January 1968, pp. 114-122 (21 April 2020)

Escher's print Waterfall used, p.114 (full page, with commentary) to illustrate perpetual motion machines. No other mention is made of Escher in the text. Oddly, the article itself gives a date of 1967.

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 tessellation 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 Voyage^TM 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. 'Self-Similar Sets 5. Integer Matrices and Fractal Tilings of Rn'. Proceedings of the American Mathematical Society, 112 (1991): 549-562. (20 April 2020)

Academic. Of no practical use. From a reference in Visions, 2nd edition (Schattschneider briefly mentions Brandt in the afterword, p. 346 as regards fractal tiling). Illustrated with various self similar images. Incidentally, Bandt wrote a series of at least seven such numbered articles, beginning in 1989 to 1992; not investigated further.

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’. 6^th 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 20^th 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.

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

Caldwell, Clare E. ‘The mind's eye.’ Journal Neurology Neurosurgery and Psychiatry, 2014 October, Volume 85, Issue 10, p.1064, Editorial Commentary (12 August 2019)

Of historical tiling interest. On the Eliseevichi, Russia, tusk artefact, with an old, c. 12,000-15,000 BC tiling of hexagons.

In short, this editorial commentary paves the way for G. D. Schott’s more expansive article in the same journal. A feature of Caldwell’s piece is that, bizarrely, of introducing the Russian artist Wassily Kandinsky into the topic. I see no connection or need! Kandinsky has no interest in tiling. Geometry, in a broad sense, yes, but not tiling. If he is included, why not many others? Further, the outlet seems a little odd; why a medical journal? No discussion of the historical mathematical aspect.

Callingham, Rosemary. ‘Primary Students Understanding of Tessellation: An Initial Exploration. Proceedings of the 28^th 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, January 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.

Chung, Ping Ngai, 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, pp. 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)

Quite how to best describe this work is unclear; it is not an article per se, although it is leaning towards this. Nonetheless, l thus placed here as an article. A translated description reads:

Lothar Collatz: Geometric ornaments. Lecture and speeches on the occasion of the award of the honorary doctorate by the Faculty of Science on November 12, 1985, Augsburg 1986

From a reference in Nenad Trinajstić. Text is all in German, with no translation. Replete with tiling diagrams, full of interest, albeit effectively with my having only the most minimal German the text is unreadable. Pp. 2-13 are all text, and can be disregarded. To what extent this is original is unclear.

Of most interest are the Cairo tiling diagrams. Attempts are made occasionally with Escher-like tessellation with minimal detail, of an eye. However, these are mostly in effect designed as ‘overlaps’, with no skill or imagination required. One exception, of a polyomino, is that of a dog,, which is quite reasonable. Pages of interest:

Horsehead p. 25 with minimal detail, of an eye designed as ‘overlaps’, with no skill or imagination required

Dogbone and applecore (of recent 2019 interest) p. 26

Human figure p. 27 human figures, designed as ‘overlaps’, with no skill or imagination required

German street paving tile, p. 27

Cairo tiling diagram, p. 29

Dog, with minimal detail, of an eye, of a polyomino, which is quite reasonable, p. 29

Cairo tiling diagram, p. 45

Of note is the bibliography for an interesting reference of a listing of 20 entries, 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 and D. C. Kay. ‘Solution to Problem 5328’. The American Mathematical Monthly, Vol. 73, No. 8, Oct., 1966 pp. 903–904 (7 March 2013)

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

Academic. Despite an ostensibly popular account by the title, too advanced for me. No diagrams.

————. ‘Triangulated Polygons and Frieze Patterns (Continued)’. The Mathematical Gazette Vol. 57, No. 401, October 1973 pp. 175–183. (8 July 2019). NOT SEEN

Academic. A follow-up to the above.

Cousineau, Guy. ‘Tilings as Programming Exercise’. Theoretical Computer Science 281 (2002) pp. 207–217 (5 April 2012)

Academic. Mostly obscure programming. Illustrated with Escher's Circle Limit III

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

The first of six articles by Cromwell, largely featuring Islamic tiling, some of more interest than others.

Popular account. Of peripheral interest only (being on knots). This article (biography) mentions his interest in M. C. Escher and impossible figures.

————. ‘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.

————. ‘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-periodicity is found.

————. ‘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. The inclusion of the word ‘stars’ in the title is questionable, the premise is not of a dedicated study of stars as such. Readable with reservation.

————. ‘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.

