Abas, Jan. ‘Islamic Patterns:
The Spark in Escher’s Genius’. In Coxeter, et al, Eds. M.C. Escher: Art
and Science. Amsterdam:
North-Holland 1986. pp. 100-112 (30 April 1994). Abbas, Masooma. ‘Ornamental Jālīs of the Mughals and Their
Precursors’. 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’. Occasional Escher: p. 32 Aboulfotouh, Hossam M. K. and
Gamal A. Abdelhameid. ‘Retrieving the Design Method of the Islamic Decagonal
Girih Patterns’. Proceedings of the 3 Acevedo, Victor. ‘Space Time
with M.C. Escher and R. Buckminster Fuller’. In Doris Schattschneider and
Michele Emmer, Eds. Adams, Colin C. ‘Tilings of Space by Knotted Tiles’. 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’. 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’.
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) Tribute and praise for Klamkin. General interest as to Klamkin. Alexanderson was a noted mathematician, although his work, save for one (joint) article (Kürschak’s Tile), is largely peripheral to tiling. Wikipedia: Gerald Lee Alexanderson (1933–2020) was an American mathematician. He was the Michael & Elizabeth Valeriote Professor of Science at Santa Clara University, and in 1997–1998 was president of the Mathematical Association of America. He was also president of The Fibonacci Association from 1980 to 1984.
Alexanderson, G. L. and Leonard F. Klosinki. ‘Mathematicians’ Visiting Cards’. The Mathematical Intelligencer (Mathematical Communities) Vol. 25, No. 4, 2003 (22 December 2011) Includes MacMahon’s visiting card.
Alexanderson, G. L. and K. Seydal. ‘Kürschak’s Tile’. Mathematical Gazette 62 1978 pp. 192-196 From a reference in Tilings and Patterns (Note that this is the only such reference to Alexander’s works here). Semi-popular.
Alexanderson, Gerald L. and John E. Wetzel. ‘Simple Partitions of Space’. Mathematics Magazine Vol. 51, No. 4 September 1978 (6 March 2013) Of an academic nature. Perhaps the title deceived me into thinking it would be simple…Of no practical use.
————. ‘Divisions of Space by Parallels’. Transactions of the American Mathematical Society, Vol. 291, Number 1, September 1985 (6 March 2013) Of an academic nature. Perhaps the title deceived me into thinking it would be simple… Of no practical use.
Alexanderson, Gerald. L. and Peter Ross. ‘Twentieth Century Gems from Mathematics Magazine’, Vol. 78, No. 2, pp. 110-123. (Date?) Of general interest and at a popular level. Includes a history of the Magazine.
Albers, Don. ‘Mathematical Games and Beyond’. Part II of an interview with Martin Gardner
Albright, Thomas. ‘Visuals-Escher’. 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
Aljamili, Ahmad M. and Ebad Banassi. ‘Grid Method Classification of Islamic Geometry Patterns’. In M. Sarfraz eds 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’. 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’. 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’. 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). 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. ‘ From Andrew Crompton’s
reference in his article
Anonymous. ‘The Geometer’s Sketchpad Workshop Guide’. 2002 Key Curriculum Press.
Anonymous. ‘Geometric Investigations on the Voyage 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. ‘Speaking of Pictures’ is a generic term,
Anonymous.
Anonymous.
Anonymous. H. S. M. Coxeter. Biography
Anonymous. 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’.
Anonymous. Ad by the AtlanticRichField Company, p. 97
(Relativity). In Prodhoretz. ‘The Literary Light as Eternal Flame’.
Anon. ‘The Lure of Puzzle Inventing’. From a reference in Williams. Minor mathematics. Loyd, 14-15, Jigsaws, wedge tromino.
Anonymous. ‘Unreal Reality, Real Unreality’. From a reference in a letter
of Cornelius van. Roosevelt in
Appel, Kenneth and Wolfgang Haken. ‘Every Planar Map is Four
Colorable’. From a reference in
Appel, K. and W. Haken. ‘The Four Color Proof Suffices’. 8, No. 1, 1986, pp. 10-20 (9 December 2011) Academic nature throughout.
ApSimon H. G. ‘Almost Regular Polyhedra’. Academic nature throughout.
————. ‘Periodic
forests whose largest clearings are of size 3’. From a reference in
————. ‘Periodic
forests whose largest clearings are of size n 4’. 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’. Brief Escher discussion and illustration.
Baboud, Roland. ‘Pavages De Pentagones’. Cairo-like diagram, David Wells mentioned, Rice, James, Stein, new types of pentagon.
Baeyer, Hans C. Von. ‘Impossible Crystals’. Article on Quasicrystals, Penrose tiles.
Badoureau, M. A. ‘Mèmoire sur les figures isoscèles’. From a reference in
Bagina, O. ‘Tiling the Plane with Congruent Equilateral
Convex Pentagons’. 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
————. ‘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.
Bakos, T. ‘2801 On Note 2530’ (Correspondence on C. Dudley Langford Cairo tile reference)’. 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'. Academic. Of no practical use. From a reference in Bandt, C. and P. Gummelt. ‘Fractal Penrose tilings
I. Construction and matching rules’. 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’. From a reference in . ————. ‘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
Barcellos, Anthony. ‘A Conservation with Martin Gardner’. Gardner interview, illuminating. I had no inkling of this journal to as late as February 2013!
Baracs, Janos, ‘Juxtapositions’. On polyhedra, with Cairo-like similarities. Has a Cairo tiling p. 65.
Baracs, Janos et al. ‘Habitat polyhedrique’.
Barnette, David W. ‘The Graphs of Polytopes With Involutory
Automorphisms’. From a reference in
Bar-On, Ehud. ‘A Programming Approach to Mathematics’. 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 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’. From a reference in
————. ‘Algebraic Theory of Brick Packing II’. From a reference in
Barrowcliff, Vikki. ‘My experience as an NQT Head of Year’. Non-tessellating article, with a one-line mention of Escher, p. ?, no illustrations.
Basin, S. L. ‘The Fibonacci Sequence as it Appears in
Nature’. Re golden section. From a reference in Livio.
Baylis. John. 73.44 ‘Fault lines and the pigeon-hole
principle’. Polyominoes.
Bays, Carter. ‘Cellular Automata
in the Triangular Tessellation’. Cairo aspect p. 148
————. ‘Further Notes on the
Game of Three Dimensional Life’. ————. ‘A Note on the Game of Life in Hexagonal and
Pentagonal Tessellations’. Beard, R. S. ‘The Fibonacci Drawing Board Design of the Great Pyramid of Gizeh’
Re golden section. From a reference in Livio.
Beard, Robert. S. (Colonel) Article as reprinted in his 1973 book ‘Patterns in Space’.
Beech, Martin. ‘Escher’s Stars’. Discussion of polyhedra used in Escher’s prints.
Beevers, Brian. ‘Filling the Gap’. 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’. On Paul Thurán and Anthony Hill.
Bellos, Alex. ‘Magic numbers: A meeting of mathemagical tricksters’.
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
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
Belov, N. V., N. N. Neronova, T. S. Smirnova. ‘The 1651
Shubnikov Groups’ (Dichromatic Space Groups.
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
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
Benedikt, M. L. ‘On Mapping the World in a Mirror’. 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 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?’ 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,
Bilney, Bruce. ‘Ozzie The Magic Kangaroo’. 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’. 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’. Academic. Found upon a search
for tessellation in
Bollabas, Bela. ‘Filling the Plane with Congruent Convex
Hexagons Without Overlapping’. 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’. 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’. 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’. 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’. 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’. 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’. Somewhat advanced, although there is the occasional diagram of interest
Boyd, David. W. ‘The disk-packing constant’. From a reference in
————.‘The osculatory packing of
a three dimensional sphere’. From a reference in
Brade, Sam. ‘Create interlocking motifs’. Tutorial on drawing tessellations of his cluster puzzle.
Bradley, H.C. and E. B. Escott. Problem 2799.
————. Problem 2933 1921, 467 Dudeney’s problem, 1902 Solution pp. 147-148 (Not in Frederickson).
————. Problem 3048. Form a reference in Frederickson.
Bravais, A. ‘Mémoire sur les systèmes formés par des points
distributés regilièrement sur un plan ou dans l’espace’. From a reference in
Breen, Marilyn Some tilings of the plane whose singular
point form a perfect set. From a reference in
————.‘Tilings Whose Members Have Finitely Many Neighbors’. Academic throughout, of no
practical purpose. Minimal diagrams. From bibliography in
Brecque, Mort La. ‘Quasicrystals. Opening the Door to
Forbidden Symmetries’. 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’. From a reference in
Brisse, François. ‘La Symétrie Bidimensionelle et le
Canada’. 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’. 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
Broos, C. ‘M. C. Escher’. In A minor ‘article’ (apparently
not previously referenced) on Escher in the context of Dutch painters of the 20 Presumably ‘C. Broos’ is he
same person as C. H. A. Broos as the author in an article in
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’. Non-tessellating article,
albeit with a slant towards
Browne, Cameron. ‘Duotone Truchet-like tilings’. Of general interest. Can be described as Truchet tiles brought up to date.
Bruckman, P. S. ‘Constantly Mean’. Re golden section. From a reference in Livio.
Buchman, E. ‘The impossibility of tiling a convex region
with unequal equilateral triangles’. From a reference in
Bunch, Phillipa. Various popular logical
puzzles of no real interest. However, included here as it has Escher’s
Burgiel, H and M. Salomone. ‘Logarithmic spirals and
projective geometry in M.C. Escher’s Path of Life III’.
————. ‘How to lose at Tetris’. Of general interest, of both popular and an academic nature.
Burn, Bob. Note 74.45. ‘The Orton-Flower tessellation’. Brief follow-up comments on Orton and Flower’s article, itself in the Gazette.
Burt, M. ‘The wandering vertex method’. One of four references in
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.
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 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’. Use of Escher's
Carrasco, Marisa, Svetlana M. Katz, Julia Winter. ‘Multidimensional
scaling and experimental aesthetics: Escher’s prints as a case study’.
Casselman, Bill. ‘On the Dissecting Table’. Is this a strictly on-line journal? I’m uncertain. On Henry Perigal’s Pythagoras dissections.
Chapelot, Pierre. 'Une Decouverte: le visionnaire Escher.' 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’.
Chavey, D. ‘Tilings by Regular Polygons’ – II. A Catalog of
Tilings.
Chen, Elizabeth R. ‘A Dense Packing of Regular Tetrahedra’. 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’. 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’. 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’. Academic throughout, of no use.
Chow, William W. ‘Automatic Generation of Interlocking
Shapes’. 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’. 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’. From a reference in
Chu, I-Ping. ‘Tiling
Deficient Boards with Trominoes’. Begins simply, then becomes academic.
Chung, Ping Ngai, Miguel A. Fernandez, Niralee Shah, Luis Sordo Viers and Elena Wilker. ‘Perimeter-minimizing pentagonal tilings’. 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’. Recreational polyominoes.
Cipra, Barry A. ‘Packing Pyramids: Is the Space Race Over?’.
On packing tetrahedral, largely a popular account.
Clarke, Eric. ‘Meaning and the Specification of Motion in
Music’. Non-tessellating article, with a one-line mention of Escher, p. 217, no illustrations.
Clason, Robert G. ‘A Family of Golden Triangle Tile
Patterns’. Many interesting ‘simple’ diagrams.
Clauss, Judith Enz.
‘Pentagonal Tessellations’. 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’. Cairo diagram p. 18, and much interesting discussion arising from this.
Coffin, Stewart. ‘Polyomino Problems to Confuse Computers’. 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’. General interest, academic.
Conlan, James P. ‘Derived
Tilings’. From a reference in
Conway, J. H and D. C. Kay. ‘Solution to Problem 5328’. 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’. Not mentioned in any of Frederickson’s three books.
Cotter, J. R.
‘Pythagoras’s Theorem as a Repeating Pattern’.
Coxeter, H. S. M. ‘The Polytopes with Regular-Prismatic
Vertex Figures’. 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
————. ‘Twelve points in PG(5,3) with 95040
self-transformations’. 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’. General historic interest
————. ‘Regular compound tessellations of the hyperbolic
plane’. Has many circle limit type diagrams, of various forms. Academic, of no real interest; far too advanced for me.
————. Review of
————. ‘The Problem of Apollonius’. 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’. ‘Typical Coxeter’, too advanced.
———— ‘Kepler and Mathematics’.
In 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
————. ‘The Mathematical Implications of Escher’s Prints’. 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’’.
————. ‘Virus Macromolecules and Geodesic Domes’. In
————. ‘Regular and Semi-Regular Polytopes II’. Of an academic nature
throughout, minimal diagram! Of no practical use. From a reference in
————. ‘The Seventeen Black and White Frieze Types’. ‘Typical Coxeter’, too advanced.
————. ‘Escher’s Lizards’. In: An analysis of two of Escher's lizard tessellation, academic from start to finish.
————. ‘Cyclomic integers, nondiscrete tessellations, and
quaiscrystals’. Academic, of no use.
————. ‘Escher’s Fondness for Animals’. In
Coxeter, H. S. M; M. S. Longuet-Higgins and J. C. P. Miller.
‘Uniform Polyhedra’. From a reference in
Craig, E. J. ‘Phenomenal Geometry’. Use of Escher’s print
Crompton, Andrew. ‘Lifelike Tessellations’. In 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’. From a reference in 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.
————.
Cruikshank, Garry. ‘The Bizarre History of Tessellated Tiles’.
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’. Polyhedral lampshade.
————.
‘Unitary
Construction of Certain Polyhedra’.
————.