Cromwell, Peter R. 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. Of recent (September 2019) interest due to a Twitter posting of mine on a Hidezaku Nomura kite tiling, of which upon Vincent Pantaloni retweeting, gained considerable interest, expanding to a ‘Pythagorean’ tiling of kites and squares. Cromwell and Beltrami title this as ‘Whirling Kites’, based upon a tiling at the Friday Mosque, Isfahan, central Iran. This article I consider to be the best, by far, broadly accessible, with occasional advanced maths. Unfortunately, their terminology does not appear to have gained traction, as a search upon the term, and with tessellation added, nothing related shows. John Golden has studied this, with a very nice GeoGebra animation.

The extent of Elisabetta Beltrami’s input is unclear, but likely of a decidedly lesser nature. She was a colleague of Cromwell at the University of Liverpool.

Cromwell, an advanced mathematician, in addition to his serious work, also has an interest in many recreational aspects, and in particular on Islamic tiling, of which he has numerous papers. However, although all are of a broad popular level, and indeed of interest, I only pay lip service to his analysis. As such, I find Islamic designs broadly intractable; there is so much variation that it is overwhelming, although I retain a passing (albeit passive) interest. He also has articles such as Borromean Rings and Celtic Knotwork. His interests also extend to impossible figures and Escher although I have not seen anything from his in this field.

Crowe, Donald. 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!

Crowe, Donald W. ‘The Geometry of African Art III. The Smoking Pipes of Begho’. In Chandler Davis, (Editor), B. Grünbaum (Editor), F. A. Sherk (Editor). The Geometric Vein: The Coxeter Festschrift. Springer, 1981 First edition, pp. 177–189. (16 June PDF)

From a reference in Tilings and Patterns. Skim read this (and the whole book). Essentially on symmetry classifications. Of a more popular level than most of the book, but still nothing of any real interest to me.

————. ‘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!

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.

————. ‘Jaap Seidel 80’. In special issue dedicated to Dr Jaap Seidel on the occasion of his 80th birthday, Oisterwijk, 1999. Designs, Codes, and Cryptography. 21 (1-3) (2000), 7-10. (30 November 2020)
A brief, albeit significant mention of Escher in relation to his exhibit at the 1954 International Congress of Mathematicians Conference, Amsterdam, p. 9: 1954 had the International Congress of Mathematicians at Amsterdam. Jaap was leader of the entertainment committee. We got the idea to attach an Escher exhibition to the congress. It was a great success. A great thing for Escher too: having it in the prestigious Stedelijk Museum, it gave him a recognition that he did not have before. And it brought him into contact with scientists from all over the world. In particular with Coxeter and young Penrose. The effect on his work is easily seen: he had learned about the circle groups.

————. ‘Jaap Seidel, a friend’. NAW 5/2 nr. 3 September 2001 pp. 204-206 (30 November 2020)

Nieuw Archief voor Wiskunde (New Archive for Mathematics)

Tribute to Jaap Seidel, who died on May 8th, 2001 at the age of 81. This leans heavily on his earlier ‘Jaap Seidel 80’ celebration where the full text can be found.

Escher, as above.

Wikipedia:
The New Archives for Mathematics is a publication of the Royal Mathematical Society and is published four times a year.

The journal is aimed at anyone who is professionally involved in mathematics, as an academic or industrial researcher, student, teacher, journalist or policy maker. Its aim is to report on developments in mathematics in general and in Dutch mathematics in particular.

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, Doris Schattschneider, and 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.

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 Canada Journal, 3, January 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 a 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. 2^nd 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. 2^nd 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 70^th 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 70^th 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-poule, for successive Bridges conferences. Each article is broadly a combination of popular and scholarly accounts, and at times he uses his superior mathematical ability to write somewhat techically, far beyond my understanding. Be that as it may, these can be considered as a must read, and they leave writings by others far in the shade.
A most pleasing discussion, although technical in places, but still largley popular, with some pleasing innovations, such as the houndstooth appearing in different orientations. Much mention of Heesch and Keinzle.

————. ‘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.

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.

————. ‘A Novel Line Fractal Pied de Poule (Houndstooth)’ Bridges 2015 Conference Proceedings
In short, houndstooth as a line fractal. Somewhat advanced in nature.

————. ‘Pied de Pulse: Packing Embroidered 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.

————. ‘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 cellular automatons. Somewhat advanced in nature.