‘Deltahedra’. Largely popular account.
————.
Notes 63.20. p3ml or p31m? On matters of incorrect usage of the terms.
Also see letters and reviews.
Curl, Robert F and Richard E Smalley. ‘Fullerenes’.
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?’ 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?’ Of no real interest. One of
four references in
————. ‘Equitransitive Tilings, or How to Discover New
Mathematics’. Largely academic, with occasional diagrams at a popular level.
Danzer, Ludwig, Grattan Murphy and John Reay. ‘Translational
Prototiles on a Lattice’. Largely academic, of no practical use.
Dauben, Joseph W. ‘Personal Reflections of Dirk Jan Struik’.
David, Guy, and Carlos Tomei. ‘The Problem of the
Calissons’. Packing problem caused by a French sweet of the name (calissons).
Davies, Roy. O. ‘Replicating Boots’. From a reference in
Dawson, T. R. ‘Ornamental Squares and Triangles’. Of number theory rather than of tiles! Bruijn, N. G. de. ‘Updown generation of Penrose patterns’. 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’. 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) ————. ‘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 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’. Non-tessellating article, with a one-line mention of Escher, p. 88, no illustrations.
Dehn, M. ‘Ueber den Rauminhalt’. From a reference in
Dejean, Françoise. ‘Sur un
Théorème de Thue’ From a reference in
Dekking, F. M. ‘Replicating
Superfigures and Endomorphisms of Free Groups’. From a reference in
Delgado, Olaf, Daniel Huson and Elizaveta Zamorzaeva.‘The
Classification of 2-Isohedral Tilings of The Plane’. 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) 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) 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’. From a reference in
Dendra, Daniel. ‘Augmented Culture’. 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. 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’. 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’. Academic. Dissections (subdivisions) of the pentagon by pentagons. Highly technical. Of limited interest.
Dixon, Robert. ‘Pentasnow’. Fractal-type patterns as formed by successive divisions of a regular pentagon, giving rise to snowflake appearance, hence the title.
————. From a reference in Abbas.
Doczi, György. Seen and Unseen Symmetries: A Picture Essay.
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’.
Dolbilin, N. ‘The Countability of a Tiling Family and The
Periodicity of a Tiling’. 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. Dorwart, Harold L. Configurations: ‘A Case Study in
Mathematical Beauty’. Academic.
Dostal, Milos and Ralph Tindell ‘The Jordan curve theorem
revisited’. From a reference in
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’.
Dress, Andreas W. M. and Daniel H. Huson. ‘Heaven and Hell
Tilings’. In Academic, with occasional simple tiling diagrams. Escher Heaven and Hell p. 26.
Driver, Denis. ‘Edging Towards Escher’. 11-15? (17 February 2013) A little obscure at times.
Drost, John P. ‘The Vortex Tessellation’. 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’. From a reference in E. B. Versluis, M. C. Escher: Art and Science.
Dudeney, Henry. E. All columns from Perplexity. In
Duijvestijn, A. J. W., P.J. Federicko and P. Leeuw. Compound
Perfect Squares’. From a reference in
Dunham, Douglas. ‘Hyperbolic Symmetry’. From the symmetry ‘special edition’ of the journal. Academic.
————. ‘A Tale Both Shocking and Hyperbolic’.
————. ‘A Family of Circle Limit III Escher Patterns’
————. ‘Creating Repeating Hyperbolic Patterns—Old and New’.
In
————. ‘Artistic Patterns in Hyperbolic Geometry’. In Bridges 1999. 239-250 ————. ‘Families of Escher
Patterns’. In Doris Schattschneider and Michele Emmer, Eds. Dunn, J. A. ‘Tessellations with Pentagons’. 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
Earle, Robert. ‘On the Campus’. From Google books. Shows Verbum
Eba, P. N. ‘Space-Filling with Solid Polyominoes’.
Eberhart, Mark. ‘Classics’. Has a Cairo tiling diagram,
without attribution, found in a excerpt page on C. S. Smith’s book
Edmonds,
Allan L, John H. Ewing, and Ravi S. Kulkarni.
‘Torsion Free Subgroups of Fuchsian Groups and Tessellations of Surfaces’. From a reference in
Eggleton, R. B. ‘Tiling the Plane with Triangles’. Academic, of no use (27 September 2013). Einsenstein, Jane and Arthur L.
Loeb. ‘Rotations and Notations’. In Doris Schattschneider and Michele Emmer,
Eds. Ellard, David. ‘Poly-iamond enumeration’. From a reference in
Emmer, M. ‘Visual Art and Mathematics: The Moebius Band’. Largely a historical account.
————. ‘Comments on the Note by Jean C. Rush on the Appeal of M. C. Escher’s Pictures’.
————. ‘Art and Mathematics: The Platonic Solids’. The title is a little
imprecise; this is a
————. ‘Comments on A. L. Loeb’s Correspondence with the
Graphic Artist M. C. Escher’. 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. ————. ‘Soap Bubbles in Art and Science: From the Past to the
Future of Math Art’. Largely a historical account.
————. ‘The ‘Belvedere’ by
Escher: A Modest Hypothesis’. In 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. Some nice stories on Ravello.
————. ‘Mathematics and Art: Bill and Escher’.
————. ‘Homage to Escher’. 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’.
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’. Of little direct interest.
Engel, Peter. ‘A Paper Folder’s Finding’. 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’. 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’. From a reference in
Ehrlich, Paul. ‘Eco-Catastrophe!’ Use of Escher’s
Ernst, Bruno. ‘Selection is Distortion’. In Coxeter et al,
Eds.
————. ‘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’. General interest
————. ‘Educating a Mathematical Genius: Alan Turing at Sherborne School’. General interest
Epstein, David. ‘Geometers unify their ideas of space’. Some advanced concepts of
hyperbolic geometry illustrated with Escher’s
Escher, George A. 'Letter to the Editor.' 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. Escher, M. C. ‘Approaches to Infinity’. In
Faddeev, D. K. Boris Nikolaevič Delone (on the occasion of
his 70 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 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. Farrell, Margaret A and Ernest R. Ranucci. ‘On the
Occasional Incompatibility of Algebra and Geometry’. Mooting semi-regular tessellations; somewhat advanced, of little direct interest.
Fathauer, Robert W. ‘Recognizable Motif tilings Based on
Post-Escher Mathematics’. In 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. Fedorov, E. S. ‘Systèmes des
planygones. 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, ————. ‘Descending A Staircase’. Aplimat Proceedings 2018. Slovak University of Technology in Bratislava, Faculty of Mechanical Engineering. (October 2018) Feijs, Loe M. G. and Marina
Toeters. ‘Constructing and Applying the Fractal Pied de Poule (Houndstooth)’.
In Proceedings of ————. ‘A Novel Line Fractal Pied de Poule (Houndstooth)’ Bridges 2015 Conference Proceedings ————. ‘Pied de Pulse: Packing Embroidered Circles and Coil Actuators in Pied
de Poule (Houndstooth)’. Bridges Finland Conference Proceedings, 2016 415-418 ————. ‘A Cellular Automaton for Pied-de-Poule (Houndstooth)’. Proceedings of
————. ‘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. 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, 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.
————. ‘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. Has a Cairo tiling diagram on p. 274, with a possible later reference by Richard K. Guy to a sighting at the Taj Mahal! However, upon enquiring with Guy, he does not exactly recall this. Likely he was mistaken.
————. ‘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?’. Ferguson, Helaman. ‘A Circle
Limit in Stone’. In Doris Schattschneider and Michele Emmer, Eds. Field, J. V. ‘Kepler’s Star Polyhedra’. 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’. 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 406-410 Re golden section. From a reference in Livio.
Fontaine, A. and G. E. Martin. 'Tetramorphic and
Pentamorphic Prototiles.'
————. ‘An Enneamorphic Prototile’. ('A note')
————. 'Polymorphic Polyominoes’.
————. 'Polymorphic Prototiles’.
Forseth, Scott L. ‘Solid Polyomino Constructions’.
Fosnaugh, Linda S. and Marvin E. Harrell. ‘Covering the
Plane with Rep-Tiles’.
Fowler, D. H. ‘A Generalisation of the Golden Section’. 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’. Upon seeing a reference to
equilateral pentagon tilings (i.e Cairo tiling) in Frank’s preface to
————. ‘Complex Alloy Structures as Sphere Packings. II.
Analysis and Classifications of Representative Structures’. See above.
Fraser, James A. ‘A
New Visual Illusion of Direction’. 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’.
————. ‘A New Wrinkle on an Old Folding Problem’. Heavily academic, of little direct interest.
————. ‘The Heptagon to the Square, and Other Wild Twists’
————. ‘Designing a Table Both Swinging and Stable’. Both popular and academic. Builds upon Dudeney’s dissection of triangle to square.
————. ‘Casting Light on Cube Dissections’. Heavily academic, of little direct interest.
————. ‘The Manifold Beauty of Piano-hinged Dissections’. In
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?’ 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’.
————. ‘Pattern Design research Using the Penrose Pattern (I)’
————. ‘Pattern Design research Using the Penrose Pattern (II)’
————. ‘Sampling Survey of ‘17 Kinds of Wallpaper Patterns’ in Marketing’.
————. Figurations of Mine on 17 Kinds of Wallpaper Patterns”
Fukada, Hiroshi, et al (Including Schattschneider). ‘Polyominoes
and Polyiamonds as Fundamental Domains of Isohedral Tilings with Rotational Symmetry’
(submitted to Somewhat advanced.
Fukuda, Hiroshi, Michio Shimizu, and Gisaku Nakamura. ‘New Gosper Space Filling Curves’. Conference paper 2001 (4 August 2016)
Fulton, Chandler.
‘Tessellations’. Academic, of no practical use.
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’. Of a broadly popular level.
Gailunas, Paul. ‘Spiral Tilings’. In Nice treatment indeed.
————. ‘Some Unusual Space-Filling Solids’.
Garcia, Paul. ‘The Mathematical Pastimes of Major Percy
Alexander MacMahon’. Part 1 - Slab Stacking. 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’. 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.
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’. A discussion of Coxeter and his new book, Introduction to Geometry.
Gardner, M. ‘The Eerie Mathematical Art of Maurits C. Escher’.
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’.
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’. 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 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’. 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.
————. ‘A Quarter Century of Recreational Mathematics’. A May 2010 reprint in honour of Martin Gardner. Includes reference to Polyominoes and Penrose tiles.
————. ‘Is Mathematics ‘Out There’?, Also see Reuben Hersch for a rebuttal of this piece.
————. ‘Around the Solar System’. General math puzzle column. I had no inkling of this journal to as late as February 2013!
————. ‘The Game of Hip’.
————. ‘The Ant on a 1 x 1 x 2’.
————. ‘Talkative Eve’.
————. ‘Some New Discoveries About 3 x 3 Magic Squares’.
————. ‘Ten Amazing Mathematical Tricks’.
————. ‘Chess Queens and
Maximum Unattacked Cells’.
————. ‘Curious Counts’.
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’. 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’.
Gerdes, Paulus. ‘On the Geometry of Celtic knots and their
Lunda-designs’. General interest.
Gerwein, P. ‘Zerschneidung jeder beliebigen Anzahl von
gleichen geradlininngen Figuren in dieselben Stücke’.
Gethner, Ellen, Doris Schattschneider, Steve Passiouras, and
J. Joseph Fowler. ‘Combinatorial Enumeration of 2 x 2 Ribbon Patterns’. 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’. 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’. 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’. Folding A4 paper to polyhedra; only of minor interest.
————. ‘Paper Patterns 3 – With
Circles’. 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’. 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’.
Further tessellation arising from weaving.
————. ‘Mathematics in a
Matchbox’ Part 1. 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. ‘Squashed matchbox’ observations.
————. ‘Polyhedra from A Sized
Paper’. Folding polyhedra from A sized paper, tetrahedron, Pelican cube, truncated icosahedron (football). Of general interest.
————. ‘Window Patterns’. From overlapping rectangles, no tessellation.
————. ‘Using Books to Construct
Shapes on the Blackboard’. Novel idea!
————. ‘Tangrami Square – A paper folding
puzzle’. Folding shapes, no tessellation; lightweight.
————. The Paper Roller’. Premise of ‘Owzat’, which I recall from my school days.
Gibbs, William and Mphrolet Sihlabela. String figures from around the world. General interest.
Giganti, Paul, and Mary Jo Cittadino. ‘The Art
of Tessellation’. 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’. One of four references in
Giles, Jack, Jr. ‘Infinite-Level Replicating
Dissections of Plane Figures’. From a reference in
————. ‘Construction of
Replicating Superfigures’. From a reference in
————. ‘Superfigures Replicating
with Polar Symmetry’. From a reference in
Ginzburg, Ralph (editor). 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’. 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 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
————. ‘More Origami Solids’. In
————. ‘Signs of Significance’. In
————. ‘Net Results’. In Polyhedra nets.
————. ‘Aperiodic Tiling’. In Arguably the article of most of interest. Penrose, Voderberg, Robinson, Ammann. Also see the following article on Penrose tiling
————. ‘Penrose Tiling’. In Penrose, Quasicrystals, Wang.
————. ‘Fourier Polygons’. In
————. ‘Celtic Knotwork, Part I’. In
————. ‘Celtic Knots,
Part 2’. In
————. ‘Celtic Knots,
Part 3’. In
————. ‘An Open and Shut Case’. In
————. ‘String Crossings’. In String art.
————. ‘Hierarchical Textures’. In
————. ‘Crop Art, Part 1’. In Some advanced maths in places.