————. ‘Cellular Automata-Based Generative Design of Pied-de-poule Patterns using Emergent Behavior: Case Study of how Fashion Pieces can Help to Understand Modern Complexity’ International Journal of Design Vol. 12 No. 3, 2018 pp. 127-144.
More precisely, Pied-de-poule, also called houndstooth, is the textile pattern produced by weaving black and white yarns in alternating blocks, both in the warp and the weft, using a balanced twill binding such that the block size is twice the float length, thus appearing as shown in Figure 1 (which indicates how the weaving works. Source: Wikimedia) and Figure 2 (which shows that certain variations exist). (following on from quoting Franklin)

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.

Fejes Tóth, G. ‘Über Parkettierungen konstanter Nachbarnzahl’. Studia Scientiarum Math Hungaricae 6, 1971, pp. 133–135 (24 May 2021). Translate: About tiling of constant number of neighbors From a reference in Tilings and Patterns. Academic, of no practical use; no ‘normal’ tiling diagrams as such.

Fejes Tóth, L. ‘On shortest nets with meshes of equal area’. Acta Mathematica Academiae Scientiarum Hungaricae 11, 1960, pp. 363–370. (24 May 2021).

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

Note that L. Fejes Tóth was a prolific author, and of which I have a whole host (15) of other papers by him, all of which are of an academic nature, but are of no practical use; therefore, these are not listed here. The listing here includes that as found in Tilings and Patterns. And on occasion semi-popular, generously descibed at times.

————. ‘Scheibenpackungen konstanter nachbarnzahl’. Acta Mathematica Academiae Scientiarum Hungaricae 20, 1969, pp. 343–351. (24 May 2021). In German. Translated: Disk packs of constant number of neighbors.
From a reference in Tilings and Patterns. Of no practical use; of 15 figures, with Figs. 1-9 of simple tilings already well known.

————. ‘Tessellation of the Plane with Convex Polygons Having a Constant Number of Neighbours’. American Mathematical Monthly, 82, 1975, pp. 273–276 (7 March 2013)

From a reference in Tilings and Patterns. Although the premise is straightforward, I do not understand the tenure of Fejes Toth’s paper. Occasionally academic.

————. ‘Symmetry Induced by Economy’. Computers and Mathematics with Applications. Vol. 12B, Nos 1/2, 1986, pp. 83–99 (2 October 2013)

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

————. 'Isoperimetric problems concerning tessellations'. Acta Mathematica Academiae Scientiarum Hungaricae 14, 1963, pp. 343–351. (24 May 2021)

From a reference in Tilings and Patterns. Academic, of no practical use; no ‘normal’ tiling diagrams as such.

————. ‘What the Bees Know and What They Do Not Know’. AMS Bulletin 70 (4) July 1964 pp. 468–471. (8 March 2012)

Both popular and academic in parts. Various aspects, noticeably on isoperimetric aspects.

————. 'Remarks on a theorem of R. M. Robinson’. Studia Scientiarum Math Hungaricae 4, 1969, pp. 441–445 (24 May 2021) From a reference in Tilings and Patterns. Largely academic.

Fejes Tóth, L. and A. Heppes. ‘Multi-saturated packings of circles’. Studia Scientiarum Math Hungaricae 15, 1980, pp. 303–307 (24 May 2021) From a reference in Tilings and Patterns. Circle packing. Largely academic.

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 Grünbaum. 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 appear to be what Frank is quoting. However, disappointingly, 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 Classifications of Representative 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

Gács, P. ‘Packing of convex sets in the plane with a great number of neighbours’. Acta Mathematica Academiae Scientiarum Hungaricae 23, 1972, pp. 383–388. (24 May 2021)

From a reference in Tilings and Patterns. Of no practical use; of three figures, but not of conventional tiling.

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 tessellations; shows no understanding at all. Horrendous.

Gilbert, E. N. ‘Random Subdivisions of Space into Crystals’. Annals of Mathematical Statistics. 33, 1962, 958-972. (4 May 2020)

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

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. Schattschneider 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, Quasicrystals, 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 - Preamable

Michael Goldberg (1902–1990). His 'A Mathematical Autobiography' article in Structural Topology gives 60 articles, from 1925.

Goldberg, Michael. 'Central Tesselations' [sic], Scripta Mathematica 21, 1955, Pp. 253-260. WANTED

From a reference in Tilings and Patterns.

Wikipedia

Scripta Mathematica was a quarterly journal published by Yeshiva University [New York] devoted to the philosophy, history, and expository treatment of mathematics. It was said to be, at its time, "the only mathematical magazine in the world edited by specialists for laymen."

The journal was established in 1932 under the editorship of Jekuthiel Ginsburg, a professor of mathematics at Yeshiva University, and its first issue appeared in 1933 at a subscription price of three dollars per year. It ceased publication in 1973. Notable papers published in Scripta Mathematica included work by Nobelist Percy Williams Bridgman concerning the implications for physics of set-theoretic paradoxes, and Hermann Weyl's obituary of Emmy Noether.

————. ‘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 105 - 11113 (1958), 107-l 09

————. 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.

Goldberg, Michael and B. M. Stewart. ‘A dissection problem for sets of polygons’. American Mathematical Monthly, 71, 1964, 1077-1095. (4 May 2020)

From a reference in Tilings and Patterns. Academic, of no practical use. Profusely illustrated with ‘simple’ diagrams, but the text is way beyond me.

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 Schattschneider. 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.

Goodman-Strauss, Chaim and N. J. A. Sloane. ‘A Coloring Book Approach to Finding Coordination Sequences’, Acta Crystallographica Section A: Foundations and Advances, 2019, Volume A75, pp. 121-134. (8 November 2019)

Ostensibly, and indeed essentially, on the Cairo tiling, but the premise of the authors' article is way beyond me! This indirectly refers to my research, of a 1950s beginning, but not by name.

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)

————. ‘M. C. Escher en zijn experimenten: een uitzonderlijk graphicus' De vrije bladen 17, no. 5 (May 1940): 3-32.

Translated as ‘M. C. Escher and his experiments: an exceptional graphic artist’. A translation is included in the Ronald J. Loveland thesis.

Graustein, W. C. ‘On the average number of sides of polygons of a net’. Annals of Math 2 32, 1931, 149-153. (4 May 2020)

From a reference in Tilings and Patterns, p. 163. Of a mathematical biological premise. Popular to begin with, then academic; of no practical use.

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.

Griswold, R. E. ‘When a fabric hangs together (or doesn’t)’. Web Technical Report, Computer Science Department, University of Arizona, 2004. (19 February 2019)

Of weave interest.

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 Society. Vol. 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 Generalization of Theorems of Kirszbraun and Minty’. 812-814. Vol. 13, No. 5, October 1962 (26 February 2013)

Of an academic nature throughout; of no use.

Grünbaum, B. and G. C. Shephard. ‘Do maximal line-generated triangulations of the plane exist?’ American Mathematical Monthly, 85, 1978, 37-41. (4 May 2020)

From a reference in Tilings and Patterns. Academic, of no practical 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 an indirect reference in Tilings & Patterns. Obtained on the premise of it consisting of ‘popular tiling’. However, somewhat of a let down as regards tiling content, although tiling is indeed shown, but is rather of ‘technical weave matters’, 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, Grünbaum 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 here could otherwise be wasted on matters largely inconsequential to my researches and understanding.

The references list works on fabrics and weaving by Albers, Fox and Haton, Nisbet, Oelsner, Pizzuto and d’Alessandro, Strong and Watson, of which at the time (1996) I neglected to pursue, being under the impression, from p.? that these were generic instances, of no particular importance, being one of many. However, upon a latter day return (of houndstooth studies, 2019) I now find this not to be so! Grunbaum and Shephard seem to have chosen specially; the books here, some found on cs.arizona.edu, seem to have been selected, of high quality.

Mentions: Nisbet p. 154.

————. ‘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) fabric 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, B., J. C. P. Miller and G. C. Shephard. ‘Uniform Tilings with Hollow Tiles’. In Chandler Davis, (Editor), B. Grünbaum (Editor), F. A. Sherk (Editor). The Geometric Vein: The Coxeter Festschrift. Springer, 1981 First edition, pp. 17-64 (16 June PDF)

From a reference in Tilings and Patterns. Skim read this (and the whole book). As to be expected, overwhelmingly academic, with only occasional text at a popular level. Gives a nice discussion on Badoureau’s work.

Note that parts of this paper also appeared in the later Tilings and Patterns, titled ‘Hollow Tiles’, pp. 632-643.

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.