————. ‘Crop Art, Part 3’. In 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’. 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 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’. From a reference in ————. 'Tetrahedra Equivalent to Cubes by Dissection'. ————. 3162. ‘A duplication of the cube by dissection and a
hinged linkage’. From a reference in
————. ‘On the Original Malfatti Problem’. Of an academic nature throughout; of no practical use.
————. ‘On the Densest Packing of Equal Spheres in a Cube’. Of an academic nature throughout; of no practical use.
————. ‘Maximising the Smallest Triangle Made by N Points in
a Square’. Of an academic nature throughout; of no practical use.
————. ‘New Rectifiable Tetrahedra’. From a reference in
————. ‘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’. Largely of an academic nature throughout; of no practical use.
————. ‘On the Space filling Decahedra’.
————. ‘On the Minimum Track of a Moved Line Segment’. 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. A follow-up arising from her book.
Goodman-Strauss, Chaim. ‘Compass and Straightedge in the
Poincaré Disk’. 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 ————. ‘Compass and Straightedge in the Poincaré Disk’,
————. ‘The Packing of Equal Circles in a Square’. 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’. Largely philosophical semantic
commentaries way beyond me (Goldstein is a philosopher). Profusely illustrated
with Escher's
Golomb, Solomon W. ‘Replicating Figures in the Plane’. Generally a popular account, drifts towards academic. C. Dudley Langford inspired.
————. ‘Tiling with Polyominoes’ From a reference in
————. ‘Tiling with Sets of
Polyominoes’ From a reference in
————. ‘Checker Boards and Polyominoes’. Semi-academic.
Golomb, Solomon W. and Lloyd R. Welch. ‘Perfect Codes in the
Lee Metric and the Packing of Polyominoes’. Academic.
Gómez, R. Pérez-. ‘The Four Regular Mosaics Missing in the Alhambra’.
Gordon, Basil. ‘Tilings of Lattice Points in Euclidean
n-Space’. From a reference in
Granger, Tim. ‘Math Is Art’. ‘How to’ on Escher-like art, with all the usual teacher shortcomings.
's-Gravesande, G. H. ‘Nieuw werk van M. C. Escher’. ————. ‘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’. From a reference in
Gregory, Richard L. and Priscilla Heard. ‘Border locking and
the Café Wall illusion’ [sic]. 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’. 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’. From a reference in
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-
————. ‘Musings on an example of Danzer’s’. Academic, not of tiling, of limited interest.
————.
‘Levels of Orderliness: Global and Local Symmetry’. In Portland Press, London 2002. Vol. I, pp. 51 – 61. Mention of Peter Raedschelders.
Grünbaum, B. and G. C. Shephard. ‘A Variant of Helly’s Theorem’. 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’. From a reference in 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? 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)
————. 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’. Of an academic nature throughout; of no use.
————. ‘Rotation and Winding Numbers For Planar Polygons and
Curves’. Of an academic nature throughout; of no use.
————. ‘A new Ceva-type theorem’. Of an academic nature throughout; of no use.
————. BG. TM On components in some families of Sers 607-
————. BG. MK Euler’s Ratio Sum Theorem and Generalisations
————. ‘The Emperor’s New Clothes: Full Regalia, G String, or
Nothing?’ 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’. As kindly supplied in a mail by Grünbaum, 18 September 1989.
————. ‘Hypersymmetric Tiles’. As kindly supplied by Grünbaum following correspondence, 18 September 1989.
————. Contains Koloman Moser’s tilings, MacMahon, Pólya, Delone, Heesch.
————. ‘The Bilinski Dodecahedron, and Assorted
Parallelohedra, Zonohedra, Monohedra, Isozonohedra, and Otherhedra’. Leaning heavily towards the academic. Polyhedra, no tiling as such.
————. ‘A Relative of Napoleon’s Theorem’. Advanced.
————. ‘A starshaped [sic] polyhedron with no net’. Advanced.
————. ‘Families of point-touching squares’. 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’ From a reference in
Grünbaum, Branko, and G. C. Shephard. ‘Perfect Colorings of
Transitive Tilings and Patterns in the Plane’. Largely academic, occasional Escher reference, shows a different presentation of Laves diagrams.
————. ‘Tilings by Regular Polygons’. Broadly accessible, with reservation! A ‘follow up’, which I don’t have, is 51, (1978), 205-206).
————. ‘Regular polyhedra – old and new’. Academic.
————. ‘The Ninety-One
types of Isogonal Tilings in the Plane’. Academic. Mostly too technical for me, despite noticeable numerous diagrams.
————. ‘Isohedral tilings of the plane by polygons’. Comment.
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’. Largely academic throughout. Many ‘simple’ diagrams; however, all of this no practical use, being academic.
————. ‘Is there an all-purpose tile?’ 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’.
————. ‘Ceva, Menelaus, and the Area Principle’. Advanced.
————. ‘What Symmetry Groups are Present in the Alhambra?’ 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’. Has dimorphic tiling and
spiral tiling forerunners to appearing in
Grünbaum, B. and Z. Grünbaum, G. C. Shephard. ‘Symmetry in
Moorish and other Ornaments’. 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’.
Gutiérrez, Angel. ‘An Experience with M. C. Escher and the
Tessellations’, 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
Haag, F. ‘Die regelmässigen Planteilungen’. (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
————. ‘Die regelmässigen Planteilungen
und Punktsysteme’. 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
————. ‘Die 17 regelmässigen Planteilungen und Punktsysteme’. Essentially, just as note rather than an article, one diagram, of staggered rectangles.
————. ‘Die pentagonale Anordung von sich berührenden Kriesen in der Ebene’. (Quoted in Bradley, Schattschneider). Has Cairo tiling in the form of circle packing.
————. ‘Die Planigone von Fedorow’ (Federov?). 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
————. ’Die Symmetrie
verhältnisse einer regelmässigen Planteilung’. 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’.
Haake, A. ‘The Role of Symmetry in Javanese Batik Patterns’.
Of limited interest.
Hadwiger, H. Z’eregungsgleichheit ebener polygone’.
Hales, Thomas C. ‘The Status of the Kepler Conjecture’. Academic.
————. ‘Sphere packing, 1. Of an academic nature throughout; the tone of the paper is way beyond me. Of no practical use.
————. ‘Sphere Packing 3. ‘Extremal Cases’. Of an academic nature throughout; the tone of the paper is way beyond me. Of no practical use.
————. ‘Sphere Packings, 4. Detailed Bounds’. 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’.
Of an academic nature throughout; the tone of the paper is way beyond me. Of no practical use.
————. ‘Historical Overview of the Kepler Conjecture’. 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’. 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. 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’ From a reference in
Hannas, Linda. ‘Two centuries of jigsaws’. 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 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’
See above. Profusion of diagrams!
————. ‘The
Fundamentals of Pattern Structure: Part III:
The Use of Symmetry Classification as an Analytical Tool’. 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’. Popular account.
Harborth, H. 'Prescribed numbers of tiles and tilings'. n-morphic tilings, in response to Grünbaum and Shepherd’s patch determined article.
————. ‘Konvexe Fünfecke in ebenen Punktmengen’. (in German) Academic, one diagram.
Hargittai, István. ‘Limits of Perfection’. 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 ————. ‘Quasicrystal Sculpture in Bad Ragaz’. (Mathematical
Tourist)
————. ‘Octagons Abound’. (Mathematical Tourist) Includes pavement.
————. ‘Fullerene Geometry under the Lion's Paw’.
(Mathematical Tourist) Chinese theme.
————. ‘Lifelong Symmetry: A Conversation with H. S. M.
Coxeter’. Interview with Coxeter. Minor Escher references, pp. 36, 38-39.
————. ‘Sacred Star Polyhedron’. (Mathematical Tourist)
————. ‘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’. Largely popular account. Brief mention of Escher pp. 699 and 705. (19 April 2017)
Hargittai, István. ‘John Conway – Mathematician of Symmetry
and Everything Else’. Interview.
Hargittai, István and Magdolna Hargittai. ‘Symmetries of
Opposites: Antisymmetry’ 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’.
Harries, John, G. ‘Symmetry and Notation: Regularity and Symmetry in
Notated Computer Graphics’. From the symmetry ‘special edition’ of the journal. Obscure.
Harrison, Wendy Cealey. ‘Madness and historicity: Foucault
and Derrida, Artaud and Descartes’. Non-tessellating article, with a one-line mention of Escher, p. 80, no illustrations.
Hart, George W. ‘Creating a Mathematical Museum on your Desktop’. 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’. From a reference in
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
Hayward, Roger. ‘The Jigsaw Puzzle and
the Inventive Mind’. Hedian, H. ‘The Golden Section and the Artist’. Re golden section. From a reference in Livio. Heesch, H. ‘Über topologisch gleichwertige
Kristallbindungen’. (About topologically equivalent crystal bonds) ————. ‘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’. Somewhat advanced concepts for the intended audience! Uses Escher print Circle Limit, p. 27.
Hensley, Douglas. ‘Fibonacci Tiling and Hyperbolas’. From a reference in
Henry, Richard. ‘Pattern and Contemplation: Exploring the Geometric Art of Iran’ (2010) Public lecture given by Richard Henry
Henry,
Bruce. ‘Polyominoes’. Brief look at combining polyominoes into pre-determined configurations.
Heppes, A. ‘Solid Circle-Packings in the Euclidean Plane’. 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’. Square packing. Largely of an
academic nature throughout, with not a single diagram! Of no practical use.
From a reference in
Herrmann, Heinz. ‘Asymmetry and Symmetry in Cellular
Organization’ From the symmetry ‘special edition’ of the journal. Obscure.
Hersh, Reuben. Reply to Martin Gardner. Opinions Column. An open letter to Martin Gardner (25 November 2011) Vol. 23, No. 2, 2001 3-5
Hilbert,
David. ‘Mathematical Problems’.
Hill, Anthony. ‘Art
and Mathesis: Mondrian’s Structures’. Philosophical musings; obscure.
————.
‘A View of Non-Figurative Art and Mathematics and an Analysis of a Structural
Relief’. Philosophical musings; obscure.
Hilton, Peter, and Jean Pedersen. ‘Comments on Grünbaum’s
Article’.
————. ‘Symmetry in Mathematics’. From the symmetry ‘special edition’ of the journal. Obscure.
Hippel,
von Frank N. ‘Arthur von Hippel: The Scientist and the Man’. 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’. Somewhat technical in places, but still of considerable interest in regard of Cairo-type tiling.
Hirschhorn, M. D. ‘Tessellations with Convex Equilateral Pentagons’.
Parabola is a Australian
mathematics magazine, first appearing in July 1964. In 2005 it merged with
another mathematical magazine for high school students, 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’.
Considerable Cairo-esque type pentagons.
————. 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 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 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.
Hoogewerff,
G. J. 'M. C. Escher, grafisch kunstenaar.' Trans. M. C. Escher, Graphic Artist Elsevier’s Geiliustreerd Maandschrift = Elsevier's Illustrated Monthly Journal 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’. 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’. From a reference in
Hollingsworth, Caroline. ‘Polyominoes: An Unsolved Problem’.
Determining the numbers possible.
Hollist, J. Taylor. ‘Escher Correspondence in the Roosevelt Collection’. 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.
————. 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. Houle, Kelly. ‘Portrait of
Escher: Behind the Mirror’. In Doris Schattschneider and Michele Emmer, Eds. Huff, William S. ‘Students' work from the Basic Design
Studios of William S. Huff’. In Parquet deformations by Huffs students. Delightful. Works by
Jacqueline Damino Right, Huff, William S. ‘Simulacra in
non-reorientable surfaces-experienced in timing’. In Chapter 4, ‘Spatial
Lines’, Patricia Muñoz, compiler Huff, William. ‘The Landscape Handscroll and the
Parquet Deformation’, 307-314. In This has four new parquet deformations by ‘new people’, namely: Alexander Gelenscer; Pamela McCracken; Loretta Fontaine; Bryce Bixby;
————. ‘Defining Basic Design as a Discipline’. In Hughes, Anne. ‘Escher’s Sense
of Wonder’. In Coxeter, et al, Eds. Huson, H. Daniel. ‘The Generation and Classification of
Tile- 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."
Huylebrouck, Dirk Antonio Buitrago and Encarnación Reyes
Iglesias. ‘Octagonal Geometry of the Cimborio in Burgos Cathedral’. Has mention of the Cordovan proportion (no pentagons though).
Inchbald, Guy. ‘Five space-filling polyhedra’, (18 February 2013) Cairo-esque aspects with bisymmetric hendecahedron.
İldeş, Güslseren. ‘An Analysis for the works of Escher and
Their Use In Art Education’.
Jablan, Slavik Vlado.
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’. 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’. Reference from Golomb.
Jendrol, S. and E. Jucovic. ‘On a Conjecture by B.
Grünbaum’. From a reference in
Johnson, Paul B. ‘Stacking Colored Cubes’.
Jones, Christopher B. ‘Periodic tilings with vertices of
species number 3’. 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’. From a reference in
Juhel, Alain. ‘Prince of Samarqand Stars’. On Ulugh-Beg.
Jung, Hwa Yol. ‘Transversality, Harmony, and Humanity between Heaven and Earth’
Non-tessellating article, with a one-line mention of Escher, p. 101, no illustrations.
Kahan, Steven
J. ‘Eight blocks to Madness’ – A logical solution. Kaiser,
Barbara. ‘Explorations with Tessellating Polygons’. Kanon, Joseph. ‘The Saturday Review December’ 16 1972 ** (29 July 2015) Sphere Spirals
Kaplan, Craig. S. ‘Computer Generated
Islamic Star Patterns’. In 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 Most informative, with a considered approach to life-like tiling. Some very clear-cut thinking. Overwhelmingly accessible.