Grünbaum, B. and G. C. Shephard, ‘Spherical Tilings with Transitivity Properties’. In Chandler Davis, (Editor), B. Grünbaum (Editor), F. A. Sherk (Editor). The Geometric Vein: The Coxeter Festschrift. Springer, 1981 First edition, pp. 65-98 (16 June PDF)

From a reference in Tilings and Patterns. Skim read this (and the whole book). As to be expected, overwhelmingly academic, with only occasional text at a popular level. Gives a nice discussion (p. 70) on Escher’s spherical tilings.

————. ‘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 square and equilateral 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 Schattschneider. 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 Hirschhorn 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)

Wikipedia: The predecessor of the magazine [Elsevier Weekblad], Elsevier's Geïllustreerd Maandschrift (Elsevier's Illustrated Monthly), was first issued in January 1891 and was modelled after Harper's Magazine. It was published by J.G. Robbers and his Elsevier company, which had been founded in 1880 and took its name from the famous (but unrelated) Elzevir family of the 16th to 18th centuries. In 1940, the magazine was prohibited by the German authorities, who occupied the Netherlands at the time, and the last issue of the magazine was issued in December that year.

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. 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 useful 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 produced 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 collaborator 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 tessellations, with usual teacher shortcomings.

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 27^th 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.

Kemp, Martin. ‘Science in culture: A trick of the tiles’. Nature 436, p. 332, 2005. (15 April 2020)

On Penrose tiling, despite the title. From the abstract or introduction:

Penrose tiling is realized on a huge scale in Perth to give a perceptual feast for the eyes. Geometry in Western art predominantly involves space and proportion. But in other cultures, most notably Islamic, Chinese and Japanese, artistic geometry flowered most conspicuously in flat patterns, above all in the invention of striking tessellations in tiling, mosaics and textile designs...

I find Kemp, an authority on Leonardo, a most impressive figure, of which I have long been aware of him from his book The Science of Art. Optical Themes in Western Art from Brunelleschi to Seurat, from 1993(?). I now see (2020) on his website that he has a whole range of 206 science publications to his name, many of likely interest. Pleasingly, although he doesn't mention this as such, he has made this, and others (about half) available on Researchgate.

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)

The second of two papers of a like name by Kershner. Note that there are slight, but subtle differences to the two papers.

One of the most important tiling papers, and largely accessible. More exactly from a basic introduction in which polygon will tile, he then concentrates on pentagons and hexagon tiling. Mentions Reinhardt’s thesis and his role in the investigation.

Also of note

(i) Implies a par hexagon (Edward Kasner so first named), p. 6.

(ii) Gives a proof that no convex tiling polygon can have more than six sides. Is the proof that everyone recalls, but can't name where it was given? Niven (again?) proved this:

Niven, Ivan. ‘Convex Polygons that Cannot Tile the Plane’. The American Mathematical Monthly Vol. 85, No. 10 (Dec., 1978), pp. 785-792

Also see:

M. S. Klamkin and A. Liu. ‘Note on a result of Niven on Impossible Tessellations’. American Mathematical Monthly 87, October 1980, pp. 651-653

Updated 7 December 2020

————. ‘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, or even vaguely understandable! Nothing of tiling as such. Gives the law of sines and law of cosines for pentagons and hexagons. The article begins by referencing his ‘On paving the Plane’ article, but the relevancy is not clear.

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.

————. ‘Note on a result of Niven on Impossible Tessellations’. American Mathematical Monthly 87, October 1980, pp. 651-653 (4 May 2020; I recall I may have seen this earlier)

From a reference in Tilings and Patterns. Academic, of no practical use. Uses Martin Gardner's heptagon tiling.

————. ‘Simultaneous Generalizations of the Theorems of Ceva and Menelaus’. Mathematics Magazine 65, 1992, 48-52. (21 April 2020)

Academic. From a reference in the second edition of Visions, p. 89. Two mentions of Escher as regards his investigations and findings, pp. 51-52 in the context of Ceva's theorem.

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's 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)

König, D. ‘Über eine Schlussweise aus dem Endlichen ins Unendliche’. Acta Litt. Sci. Szeged 3 1927, pp. 121–130 (26 May 2021). Translated: About a conclusion from the finite to the infinite.

From a reference in Tilings and Patterns. Of no practical use; the entire article is without any diagrams!