————. ‘Islamic Star Patterns from Polygons in Contact’. 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’.
————. ‘The trouble with five’. For the online 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’. 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’. Cairo tiling p. 923 in the context of the Laves tiling; but as such, inconsequential.
————. ‘The Geometry of Coastlines’. From the symmetry ‘special edition’ of the journal. Obscure.
Kazarinoff, N. D and Roger
Weitzenkamp. ‘Squaring Rectangles and Squares’. From a reference in
Kazancigil, Ali ed. By Emerita S. Quito: ‘Value as a factor
in social action’, p. 605. Use of Escher’s print
Keeton, Greg. 'The Artist who Aims to Tease.' 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’. 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?’ Minor Dudeney reference.
Kershner, R. B. ‘On Paving the Plane’. Of significance re the distinct convex pentagon types. Gives eight of the convex pentagon types then known. ————.
‘On Paving the Plane’. 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’. 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. Kindt,
Martin. ‘Wat te bewijzen is’ (in Dutch) (38) (translated ‘What is to be proved’).
Kingston, J.
Maurice. ‘Mosaics by Reflection’. Klamkin, M. S. and A. Liu. ‘Polyominoes
on the Infinite Checkerboard’ From a reference in ————. ‘Note on a result of Niven on Impossible Tessellations’. 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’. ————. ‘A Packing Theory’. Of an largely academic nature throughout.
————. ‘Packing a Rectangle with
Congruent From a reference in
Klarner, David A. and Ronald. L. Rivest. ‘Asymptotic Bounds
for the Number of Convex From a reference in
Klarner, David A. and Spyros S. Magliveras. ‘The Number of Tilings
of a Block with Blocks’. Of an largely academic nature throughout.
Klein, Felix. ‘Vergleichende Betrachtungen uber neuere
geometrische Forschungen’. From a reference in
————. ‘A Comparative Review of Recent Researches in Geometry’.
From a reference in Knoll, Eva. ‘Life After Escher:
A (Young) Artist’s Journey’. In Doris Schattschneider and Michele Emmer, Eds. 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. 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.
Krishnamurti, R and P. H. O’N. Roe. ‘On the generation and
enumeration of tessellation designs’. From a reference in
Krivý, Maroš. ‘Towards a critique of cybernetic urbanism:
The smart city and the society of control’. 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?’ References to Kepler and Penrose.
Kulpa, Zenon. ‘Are Impossible Figures Possible?’ Although not strictly of a
mathematical nature, included here as it is occasionally quoted in impossible
object matters. From a reference in
Kuratowski,
Casimir. ‘Sur les coupures irréductibles du plan’. From a reference in
Kvern, Olav, M. ‘Eschersketch – An Adventure in the World of Tessellations’. Desktop Science. Adobe magazine 43-46 Winter 1998-1999. (10 September 2007) Tessellation tutorial.
Lagae, Ares and Philip Dutré. ‘Tile Packing Problems’. 2006 Broadly, edge tile colouring, Wang tiles, only of peripheral interest.
Lalvani, Haresh.
‘Structures on hyper-structures’. One of four references in
————. ‘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 Escher prints Ascending and Descending p. 35, Waterfall p. 37
Langford, C.
Dudley. ‘Correspondence’. Drawing
readers attention to MacMahon’s Cairo
tiling picture in
————. ‘Note 1464 Uses
of a Geometrical Puzzle’. From a reference in Golomb. On Rep-tiles.
————. ‘Note 2793. A conundrum for Form VI’. From a reference in Golomb.
————. ‘Note 2864. A Chess-board Puzzle’. From a reference in Golomb.
————. ‘To pentasect a pentagon’. From
a reference in
————. ‘Tiling Patterns for Regular Polygons’. From
a reference in
————. ‘On dissecting the dodecahedron’. From
a reference in
————. ‘Polygon dissections’. From
a reference in
————. 1538. ‘Tangrams and incommensurables’
————. ‘Super Magic Squares’.
————. 3133. ‘Some teaching
points’.
————. ‘Some missing figure
problems and coded sums’.
Also see obituary by E.A. Maxwell.
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. Landwehr, Klaus. ‘Visual Discrimination of the 17 Plane
Symmetry Groups’. 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’. Re golden section. From a reference in Livio.
Laves,
F. ‘Ebenenteilung in Wirkungsbereiche’. 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 ‘Ebenenteilung in Wirkungsbereiche’ = Level division in impact areas.
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
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’.
————. ‘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. Léger, Jean-François. ‘M.C.
Escher at the Museum: An Educator’s Perspective’. In Doris Schattschneider and
Michele Emmer, Eds. 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) Lenstra, Hendrik
and Gerard van der Geer. ‘The On Lenstra, also see interview with Jeannes Daems.
Levine, Dov and Paul Joseph Steinhardt. ‘Quasicrystals: A
New Class of Ordered Structures’. Quoted by Grünbaum in
Levy, Silvio. ‘Automatic
Generation of Hyperbolic Tilings’. 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’. From
a reference in
————. ‘Dissecting the Decagon’. From
a reference in
Liu, Yang and Godfried
Toussaint. ‘Unravelling Roman mosaic meander patterns: a simple algorithm for
their generation’. 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’. Of general interest.
Liversidge, Anthony. Interview with Roger Penrose. Minor references to Penrose tiles in an article/interview mostly about cosmological matters.
Lloyd, D. R. ‘How old are the Platonic Solids?’
Loeb, Arthur. L. ‘The Architecture of Crystals’. In 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
Locher, G. W. ‘Structural Sensation’. In
Locher. P and C. Nodine. ‘The Perceptual Value of Symmetry’.
In 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’. (26 March 2013)
Loe, Brian J. ‘Penrose Tiling in Northfield, Minnesota’.
(Mathematical Tourist)
Loeb, A. L. ‘Structure and patterns in science and art’.
————. ‘On My Meetings and Correspondence between 1960 and
1971 with the Graphic Artist M.C. Escher’. 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’. Interesting pentagon and skew pentagon tiling discussion, pp. 67-69
————. ‘Symmetry in Court and Country Dance’. General interest.
————. ‘Some Personal Recollection of M.C. Escher’.
————. ‘The Magic of the Pentangle: Dynamic Symmetry From
Merlin to Penrose’. Obscure.
Lord, Nick. ‘Constructing the 15
Lord, Eric A. and S. Ranganathan. ‘Truchet Tilings and their
Generalisations’. Note that Lord, a
metallurgist, based in India, has published widely (sometimes with others,
namely S. Ranganathan), of both popular (in the Indian Mention of Cyril Stanley Smith’s part in bring to prominence Truchet tiles.
Lord, Eric A. ‘Quasicrystals and Penrose patterns’. 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. Semi academic/popular.
Lowman, E. A. ‘Some Striking Proportions in the Music of
Bela Bartók’. Re golden section. From a reference in Livio.
Lu, Peter J. and Paul J. Steinhardt. ‘Decagonal and
Quasi-Crystalline Tilings in Medievel Islamic Architiecture’.
Lück, Reinhard. ‘Dürer–Kepler–Penrose, the development
of pentagon tilings’. This quotes Escher’s
‘monstrum’, albeit the attribution is not categorically stated. However, in the
bibliography, Kepler’s Note that the book
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.
Maass. John. ‘The Stately Mansions of the Imagination’. In 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’.
————. ‘The Dissection of Rectilinear Figures’.
————. ‘The Dissection of Rectilinear Figures concluded’.
————. ‘The Dissection of Rectilinear Figures’. 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)’.
————. ‘The Dissection of Rectilinear Figures (continued)
Macbeath, A. M. ‘The classification of non-Euclidean plane crystallographic groups’.
From a reference in
MacGillavry, Caroline H. ‘The Symmetry of M. C. Escher’s ‘Impossible’
Images’. Popular account of Escher's prints as regards symmetry.
Macmillan, R. H. ‘Pyramids and Pavements: some thoughts from
Cairo’. Highly significant as regards the Cairo tiling discussion, in depth, the fourth discussion (1979), after Dunn (1971), Gardner (1975) and Schattschneider (1978), and the second in situ account. Also of note is the collinearity aspect, first discussed.
Mackay, Alan L [15] ‘Extensions of space group theory’ As regards Mackay and his articles, most of these are of an academic title, and so are mostly hard, if not impossible to obtain. However, as of 6 September 2016, I found a website making a substantial proportion of his many papers and others available, as a pdf. These are described as: (i) scientific publications, (ii) miscellaneous publications, (iii) Anecdotal Evidence" columns from "The Sciences" , (iv)indirect material, (v) book reviews, (vi) unpublished papers etc http://met.iisc.ernet.in/~lord/webfiles/Alan/Alanpapers.html However, despite being warmly welcomed, these are mostly of limited use and interest, as to be expected given the source, as these are highly academic, way beyond my understanding, not to mention of diverging specialties. However, on occasion, there are indeed crossovers with tiling, and of which some, with diagrams, are broadly followable; for instance, Penrose and pentagon tilings feature strongly. To add all these available listing here, of about 250 available papers seems somewhat over officious, given the lack of need. Therefore, I here thus list just those of interest, and where quoted in tiling matters, such as with Grünbaum. Incidentally, Grünbaum lists seven articles of his, all of which I now have. Also see his review of
————. [42] ‘The structure of structure: some problems in solid state chemistry’,
————. ‘Crystal Symmetry’.
————. ‘Bending the Rules. Crystallography, Art and Design’ (lecture 1997/98) (2010) Also see his review of
————. ‘De Nive Quinquangula: on the pentagonal snowflake’. [in Russian] Kristallografiya, pp. 909-918. English version: Soviet Physics–Crystallography 26 (1981) 517-522 (31 January 2011) Mentioned in Grünbaum bibliography.
————. ‘But What is Symmetry?’ From the symmetry ‘special edition’ of the journal. Obscure.
‘Generalised Crystallography’. From the symmetry ‘special edition’ of the journal. Obscure.
MacMahon P. A. ‘On
Play “a outrance”’. From a Garcia reference in
————. ‘On the Thirty Cubes that can be constructed with Six
differently Coloured Squares’. From a Garcia reference in
————. ‘The design
of repeating patterns for decorative work’. From a Garcia reference in As is typical of MacMahon, his favoured method of showing tessellations is of a single diagram, notably pp. 576-577, rather than, as he puts it, ‘assemblages’. p.577 also has tiles derived from the Cairo tilings.
MacMahon, P. A. and
W. P. D. MacMahon. ‘The Design of Repeating Patterns’. Part I. N.B. There is no
Part II) From a Garcia reference in
MacMahon, W. P. D. ‘The
theory of closed repeating polygons in Euclidean space of two dimensions’. From a Garcia reference in This is noteworthy on account that although a tiling paper, no tilings are actually shown! Instead, single diagrams are shown, with the presumption of a tiling. MacMahon refers to ‘contact system of assemblages’, of which this is presumed to tessellate. the text in general is too difficult for me to follow.
Note
that I have many other MacMahon papers from Garcia’s bibliography, but these
are of an academic nature of no practical use, and so rather than a pointless
listing are not shown here. He
particularly favoured
‘MacMahon mentions’ in:
Macnab, Maggie. ‘Decoding Design. Understanding and Using Symbols In Visual Communication’ (1 June 2011) On logos, with a leaning towards symmetry.
Mackinnon, Nick.
‘Some thoughts on polyomino tiles’. ‘Idiot-proof tiles’. See the Grünbaum follow-up on the concept.
Macknik, Stephen L. and Susana Martinez-Conde. ‘Sculpting
the Impossible: Solid Renditions of Visual Illusions’. Popular account; use is made of two of Escher’s prints, Waterfall and Belvedere.
————. ‘The Portrait of Fra Luca Pacioli’. The defining work on the well-known painting.
————. ‘Polyomino Tessellations: A Class project’. School children project.
Madachy, Joseph S.
‘Recreational Mathematics’. On polyominoes.
Majewski, Miroslaw and Jiwan Yang. ‘A Journey through Chinese Lattice Designs an introduction to Chinese’. Journal source not given (22 March 2012)
Makovicky, Emil.
‘Ornamental Brickwork. Theoretical and applied Symmetrology and Classification
of Patterns’. Of limited, but still general interest. Of note is an interesting tessellation, with many stackings, p. 962.
————. ‘Symmetrology of Art: Coloured and Generalised
Symmetries’.
————. ‘800-year-old pentagonal tiling from Maragha, Iran,
and the new varieties of aperiodic tiling it inspired’. 67-86. In István
Hargittai, On a (far fetched) premise of Penrose aperiodic tiling.
Makovicky, Emil and M. Mackovicky. Arabic Geometrical Patterns – a treasury for crystallographic teaching. t Jahrbook fur mineralogy Monatshefte, 2 58-68, 1977. WANTED From a reference in Abbas.
Malcolm, Paul S. ‘Braided Polyhedra’. Simple braiding. Maldacena, Juan. ‘The Illusion Illusion of Gravity. The force of gravity and one of the dimensions of space might be generated out of the peculiar interactions of particles and fields existing in a lower-dimensional realm’. Scientific American, November 2005, 56-63. Use of Escher’s Circle Limit IV in various ways, 59-61 to illustrate his premise. ‘Popular’ account from a renowned expert in the field of theoretical physics, without equations, albeit still ‘advanced’. Juan Martín Maldacena, born 1968 in Buenos Aires, Argentina is a theoretical physicist. Among his many discoveries, the most famous one is the most reliable realization of the holographic principle – namely the AdS/CFT correspondence, the conjecture about the equivalence of string theory on Anti-de Sitter (AdS) space, and a conformal field theory defined on the boundary of the AdS space. Malek, Samar and Chris Williams. ‘Structural Implications of using Cairo Tiling and Hexagons in Gridshells’. Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2013 BEYOND THE LIMITS OF MAN 23-27 September, Wroclaw University of Technology, Poland J. B. Obrębski and R. Tarczewski (eds.) 1-4 (25 April 2016, but seen earlier) Academic in tone. Cairo tiling in the context of ‘gridshells’. As such, this appears to have been an academic study rather than its implementation into actual physical models/structures.
Maletsky, Evan, M. Activities: ‘Designs with Tessellations’.
Note that the article featured in a ‘special edition’ on tessellations, specifically concerning three Escher-inspired tessellation articles; (i) Ernest R. Ranucci, ‘Master of Tessellations M.C. Escher;’ (ii) Joseph L. Teeters ‘How to draw tessellations of the Escher Type’, and (iii) Evan M. Maletsky ‘Activities: Designs with Tessellations’. The article is most brief indeed, of just four pages padded by large scale diagrams and shows a typical teacher attempt at the Escher-like aspect, with two tessellations showing no understanding of the matter, with little more than detail added to a tile. This also shows an analysis of Escher's flying fish tessellation. Admittedly, the article appears slanted to children, but it should have been better. This appears to be his only attempt at Escher-like tessellations, as upon a web search (October 2017), there is no other apparent works by him in this field. Maletsky at least speaks with authority, a maths (and physics) teacher, with a NCTA lifetime achievement award of 2009.
Mallinson, Philip R. ‘Geometry and its Applications. Tessellations’. 1-74. (2010) Seems to be excerpted from a book. Very pleasing, in that the basics are covered succinctly, and then it moves on to tilings by Rice, and others, and pentagons.
Mamedov, Kh. S. ‘Crystallographic Patterns’. From the symmetry ‘special edition’ of the journal. Largely obscure, but with a small tessellation section pp. 527-529, of which although in general the tessellations are not particularly good, an exception is that of his ‘Unity’, of what appears to be a jailer and prisoner theme, of which has much to recommend it.
Mann, Casey. ‘A Tile with Surround Number 2’. On coronas, something of which I am not particularly interested in. Voderberg spiral discussion.
————. ‘Heesch’s Tiling Problem’.
————.‘Hyperbolic Regular Polygons with Notched Edges’. Of an academic nature throughout; the tone of the paper is way beyond me. Of no practical use.
Mani, P. ‘Automorphismen von polyedrischen Graphen’.
March, L. and R. Matela. ‘The animals of architecture: some
census results on From a reference in
Marck, K. W. ‘Enkele Overeenkomsten tussen het werk van M.
C. Escher en de plastiche chirugie’ (in Dutch). An English abstract is given – ‘Some similarities between the work on M.C. Escher and plastic surgery’.
Marley, Gerald C. ‘Multiple Subdivisions of the Plane’. From a reference in
Markowsky, George. ‘Misconceptions about the Golden Ratio’. Also see for contrary opinion! http://www.goldennumber.net/un-secretariat-building-golden-ratio-architecture/
Martin, G. E. ‘Polytaxic Polygons’.
May, Kenneth O: 1965: ‘Origin of the Four Color Conjecture’. ————. 'Mathematics and Art'. Popular piece on the connection between maths and art. I am unaware as to the source of this reference; it's not in the three main books on Escher. Escher pp. 570-571, with Cube with Magic Ribbons, Day and Night and Circle Limit IV (Heaven and Hell) with a brief mention in the commentary. Mey, Jos De. ‘Painting After
M.C. Escher’. In Doris Schattschneider and Michele Emmer, Eds. Marcotte, James and Matthew
Salomone. ‘Loxodromic Spirals in M. C. Escher's Sphere Surface’. Maynard, Phillip M. ‘Isohedrally compatible tilings’. http://symmerty-us.com (16 April 2014) References to isohedral aspects, with many morphic tilings, albeit the text is largely beyond me. Includes Escher-like tilings of such variations.
McAlhany, Joe. ‘A Brief History of Dissected Maps, the
Earliest Jigsaw Puzzles’. McConnell, James V. ‘Confessions of a scientific humorist’. impact of science on society, Vol. XIX, No. 3 July-September 1969, 241-252. Of James McConnell interest, re Escher-flatworms, albeit there is nothing here on Escher, but rather of his (admirable) humour. McConnell’s piece was part of a seemingly special edition on humour and science, from Unesco. (May 2019)
————. Article Title Unknown. Worm Runner’s Digest Vol. XVI No. 2, December 1974, pages unknown. WANTED Of Escher reference, at least of the cover, of which after this there are many uncertainties here. I do not have the journal in my possession, and quite where I got this reference from is unclear; I may have found it independently, although I doubt it. Be all as it may, an article in The Unesco Courier of April 1976, shows the cover of the WRD, illustrated with Escher’s Flatworm print (a topic of recent (May 2019) interest). Quite what, if indeed there is an Escher related article here is unclear.
————. ‘Worm-Breeding With Tongue in Cheek or the confessions of a scientist hoisted by his own petard’. The Unesco Courier, April 1976, pp. 12-15, 32 As such, the Escher aspect here is only of illustrations; there is not any reference in the text. More exactly, this shows shows the cover of the WRD of 1974, illustrated with Escher’s Flatworm print (a topic of recent (May 2019) interest). The Courier piece is an interesting read in many ways. There is no Escher discussion as such in it, although the Flatworms print is shown on p. 13, with the premise on flatworms (a most interesting creature, I might add. I had no idea of the fascinating science on it). As an aside, I very much enjoy McConnell’s humour. McLean, Robin K. ‘Dungeons, dragons and dice’.
————. ‘Loops of Regular Polygons’. June-July 2000 500-510 (12 March 2013) Largely academic.
————. ‘The tiling conjecture for equiangular polygons’. Academic, of no practical use.
Merow, Katherine. ‘A Toast! To Type 15!’ Popular account of the Type 15 pentagon discovery by Mann et al. Gives a history, with a small section on the discovery itself.
Meurant, Robert C. ‘A New Order in Space - Platonic and Archimedian (sic] Polyhedra and Tilings’ (17 January 2017)
Mielke, Paul T. ‘A Tiling of the Plane with Triangles’. From a reference in
Mikkonen, Yrjö. ‘Ontology
intermingling with onticity and vice versa in M.C. Escher's Reptiles’ Discussion on Escher's Reptiles print, as regard ontology and onticity. Both popular and academic in tone.
Miller, William. ‘Pentagons and Golden Triangles’.
Miller, A. William. ‘Golden Triangles, Pentagons, and
Pentagrams’.
Millington,
W. ‘Polyominoes’. Friezes, poster design, the cube and Pentominoes
Moore, Calvin, C. ‘Mathematical Sciences Research Institute Berkeley, California’.
General interest.
Molnar, V. and F. Molnar. ‘Symmetry-Making and – Breaking in Visual Art’.
In From the symmetry ‘special edition’ of the journal. Obscure.
Mozes, Shahar. ‘Aperiodic tilings’. Largely of an academic nature throughout, with not a single diagram! Of no practical use.
Muirhead, R. F. ‘On superposition by the aid of Dissection’. Publication not known.109-112 How I came about this reference is forgotten. whatever, it is not of any great significance, and is academic in tenure, with not a single diagram!
Mueller, Conrad George and Mae Rudolph (consulting editors
René Dubos, Henry Margenau and C. P. Snow). Eight chapters, all full of
general interest. Escher’s
Muscat,
Jean-Paul, ‘Polygons & Stars’. LOGO type instructions/diagrams.
Myers, Joseph. ‘Tiling with Regular Star Polygons’. Publisher Unknown (24 February 2011) pp. 20-27. A part work apparently taken from a book, but which is not stated?
Naylor, Michael. ‘Nonperiodic tiling: the Irrational Numbers
of the Tiling World’. Popular. Nakamura, Makoto. ‘New
Expressions in Tessellating Art: Layered Three-Dimensional Tessellations’. In
Doris Schattschneider and Michele Emmer, Eds. Necefoğlu, Hacali. ‘Turkish Crystallographic Patterns: From Ancient to Present’ Includes Escher-like tessellations from Imameddin Amiraslan and Khudu S. Mamedov.
Nemerov, Howard. ‘The Miraculous Transformations of Maurits
Cornelis Escher’. Fairly lightweight treatment, no new insight gained. Shows seven prints (six from Mickelson Galleries, one from Roosevelt) Castrovalva, Day and Night, Reptiles, Three Worlds, Another World, Relativity, Three Spheres.
Nelson, David R. and Betrand
I. Halperin. ‘Pentagonal and
Icosahedral Order in Rapidly Cooled Metals’. Quasicrystal, academic
Neuhaeuser, Stefan, Fritz Mielert, Matthias Rippmann and
Werner Sobek. ‘Architectural and Structural Investigation of Complex Grid
Systems’. Features use of the Cairo tiling in ‘grid systems’, although mostly of the terminology used is beyond me. Cairo tiling references on p.7, and liberally illustrated throughout. The paper arose of the work with ILEK
Nicki, Richard M. et al. ‘Uncertainty and preference for
ambiguous figures, impossible figures and the drawings of M. C. Escher’.
Niman, John and Jane Norman. ‘Mathematics and Islamic Art’. American Math Monthly, Vol. 85, 489-490,1978 (18 February 2013) References of semi-regular tilings in Islamic art. Widely quoted. Discussion at a popular level, no drawings or pictures.
Niven, Ivan. ‘Convex Polygons that Cannot Tile the Plane’. Mostly academic, a few simple diagrams. The premise of this article is the old chestnut of convex polygon of seven or more sides cannot tile, widely quoted but never shown, and put succinctly Niven proves it.
Norcia, Megan A. ‘Puzzling Empire: Early Puzzles and Dissected Maps as Imperial Heuristics’.
Norgate, M. ‘Non-Convex Pentahedra’. Academic nature throughout.
Norgate, Martin. ‘Cutting Borders: Dissected Maps and the
Origins of the Jigsaw Puzzle’. Jigsaw reference.
O’Beirne, T. H. ‘Puzzles and Paradoxes’ (?):
————. ‘Puzzles and Paradoxes 44: Pentominoes and Hexiamonds’.
————. ‘Puzzles and Paradoxes 45: Some Hexiamond solutions:
and an introduction to a set of 25 remarkable points’.
————. ‘Puzzles and Paradoxes 50: Thirty-six triangles make
six hexiamonds make one triangle’.
————. ‘Puzzles and Paradoxes 51: Christmas Puzzles and
Paradoxes’.
————. ‘Puzzles and Paradoxes 55: Some tetrabolical difficulties’. New Scientist. (No. 270) 18 January 1962. 158-159. (27 March 1993)
O’Keefe, M. and B. G. Hyde. ‘Plane Nets in Crystal Chemistry’. Philosophical Transactions Royal Society London. Series A, 295 1980, 553-618 (8 March 2013) From a reference in Tilings and Patterns. A very nice paper indeed in a generalized sense, although of course, given the academic nature, much is beyond my understanding. Of particular note is two instances of the Cairo tiling, although not stated as such: P. 557, in relation to use in Mathematical Models by Cundy and Rollett and New Mathematical Pastimes by MacMahon. P. 567, a diagram, where O’Keefe and Hyde specifically name it after MacMahon, with ‘MacMahon’s net’. As such this paper seemingly marks the introduction of the term ‘MacMahon’s Net’ for the Cairo tiling, and was used again by them in their 1996 paper, but this time in addition with the Cairo association. However, this is very much an ‘unofficial’ description. Upon correspondence (2012) with him: I suspect I got ‘Cairo tiling’ from Martin Gardner who wrote several articles on pentagon tilings. He is very reliable. As to ‘MacMahon's net’, I got the MacMahon reference from Cundy & Rollet….We are mainly interested in tilings on account of the nets (graphs) they carry. Possibly, and plausibly, this by MacMahon, of 1921, was the earliest known representation, and so in a sense, it was indeed broadly justified, even though by 1980 the ‘Cairo tiling’ term was coming into popular use, although if so, it is now been left behind by my subsequent researches. Curiously, the term is used on the Cairo pentagonal tiling Wikipedia page. However, the page leaves much to be desired, including this designation. Toshikazu Sunada has also used this term. However, I do not like this at all; it seems a somewhat artificial, additional naming, and so is unnecessary. Better would simply to have credited MacMahon as the first known instance (at the time) but without naming it after him. Also see a later paper, of 1996. Biographical details: Michael O’Keefe (1942-) and Bruce Godfrey Hyde (1925-2014) are two prominent people in science, and leading lights in their fields, namely physics and solid state chemistry respectively. In this regard, both have an interest in tilings, of which the paper addresses. Although of an intended academic audience, the paper is still nonetheless has much popular interest. Aside from the Cairo tiling, there are some most interesting simple, but aesthetic tilings, deserving of study.
Oliver, June. ‘Symmetries and Tessellations’. How to Escher-guide, typical teacher lack of understanding of issues, own illustrations belying lack of knowledge. That said, a ‘spider and web’ tessellation does indeed show a little imagination, albeit of a novelty level.