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. Preamble

John Lansdown (1929–1999), a computer graphics expert and architect, wrote a series of short page (1-2 in length) articles for the Computer Bulletin journal between the years 1984–1992. His Escher and tessellation interests were decidedly mild (at least in print), and of which although such articles appeared on occasion, but decidedly rarely, as to be expected really. Indeed, strictly, there are only about three articles of direct interest. Upon long (April 2015) being familiar previously with his 1992 paper on Escher (incidentally his last), but no others, without the journal being readily available (or so I believed) I put his interest down to a brief flirtation and did not pursue this. However, upon recent (October) 2020 interest in the computer art of William Kolomyjec (as part of fractals investigations with a collaboration with Peichang Ouyang), I stumbled across the entire run of the articles as freely available PDFs at a Birkbeck College site. Therefore, knowing of his Escher/tessellation interest, I decided to examine the archive more circumspectly, in the hope of finding more, to which I examined each issue. However, little more was found, and what there was of relatively only mild interest. Of course, I was also looking for other recreational maths aspects of interest/computer art, and of which there are some, but not too much of direct interest. However, given the free availability of the archive, it was incumbent on me to mine this, and although strictly the time was disproportionate as to any benefits gained, what with downloading, reading and then resulting admin work to make it ‘useful’. However, I do not unduly begrudge the time spent, the articles make for excellent coffee-time reading.

The articles were under the title of ‘Not only computing - but also art’, with a later minor variation of ‘almost’ rather than ‘also’. The column (in contrast to the journal itself(?), and of which I have not seen) was of a broad popular level in general. The journal is now known as ITNOW (IT now). Lansdown has a distinguished place in computer history. In 1968 he co-founded the Computer Arts Society in the UK along with Alan Sutcliffe and George Mallen. In general, the articles per se are of interest in a peripheral sense. To include all here would only inflate the page, without any benefit. Therefore, below I give the more interesting, largely restricted to tiling.

————. ‘Escher, Escher, all fall down’. Computer Bulletin March 1985, pp. 18-19. (26 October 2020)

Column on Escher p. 19. Heesch, Chow mentions.

————. 'Truth is beauty’. Computer Bulletin December 1987, pp. 16-17. (26 October 2020)

On Owen Jones’ Grammar of Ornament, before moving on to (simple) quadrilateral tiling and again non-periodic tiling Penrose tiling. Mentions a paper by a new name, Rangel-Mondragon, which I find to be ‘Computer Generation of Penrose Tilings’ (behind a paywall). McGregor & Watt..

————. ‘Escher revisited’. Computer Bulletin April/May 1992 (23 April 2015)

Ostensibly on using a computer to create Escher-like tilings, with reference to Heesch types. However, there are no Escher-like tilings as such, only tiles without this element. Recommendation of Visions of Symmetry by Schattschneider. Chow mention. McGregor & Watt..

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 ‘anomolies’.

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.

Lewis, Frederic T. ‘The Typical Shape of Polyhedral Cells in Vegetable Parenchyma and the Restoration of That Shape following Cell Division’. Proceedings of the American Academy of Arts and Sciences, Vol. 58, No. 15 (June 1923), pp. 537-552, 554 (Free JSTOR, 7 January 2020)

Of a Donald G. Wood reference (Space Enclosure Systems). Examined (in the hope of pentagon tiling), but no tiling as such. Overwhelmingly of text.

————. ‘A Further Study of the Polyhedral Shapes of Cells’. Proceedings of the American Academy of Arts and Sciences, Vol. 61, No. 1, December 1925 pp. 1-34, 36 (12 May 2020)

Although not a commonly quoted name in tiling circles, Frederic T. Lewis, a ‘mathematical biologist’, is referenced in Tilings and Patterns, p. 163, and an article in the references (albeit just once) and Donald Wood (Space Enclosure Systems). I also recall his name mentioned in the context of soap bubbles, likely by C. S. Smith. Forewarned as to possible pentagon interest, I followed him up in May 2020 on JSTOR. I see 52 references to him, not all articles, with about 20 of possible interest. Upon examining (skim reading) en masse, a parquet deformation, p. 3, whether by accident or design, attracted my attention, not by Lewis but rather by D'Arcy Thompson, sourced from On Growth and Form, 1917 edition (not 1945 edition).

Many of Lewis’s papers contain tiling diagrams of possible interest. However, I do not have the time to examine each paper in depth, and so I have saved likely articles of interest, pending finding references to these elsewhere, of which I can then refer. This article will serve as a marker to his name; I see little point in inflating this long listing with obscure references to little immediate purpose any further!

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 15^th 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. Ranganathan), of both popular (in the Indian Resonance journal) and academic, of no less than 71 articles 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 prominence 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 (Wikipedia) 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 obvious 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 Schattschneider. 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)