Ollerton, Mike. ‘Dual Tessellations’. This seems to be a small innovation of Ollerton’s devising. Mostly duals are taken from the semi-regular tilings, but here he uses a dual of a right-angled triangle tessellation, with his own notation to describe this. Orosz, István. ‘The Mirrors of
the Master’. In Doris Schattschneider and Michele Emmer, Eds. Orton, Tony. ‘From Tessellations
to Fractals’. Fractals based on a equilateral triangle. This lead to an extensive study, but somewhat overblown on my part. Orton, Tony. ‘Half-Squares, Tessellations and Quilting: Variations on a Transformational Theme’. Mathematics in School Vol. 23, No. 1, Primary School Focus (January 1994), pp. 25-28 (23 September 2019, read online, JSTOR) Simply dividing a square with a geometrical line and then tiling in a variety of ways. Nothing of any note. It seems to be an article for the sake of it
————. ‘Tessellations in the Curriculum’. Lightweight in the extreme.
————. ‘Circle Tessellations’. Tessellations derived from various grids of circles. Master copies of grids are provided.
Orton, T. and S. M. Flower. ‘Analysis of an ancient tessellation’.
The ‘Hammerhead’ tessellation, as described by the authors, of interest. Bob Burn comments on this in…
Osborn, J. A. L. ‘Amphography: The Art of Figurative Tiling’.
A brief exemplar of Osborn’s philosophy, with his own term of ‘amphography’. Usual shortcoming as to pretentiousness. For instance, a claim is made for ‘ultra realistic’ as regards a sub species of bats, but no corroborating real-life picture is shown.
————. ‘Diminishing Opportunity
in Amphography’. In Broadly philosophical
speculations by Osborn on the reducing numbers of possible future life-like
tessellation. This is noteworthy for the ridiculous statement by Osborn
‘…Escher foreclosed forever the possibility of any subsequent artist employing
this user friendly geometry for the Amphographic depiction of any even remotely
related subject matter without incurring the epithet ‘imitator, or
‘derivative’, or even plagiarist’. Absurd. So every painter copies from caveman
art? Every mathematician copies from Euclid? Every artist is copying Escher? No,
no, no; people Also see his two patents’ ‘Variably Assemblable [sic] Figurative Tiles for Games, Puzzles, And For Covering Surfaces’ and Single-Shape Variably Assemblable [sic] Figurative Tiles for Games, Puzzles, And For Covering Surfaces’. And also a self-published 14 page booklet concerning his ‘The Bats and Lizards How-To-Play book’, a guideline to ‘his’ Bats and Lizards tiles
Osborne, Harold. ‘Symmetry as an Aesthetic Factor’. From the symmetry ‘special edition’ of the journal. Obscure.
Ostromoukhov, Victor ‘Mathematical Tools for Computer-Generated Ornamental Patterns’ In Somewhat advanced. Of limited practical value.
————. Ostromoukhov, Victor. ‘Multi-Color and Artistic
Dithering’. Of no interest beyond a Cairo tile reference that cannot be seen!
A sample image
produced using a threshold matrix inspired by the
Ostromoukhov, Victor and Roger D. Hersch. ‘Artistic
Screening’. On a premise of screens, or half toning, with use made of a variety of art works; Escher (Sky and Water I), Islamic design. A little technical in places, although obviously mighty clever.
Özdural, Alpay. ‘On Interlocking Similar or Corresponding Figures and Ornamental Patterns of Cubic Equations’. 191-211 (2010). No citation for this article. Mostly of geometric constructions rather than interlocking figures. The original Arabic manuscripts are shown at the end of the article. Like most of Özdural’s writings, not a particularly easy read.
————. ‘Mathematics and Arts: Connections between Theory and
Practice in the Medieval Islamic World’. Discusses Arabic geometers of yesteryear, notably Abu’l-Wafa. Of little practical use.
————. ‘The Use of Cubic Equations in Islamic Art and Architecture’. Source unknown (30 April 2012)
Özgan, Sibel Yasemin and Mine Ökar. ‘Playing by the Rules. Design reasoning in Escher’s creativity’ (25 August 2016) In N. Gu, S. Watanabe, H. Erhan, M. Hank Haeusler, W. Huang,
R. Sosa (eds.),
Palmer, Chris K. ‘Spiral Tilings with C-curves Using
Combinatorics to Augment Tradition’. In
Palmer, Kelvin. ‘Jumble-Fits, Cluster Puzzles and Alec
Zandimer Plerp’. Essentially an announcement of the forthcoming publication of his book, rather than a discussion of cluster puzzle in general.
Paranandi, Murali. ‘Making Ripples: Rethnking pedagogy in the
digital age’ Of note is my name checks as regards my grid filling method, as well as illustrations.
Pargeter, A. R. ‘Plaited Polyhedra’. As quoted in
Parker, John. ‘Tessellations’, Topics, Cairo-esque pentagon, p. 34. Parker states that this is a footnote to the article by Clemens of the same journal.
————. ‘Dissections’. Begins of a accessible level, then moves onto academic ground.
————. ‘The Ratchet’. Discusses the various and numerous ways in which a ‘ratchet’ tessellation as shown can be composed.
————. ‘The rhomboid and its parts’.
Parviainan, Robert. ‘Connectivity Properties of Archimedean
and Laves Lattices’. Quote: The lattice D (3 A fleeting mention of the Cairo tiling in the context of a study on Laves tilings.
Paulino, Glaucio H. Arun L. Gain. ‘Bridging art and engineering using Escher-based virtual elements’. Some fearsome mathematics to explain Escher’s ‘simple’ periodic tessellations!
Pasko, Galina, Alexander Pasko, Turlif Vilbrandt, Arnaldo
Luis Lixandrão, Filho and Jorge Vicente, Lopes da Silva. ‘Ascending in Space
Dimensions: Digital Crafting of M.C. Escher’s. Graphic Art’. Only of minor interest, essentially on 3D fabrication matters, the nuances of which are beyond me. Penrose, L. S. and R. Penrose. ‘Puzzles for Christmas’. New Scientist, Volume 4, Number 110, 1580-1581, 25 December 1958. (30 May 2019) Puzzle 2 is a variant of the Penrose staircase. Puzzle 6 is on seven simple (if not more involved), geometric tilings. Solutions are on p. 1597.
Penrose, L. S. and Penrose R. ‘Impossible Objects: A Special
Type of Visual Illusion’. (Reprinted in
Penrose, R. ‘On the Cohomology of Impossible Figures’. Largely academic. Impossible tribars, p. 12. N.B. On Penrose per se, also
see interview with
————. ‘Pentaplexity. A Class of Non-Periodic Tilings of the
Plane’. Penrose tiles, a popular account. The article is a reprint from the ‘Archimedeans’ of Cambridge University, which first appeared in Eureka No. 39 (1978), 16-22.
Perigal, Henry. ‘On Geometric Dissections and
Transformations’. From a reference in
————. ‘Geometrical Dissections and Transformations. No. II’.
From a reference in
Perisho, Clarence R. ‘Colored polyhedra: a permutation
problem’. From reference in Garcia.
Petersen, Mark A. ‘The Geometry of Piero della Francesca’. General and academic.
Peterson, Ivars. ‘The Fivefold Way for Crystals’. Crystallography inclined, a little obscure in places. (From a reference in Frederickson). Ivars Peterson (1948-) is an award-winning mathematics writer. He is Director of Publications for Journals and Communications at the Mathematical Association of America. He worked for 25 years as a columnist and online editor at ————. 'Tiling to Infinity' ————. ‘Shadows and Symmetries’. Subtitled ' Quasicrystal geometry brings a new dimension to art and design. ————. Clusters and Decagons. Subtitled 'picturing complex alloy structures as overlapping atomic structure'. Popular. ————. 'A Quasicrystal Construction Kit'. Subtitled 'New rules for constructing a quasicrystal'. Popular. ————. ‘The Honeycomb Conjecture’. Subtitled 'Proving mathematically that honey bee constructors are on the right track'. ————. ‘Pieces of a polyomino puzzle’. On Karl A. Dahlke.
Pickover, Clifford. ‘Mathematics and Beauty: A Sampling of
Spirals and Strange Spirals in Science, Nature and Art’. Typical Pickover. Somewhat advanced.
————. ‘How to Design Textures Using Recursive Composite
Functions’. Somewhat advanced computer graphic art, of an equation nature.
Pill, Steve. ‘Master Techniques MC Escher’. A piece in conjunction with the contemporary Dulwich exhibit, illustrated with the prints Bond of Union, Castrovalva, Relativity and Reptiles. Nothing new, with the title insinuating technique, which is nothing of the sort.
Platt, Charles. ‘Expressing the Abstract’. In Illustrated with nine artworks: Relativity (cover), Mobius Strip, Horseman, Pegasus, Reptiles, Dragon, Liberation, Three Worlds, High and Low. An intersecting aside (p. 46) is of the Reptile print ‘as an optical illusion in a colour supplement feature’. Does anyone know what this is referring to?
Pomerance, Carl. ‘On a Tiling Problem of R. B. Eggleton’.
Popkin, Gabriel. ‘The Hidden Pattern’. An article on cosmological matter, illustrated with Escher’s Print Gallery, albeit essentially no other mention of him in the text. First saw of the day of publication, of which I noticed a tilling pattern on the cover, and so hence investigated, otherwise the reference would not have come to my attention.
Post, K. A. ‘Regular Polygons with Rational Vertices’. (8 March 2013) From a reference in
Post, Diana and Munro Meyersburg. ‘Celebrating Rachel Carson (1907-1964) In Her Centennial Year’. Rachel Carson Council Inc. March 2007 (October 2015) Use of Escher’s Metamorphosis print throughout discussion. Preston, Geoff. ‘Escher’s Delight’.
Prete, Sandro Del. ‘Between
Illusion and Reality’. In Doris Schattschneider and Michele Emmer, Eds. Propp, James. ‘A Pedestrian Approach To a Method of Conway, or, A Tale of Two
Cities’. Polyominoes, begins at a popular level, then becomes academic.
Journal of European Studies
Quadling, D. A. ‘Quadrilateral Crazy Paving’. The Mathematical Gazette Vol. 53, No. 383 (February 1969), Note 182, pp. 54-55. (8 July 2019) Somewhat confusing titled, as it is not a crazy paving in the normal sense of the word! Rather, it is on a simple quadrilateral tessellation midpoint rule, in which Quadling (one of the four inspirational drivers behind the School Mathematics Project (SMP) in the 1960s and 70s), surprisingly seems unaware of (or am I missing something?).
Radin, Charles. ‘Symmetry of Tilings of the Plane’. Radin’s numerous papers are typically highly academic, way beyond me. As he pleasingly makes these available for download, I here only record only the more, in relative terms, accessible instances.
————. ‘The Pinwheel tilings of the plane’. Of an academic nature throughout; the tone of the paper is way beyond me. Of no practical use. Raedschelders, Peter. ‘Tilings
and Other Unusual Escher-Related Prints’. In Doris Schattschneider and Michele
Emmer, Eds. Ranucci, Ernest R. ‘His Designs Come From Math Books’. Nothing of much significance. Available from the Popular Science website, although not as a print-out.
————. ‘Tiny Treasury of Tessellations’. A simple article on tessellation per se, without Escher-like aspects. Of note is that this has his own version of a Cairo tiling, albeit of different sized pentagons. Also, he uses the term ‘par hexagon’, perhaps taken for one of his references, by Kasner and Newman.
————. ‘Space-Filling in Two Dimensions’. Somewhat advanced, of little direct interest. No Escher like tessellations.
————. ‘Master of Tessellations: M. C. Escher, 1898-1972’. A brief discussion on Escher’s
tessellations and prints: Note that this article is part of a ‘special edition’ on Escher-like tessellations, by Ranucci, Teeters, and Maletsky
————. ‘Cutting Candles’. General interest, from an idea
in
————. ‘Function follows form’. General interest.
————. ‘The World of Buckminster Fuller’. (16 February 2013) General interest.
————. ‘On Skewed Regular Polygons’. General interest. In Farrell.
————. ‘Fruitful Mathematics’. Sphere packing.
Note that Ranucci was a prolific author, of mostly school-orientated material that is generally understandable, and so I trawled the JSTOR archives for any such articles, although the distinction between usefulness and those of lesser interest is no easily demarked. I have many other articles by Ranucci of no special importance to me (being non-tessellation and polyhedra), and so for this reason they are not listed in any great detail here in the main listing, instead I simply list the titles: Jungle-Gym Geometry, Isosceles, Permutation Patterns, On the Occasional Incompatibility of Algebra and Geometry, Of Shoes-and Ships-And Sealing Wax-of Barber Poles and Things, Tantalizing ternary, applications.
Rawsthorne, Daniel A. ‘Tiling Complexity of small n-ominoes’.
Fairly popular level, of general interest.
Reyes,
Encarnación. Inmaculada Fernández Benito
Redondo Buitrago, Antonia and Encarnación Reyes. ‘The
Geometry of the Cordovan Polygons’, Cairo-like tiles p. 12, based on the ‘Cordovan proportion’. Also see p. 14. Also of note is that this paper mentions a par hexagon.
————. ‘The Cordovan Proportion: Geometry, Art and Paper
Folding’.
Reeve, J. E. and J. A. Tyrrell. ‘Maestro Puzzles’. Polyominoes, polyiamonds.
Reichert, Muchael and Franz Gahler. ‘Cluster model of decagonal tilings’. Source? (Date?) Relating to Gummelt’s covering rule. Of an academic nature, leaning towards physics, albeit broadly understandable, at least of the first few pages, whereupon it rapidly becomes abstruse, and is not of any practical use.
Reid, Michael. ‘Tiling with Similar Polyominoes’. Very accessible.
Reinhardt, Karl ‘Über die Zerlegung der hyperbolischen Ebene
in konvexe Polygone’ From a reference in
Renz, Peter L. ‘Martin Gardner and Scientific American: The Magazine, Columns and the Legacy’. pp. 1-4 Reminiscences of Martin
Gardner by Renz, an editor of
Rhoads, Glenn C. ‘Planar tilings by polyominoes, polyhexes,
and polyiamonds’. Of particular interest in regards of polyhexes, of pp.336-338. Rice, Marjorie. ‘Escher-Like
Patterns from Pentagonal Tilings’. In Doris Schattschneider and Michele Emmer,
Eds. Richardson, Bill. ‘A short tale on two small tiles’. Quadrilateral tiling leading to Penrose tiles.
Richardson,
Martin. ‘Mixed Media: Holography Within Art’. Three brief references to Escher re ‘Cubic Space Division’, no illustrations, nothing of any great importance.
Richmond,
C. A. ‘Repeating Designs in Surfaces of Negative Curvature’. From a reference in
Richmond,
H. W. 1282. ‘A topological puzzle’. Dudeney utilities reference, not in Frederickson
Some new regular compound tessellations. Largely academic. Of circle limit type diagrams. Nothing of direct interest.
Rigby, John F. ‘Napoleon, Escher, and Tessellations’. On Escher’s ‘pure tiling’ conjecture. Of limited interest. ————. ‘Escher, Napoleon, Fermat,
and the Nine-point Centre’. In Doris Schattschneider and Michele Emmer, Eds. ————. ‘Precise Colourings of Regular Triangular Tilings’.
————. ‘A Turkish interlacing pattern and the golden ratio.
Whirling dervishes and a geometry lecture in Konya’. Begins at a popular level, and then become progressively academic. Still much of interest though.
————. 79.51 ‘Tiling the plane with similar polygons of two
sizes’. Largely academic throughout, no practical use.
Richert, Michael and Franz Gähler. ‘Cluster Models of Decagonal Tilings’. (2010) Penrose-like material, somewhat advanced, but of some interesting diagrams 2003.
Roberts, David L. ‘Albert Harry Wheeler (1873-1950): A Case
Study in the Stratification of American Mathematical Activity’. On Harry Wheeler’s polyhedra. This appears to be the main reference to Wheeler. I believe he first came under my orbit from Frederickson, ‘Dissections….’ pp. 141,145 (biography), 236-235. Oddly no mention is made of dissection in the Roberts article.
Roberts, Siobhan. ‘A Reclusive Artist Meets Minds with a
World-Famous Geometer: George Odom and H. S. M. (Donald) Coxeter’. On George Odom’s polyhedra, and interaction with Coxeter. Odom himself is a most interesting character.
Roberts, Siobhan and Asia
Ivić Weiss. ‘Donald in Wonderland: The Many Faceted Life of H. S. M. Coxeter’.
Robinson, E. Arthur. ‘The Dynamical Properties of Penrose
Tiling’. Of an academic nature throughout, of no practical use.
Robinson, J. O. and J. A. Wilson. ‘The Impossible Colonnade
and Other Variations of a Well-Known Figure’. Although not strictly of a mathematical nature, included here as it is occasionally quoted in impossible object matters. Of note is that I must have purposeful sought this out; I photocopied it in Hull reference library, along with an article by James Fraser. A short article of just three pages, of a popular account, with discussions on the ‘three-stick clevis’. There is nothing (unsurprisingly) tessellation related.
Robinson, Raphael M. ‘Undecidability and Nonperiodicity for
Tilings of the Plane’. From a reference in
————. ‘Multiple Tilings on n-Dimensional Space by Unit
Cubes’. From a reference in
————. ‘Undecidable Tiling Problems in the Hyperbolic Plane’.
From a reference in
Robinson, Sara. ‘M.C Escher: More Mathematics Than Meets the
Eye’. Examination of ‘Print Gallery’ type effect, with H. Lenstra quoted.
Robinson, S. A. ‘Classifying triangles and quadrilaterals’. Of an academic nature throughout, of no practical use.
Rodler, Hieronymus. On perspective, first referenced on John Coulthart’s site.
Roelofs, Rinus. ‘Tegels kleuren’ (tile colours). In An article ostensibly on the Cairo tiling (within the ‘Escher special’ edition), although it begins with Escher’s periodic drawing 3 and a sketch, from Schattschneider, p.102! ————. ‘Not the Tiles, but the Joints: A little
Bridge Between M.C. Escher and Leonardo da Vinci’. In Doris Schattschneider and
Michele Emmer, Eds. Rogers, C. A. ‘The packing of equilateral spheres’.
Rollett, A. P. ‘A Pentagonal Tessellation’. Cairo-like diagram p. 209, but without the attribution, of
interest due to so early an instance. Also included are other references to its
sighting; a school in Germany (speculating, of Villeroy and Bosch?), and
Rollings, Robert Wheadon. ‘Polyhedra expressed through the
beauty of wood’. Of general interest re polyhedra.
Rose, Bruce I. and Robert D. Stafford. ‘An Elementary Course
in Mathematical Symmetry’. From a reference in
Rosen, J. ‘Symmetry at the Foundation of Science’. From a reference in Abbas.
Rosenbaum, Joseph. Problem E721? (From
Rosenqvist, I. T.
‘The Influence of Physico-Chemical Factors upon the Mechanical Properties of
Clays’. From a reference in Locher and Schattschneider. An academic article. Some
minor use of two of Escher prints, Available from: http://www.clays.org/journal/archive/volume%209/9-1-12.pdf
Roth, Richard L. ‘Color Symmetry and Group Theory’.
Rowe, David E. ‘Coxeter on People and Polytopes’. (In ‘Years
Ago’ column). Minor Escher reference p. 30.
————. ‘Herman Weyl, the Reluctant Revolutionary’. (In ‘Years
Ago’ column). General interest.
————. ‘Puzzles and Paradoxes and Their (Sometimes)
Profounder Implications’. In remembrance of Martin Gardner. Minor Martin Gardner reference, square to rectangle paradox.
————. ‘From Königsberg to Göttingen: A Sketch of Hilbert’s
Early Career’. (In ‘Years Ago’ column). General interest.
————. ‘On Projecting the Future and Assessing the Past – the
1946 Princeton Bicentennial Conference’ Limited interest
————. Euclidean Geometry and Physical Space.
————. Felix Klein, Adolf Hurwitz, and the Jewish Question in
German Academia December 2011) In ‘Years Ago’ column). General interest.
Rózca, Erzsébet. ‘Symmetry in Muslim Arts’. From the symmetry ‘special edition’ of the journal. Obscure.
Ruane, P.N. ‘The curious rectangles of Rollett and Rees’. (2010) Of limited interest.
Rush, Jean C. ‘On the Appeal of M. C. Escher’s Pictures’. An essay by an art teacher, in which she essentially speculates on the appeal of Escher’s works. Also see another letters between Rush and Michele arising from this, referenced below.
————. ‘On the Appeal of M. C. Escher’s Pictures (Continued)’.
Letters following Rush’s article above. Debating (among other matters) on who devised the impossible triangle; Escher or Penrose.
Sachse, Dieter. M.C. Escher. In Discussion of Escher and his
prints
Sadahiro, Yukio. ‘An exploratory method for analyzing a
spatial tessellation in relation to a set of other spatial tessellations’. Unlike other articles from
Sakkal, Mamoun. ‘Intersecting squares: applied geometry in
the architecture of Timurid Samarkind’. An extensive treatment on the subject. Although much of this is beyond my immediate interests, of note is that of one of Sakkal’s procedures, of double squares, p. 87, where he composes what is in effect a parquet deformation. In his notes, he mentions Makovicky, referring to Shipibo geometric designs Symmetry: Culture Sci 22 (2011) pp. 373-389. From his web page: Born in Damascus, Dr. Mamoun Sakkal is a native of Aleppo, Syria, who immigrated to the United States in 1978. He practices Arabic type design, graphic design, and calligraphy as principal and founder of Sakkal Design in Bothell, WA.
Sallows, Lee. ‘The Lost Theorem’. Magic squares.
————. ‘More on Self-Tiling Tile Sets’. On rep-tiles.
Sallows, Lee and Martin Gardner, Richard K. Guy, Donald
Knuth. ‘Serial Isogons of 90 Degrees’. Begins at a popular level, then turns academic
Samyn, Phillipe and partners. Hotex – Village de Toile. Lacs de l’Eau d’Heure (Belgiqiue) c. 28 May 2010 (18 December 2012) Cairo tiling architecture.
Sands, A. D and S. Swierczkowski. ‘Decomposition of the line
in isometric three-point sets’. From a reference in
Sarhangi, Reza. ‘The Sky Within: Mathematical Aesthetics of
Persian Dome Interiors’ pp.145-156. In
————. Interlocking Star Polygons in Persian Architecture:
The Special Case of the Decagram in Mosaic Designs’. Premise is of a historical account. Gives a convenient time-line of surviving historical documents on Islamic designs (pp. 348-350). Various constructions.
————. All Bridges articles, from 1998 to present day
The Geometry and Art of Tesselation ISAMA 2007 223- (9 April 2014) Cairo tiling p. 226, albeit in the context of pentagon tiling possibilities; no reference is made to the Cairo aspect.
Sauer, Robert. ‘Ebene gleicheckige Polygongitter’ From a reference in
Sawada, Daiyo. ‘Symmetry and Tessellations from Rotational
Transformations on Transparencies’.
Schattschneider, Doris. ‘Tiling the Plane with Congruent
Pentagons’. Of fundamental importance concerning tiling with pentagons, full of interest, and all largely accessible. ‘Cairo tiling’ as a term is mentioned, as an Archimedean dual, p. 30, with three references: to likely Gardner’s article (as Macmillan does not get a mention in the bibliography, but it could be Dunn), Coxeter’s cover, and Escher’s usage of the tiling.
————. ‘The Plane Symmetry Groups: Their Recognition and
Notation’. Largely of an academic nature, and the subject itself is of limited interest. Two uses of Escher's tessellations, p. 440. Tilings in the form of Chinese lattice designs, pp. 444-445.
————. ‘Will it Tile? Try the Conway Criterion!’ Of both academic and popular nature. Rightly or wrongly, it has had no practical application in my studies. Figure 6 is an obvious fish, of which I haven’t found the time to compose. Needs a re-read.
————. ‘In Black And White: How To Create Perfectly Colored
Symmetric Patterns’. As such, of limited interest as regards tessellation. This borders on the popular and academic, and in relation to tessellation per se is of little value.
————. ‘The Pólya-Escher Connection’. Contains a previously
unpublished page from Escher's sketchbook, which is of some significance, in
that it shows how Escher formed his Eagle motif (PD 17), by fusing two tiles. This
formation had previously escaped me. This is all the more galling, in that the
information was available from as far back as 1987 with this article, but it
took me
————. ‘Escher: A Mathematician In Spite Of Himself’. In: This largely features aspects arising from Escher’s notebooks of 1941-1942, in which Schattschneider examines his mathematics.
————. ‘The Fascination of Tiling’. Full of interest; various aspects; Escher, Rice, pentagons, Penrose, kites and darts, rep-tiles.
————. ‘Escher’s Metaphors’. Somewhat curious; the premise
here is unclear, and there is nothing that has not been discussed before in
————. ‘Math and Art in the Mountains’. Talking about the Banff Bridges Conference of 2005.
————. ‘The Mathematical Side of M. C. Escher’. Although full of interest,
this largely covers ground already discussed in
————. Afterword. On pentagon article. source, date not recorded (24 November 2009)
————. ‘Escher’s Combinatorial Patterns’. An examination of Escher’s ‘combinatory tile’ problem. Of very little interest in itself; it’s really an instance of personal study to the person (Escher) concerned. a related piece by George Escher Essentially an update of ‘Potato printing a game for winter Evenings M.C. Escher Art and Science 9i11’ as an addendum to his article of Escher by the same author.
————. ‘Mathematics and Art’. Math Awareness Month – April 2003. Web. (6 November 2007)
————. ‘The Mystery of the MAA Logo’. On the icosahedron logo of the journal .
————. ‘M.C. Escher and the Crystallographers’.
————. ‘Lessons in Duality and Symmetry from M. C. Escher’. Bridges Leeuwarden, 2008 1-8 (August 2008) Likely
one of the lesser articles of interest from Schattschneider; there is nothing
really new or insightful here. Of interest is a reference, pp. 7-8 to Dylan
Thomas and his ‘Coast Salish’ tessellation artwork, although quite why
Schattschneider chose to promote this is unclear; his work is undeserving, in
both extent and quality. However, they seem to have some kind of association or
friendship; Schattschneider comments again on his work in a 2011 paper by him.
‘Artist profile Dylan Thomas: Coast Salish artist’.
N.B. For other papers where Schattschneider is listed
other than the main author see : (1) Ding, Ren; Doris Schattschneider, Tudor
Zamfirescu. ‘Tiling the Pentagon’. Also see Review section Note that although
Schattschneider was one of two of the lead
authors of
————. ‘Marjorie Rice and the MAA
tiling’. In
Schenk, Robert. ‘Hexagonal Jigsaw Puzzle Pieces’. 1-9. April 2016 (April 2016) Self published.
Scher, Daniel. ‘Lifting the Curtain: The Evolution of the
Geometer’s Sketchpad’. The General interest of the development.
Scherer, Karl. ‘The
impossibility of a tesselation of the plane into equilateral triangles whose
sidelengths are mutually different, one of them being minimal’. From a reference in
Schott, G. D. ‘Engraved hexagons on an Ice Age ivory: a neurological perspective on an anthropological debate’. Journal Neurology Neurosurgery and Psychiatry. 2014 October; 85 Issue 10: pp. 1174-6. NOT SEEN, WANTED https://jnnp.bmj.com/content/85/10/1174 The link gives the first page of the article, behind a paywall. Of historical tiling interest. On the Eliseevichi, Russia, tusk artefact, with an old, c. 12,000-15,000 BC tiling of hexagons. Further, the outlet seems a little odd; why a medical journal? Also see Clare E. Caldwell’s editorial commentary in the same journal. Given that I have only seen the first page, I will refrain from comment. Schott is a doctor at The National Hospital for Neurology and Neurosurgery, London, UK. Schulte, Egon. ‘The Existence of
Non-tiles and Non-facets in Three Dimensions’. From a reference in Schuster, D. H. 'A New Ambiguous Figure: A Three-Stick Clevis'.
Schwarzenberger, R. L. E. ‘The 17 Plane Symmetry Groups’. From a reference in
————. ‘Colour Symmetry’. From a reference in
Seccaroni, Claudio and Marco Spesso. ‘Architecture,
Perspective and Scenography in the Graphic Work of M.C. Escher: From Vredeman
de Vries to Luca Ronconi’. In Doris Schattschneider and Michele Emmer, Eds.
Senechal, Marjorie. ‘Color Groups’. From a reference in
————. ‘Which Tetrahedra Fill
Space?’ Largely popular account; some academic.
————. ‘Coloring Symmetrical Objects Symmetrically’. From a reference in
————. ‘Geometry and Crystal Symmetry’. From the symmetry ‘special edition’ of the journal. Obscure.
————. ‘The Algebraic Escher’. In
Largely of group theory, academic, of little practical use.
————. ‘Tiling the Torus and Other Space Forms’. Academic.
————. ‘Orderly Dispositions in Space’. March 1988. A report on a workshop meeting. (13 December 2012)
————. ‘Symmetry Revisited’. Cairo diagram as in the context of the set of 11 Laves diagrams, p. 9; as such per se, inconsequential. Of note is the poor accuracy of the Cairo drawing - it appears to show an equilateral pentagon (or is at least intended), and not the Archimedean dual!
————. ‘Tilings, quasicrystals, and Hilbert’s 18 No Cairo tiles. Escher tiling E128 (ghosts) p. 9. Mostly of two rhomb tiling. Senechal, Marjorie. ‘The
Symmetry Mystique’. In Doris Schattschneider and Michele Emmer, Eds. ————. ‘Coxeter and Friends’.
————. Note
that this is reprinted in
————. ‘The Mysterious Mr. Ammann’. (Mathematical
Communities)
————. Martin Gardner tribute (1914-2010).
————. ‘What is… a Quasicrystal?’ Somewhat advanced. ———— ‘Parallel Worlds: Escher
and Mathematics, Revisited’. In Coxeter, et al, Eds. Senechal, Marjorie and Jean Taylor.
Senechal, Marjorie and R. V. Galiulan. ‘An Introduction to
the Theory of Figures: the Geometry of E. S. Federov’.
Sequin, Carlo, H. ‘Topological tori as abstract art’. Largely academic.
————. All papers available on the web from the Bridges archive, from 1998.
Serlio, Sebastiano. From a reference in Frederickson.
Sharp, John. ‘Dürer’s Melancholy Octahedron’. For general interest on Dürer.
————. ‘The Circular Tractrix and Trudix’. Advanced. For general interest.
————. ‘Have you seen this number?’ Fibonacci sequence. Academic.
————. ‘Pictures inspired by Theo van Doesburg’. 18-19
————. All papers from the Bridges archive, from 1998.
————. ‘Golden Section Spirals’. For general interest
————. ‘Beyond the Golden Section – the Golden tip of the iceberg’. Bridges 2000, 87-98 General interest.
————. ‘Fraudulent Dissection puzzles – a tour of the
mathematics of bamboozlement’.
————. ‘Sliceform Craters’. An exploration in equations’. For general interest.
————. ‘Parabolic Tiling’. For general interest.
————. ‘D-forms and developable surfaces’. Bridges Renaissance Banff, 2005 121-128
————. ‘Beyond Su Doku’. Cairo tiling on pp. 167-169, in the context of a ‘Cairo Su Doku’.
Shechtman, D. et al. ‘Metallic Phases with Long Range Orientational Order and No Translational Symmetry’. Physical Review Letters, Vol. 53, No. 20, 12 November 1984, pp. 1951-1954 (19 June 2018) Quoted by Grünbaum in
Shefrin, Jill. ‘Make it a Pleasure and Not a Task’: Educational Games for Children in Georgian England. In Princeton Catalogue, 251-275 (24 May 2017) Jigsaw puzzle interest.
Shephard, Geoffrey C. ‘Super and
Superb Colourings of Tilings’. In Largely academic, of little practical use. Profusely illustrated though. Escher fish p. 49, lizards, birds flying fish 51.
Shorter, S. A. ‘The Mathematical Theory of the Sateen
Arrangement’. Of what I would tem as ‘motif placement, nothing here of tessellation. Of limited interest, to put it mildly.
Sibson, R. ‘Note 1485. Comments on Note 1464’ (C. Dudley
Langford proposition).
Silva, Jorge Nuno. ‘On mathematical games’ [sic]. A survey of games throughout history that can be described as ‘mathematical’ up to and including the present day. Dice and astragal, Go, Mancala board, Tangram, Alquerque board, Chess, to name but few. Of general interest, at a accessible level.
Simpson, R. ‘Locally equiangular triangulations’. From a reference in
Singmaster, David. 82.42 ‘According to Cocker’ 302-303. General historical interest.
————. ‘Covering Deleted Chessboards With Dominoes’. Academic.
Situngkic, Hokky. ‘What is the relatedness of mathematics and art and why should we care?’ (2010) Escher p. 5.
Slocum, Jerry and Dieter Gebhardt. ‘Puzzles from Catel’s Cabinet and Bestelmeier’s Magazine 1785-1823’. English translation. (11 June 2014) General puzzle history; of historical significance, the first such catalogue of this type. Tiling negligible.
Smit, Bart de and H. W. Lenstra, Jr., ‘The Mathematical Structure
of Escher’s Print Gallery’,
Smith, Cyril Stanley. ‘The Shape of Things’. Subtitled. The
comparison of crystals, soap bubbles, crazed ash trays, insect wings, living
cells and other objects demonstrates the rigorous relationship between natural
forces and forms. A recent (October 2018) revival in Smith (as regards his interest in the Cairo tiling) has led to a more intensive study of his work. However, much of extensive and lengthy writings when available are mostly outside of my mainstream interest. This being so, I will not formally list these, contenting myself with a title only for the sake of having ‘seen and noted’: ‘On Material Structure and Human History’ Cairo tiling p.
————. ‘The Tiling Patterns of Sebastien Truchet and the
Topology of Structural Hierarchy’. Smith is considered as the
person who brought to prominence Truchet’s seemingly forgotten work. Note a
special feature of Smith in the Indian journal
Smith, Cyril Stanley. ‘Structure Substructure Superstructure’.
In Kepes, G. Pages 36-37 mentions his interest in pentagons.
Sobczyk, Andrew. ‘More progress to madness via Eight Blocks’.
(26 March 2013)
Socolar, Joshua E. S. ‘Hexagonal Parquet Tilings Somewhat advanced. N.B. this is not parquet deformation per se!
Socolar, Joshua E. S. and Joan M. Taylor. ‘An aperiodic hexagonal tile’. 1-21 (2010) Somewhat advanced, of a academic nature. Of next to no practical use.
Sohncke, Leonard. ‘Die
regelmässigen ebenen Punksysteme von umbegrenzter Ausdehung’. I’m not sure why I have this; there is not a single tiling diagram in the article! Perhaps it’s a ‘seen and noted’ reference, given Sohncke’s fame.
Somerville, Duncan M. Y. ‘Semi-regular Networks of the Plane
in Absolute Geometry’. From a reference in
Sprague, R. ‘Beispiel einer Zerlegung des Quadrats in lauter
verschiedene Quadrate’. From a reference in Spilhaus, A. ‘World Ocean Maps: The Proper Places to Interrupt’. Proceedings of the American Philosophical Society 127 1983, 50-60. (4 May 2020) From a reference in Tilings and Patterns, and p. 163. Popular account, but of no practical use, with tiling minimal, if at all! No reference is made to jigsaw puzzles, see his other paper. ————. ‘Plate tectonics in Geoforms and Jigsaws’. Proceedings of the American Philosophical Society 128 No. 3, 1984, 257-269. (4 May 2020) From a reference in Tilings and Patterns. Popular and thought provoking, albeit of no practical use, with tiling minimal, if at all! Lots of references to jigsaw puzzles as a concept re tectonics.
Spiro, Michel. ‘On the Golden Ratio’. 12 Debunks the usual nonsense about the golden ratio appearing in numerous works of art; also see Markowsky and Falbo of a like mind on the subject.
Stebbins, G. L. ‘Prospects for Spaceship Man’. Although non mathematical,
included here as it contains Escher references. Use of
Steggal, J. E. A. ‘On the Number of Patterns Which Can be
Derived from Certain Elements’. From a reference in
Stein, Sherman. ‘Tiling, Packing, and Covering by Clusters’.
Academic throughout. Bare minimum of diagrams; much theory, all of no practical use.
————. ‘Algebraic Tiling’. From a reference in
————. Cutting a Polygon into Triangles of Equal Area Mathematical Entertainments in Mathematical Intelligencer. In contrast to the above, largely of a popular level, although indeed academic at times.
Steinbach, Peter. ‘Golden Fields: A Case for the Heptagon’. Golden ratio in the heptagon; largely academic.
Steinhardt, Paul Joseph. ‘Quasicrystals’. Largely popular account.
Stengel, Carol Elizabeth. ‘A Look at Regular and Semiregular
Polyhedra’. Largely popular account as regards historical account, with Plato etc.
Steward, D. ‘Trisides’. This is worth another look; I seem to recall I couldn’t understand the premise back in 1991.
Stewart, Ian. ‘Rep-Tiling the
Plane’.
————. ‘The Art of Elegant Tiling’. Minor instance of Cairo tiling, page 97, as devised by Rosemary Grazebrook.
————. ‘Polyominoes of order 3 do not exist’. From a reference in Golomb.
————. ‘The Ultimate Jigsaw Puzzle’. ————. ‘Now you see it, now you don’t. What optical illusions tell us about our brains’. New Statesman, pp. 36-41, 20 December 2013-9 January 2014 (2020) Escher mention in passing p. 38, Relativity, p. 41.
Stillwell, John C. ‘The Tessellating Art of M.C. Escher’. ‘Function’ was a mathematics magazine published by Monash University mathematics department from 1977-2004. All 140 issues are available online. Somewhat of a curious article. This begins ‘simply’ by showing ‘absurd’ overlays of grids onto Escher tilings drawn as wireframes, before then moving onto decidedly advanced hyperbolic tilings!
Stock, Daniel L. and Brian A. Wichmann. ‘Odd Spiral
tilings’. Some parts are more academic than others. Sugihara, Kokichi. ‘Computer-aided generation of Escher-like Sky and Water
tiling patterns’. ————. ‘Spatial Realization of Escher’s Impossible World’.
Sugimoto, T. ‘Classification of Convex Pentagons That Can Generate Edge-to-Edge Monohedral Tilings of The Plane’ 2012? (25 May 2012)
Sugimoto, Teruhisa and Tohru Ogawa. ‘Tiling
Properties of tilings by Convex Pentagon’. Set of 14 Convex pentagons.
————. ‘Convex Pentagonal Tiling Problem
and Properties of Nodes in Pentagonal Tilings’, 452-455
————. (2000a) ‘Tiling Problem of Convex Pentagon’,
————.
‘Systematic Study of Convex Pentagonal Tilings’, I: Case of Convex Pentagons
with Four Equal-length Edges.
Sydler,
J.-P. ‘Sur les tétrahèdres équivalents á un cube’. From
a reference in
————. ‘Conditiones nécessaries et suffisantes pour
l’équivalence des polyhèdres de l’espace Euclidien á trios dimensions’. From
a reference in
Taalman, Laura and Eugénie Hunsiscker.
‘Simplicity is not Simple’. Loosely polyhedra architecture.
Tapson, Frank. ‘Cutting and
Sticking’. Maths Resources 21-24: Dissections, the originality of which is not at all clear; likely taken from Lindgren.
————. 71.25 ‘The magic hexagon: an historical note’. Of note is that it contains an interesting discussion of Dudeney correspondence, pp. 218-220.
————. ‘Filling in Space’. Tessellations, many of which I studied, and used for bird motifs, in 1991.
————. ‘Watch Your Mathematical Language!’ General interest.
Note that Tapson has a whole
host of articles published in the
Tarnai, Tibor, Zsolt Gaspar, and Lidia Szalait. ‘Pentagon
Packing Models for "All-Pentamer" Virus Structures’. Of interest.
Taylor, Steve and Matthew Taylor. ‘Does Alicante have the
longest urban geometric illusion in the world?’ On geometric pavements; very good indeed.
Taylor, H. M. ‘On some geometrical dissections’. From a reference in
Taylor, M. V. ‘The Roman Tessellated Pavement at Stonesfield, Oxon’. Oxoniesia Vol. VI 1941. (2010) Historical account, of limited interest.
Teeters, Joseph L. ‘How to Draw Tessellations of the Escher
Type’. Note that the article featured in a ‘special edition’ on tessellations, specifically concerning three Escher-inspired tessellation articles; (i) Ernest R. Ranucci, ‘Master of Tessellations M.C. Escher;’ (ii) Joseph L. Teeters ‘How to draw tessellations of the Escher Type’, and (iii) Evan M. Maletsky ‘Activities: Designs with Tessellations’. That by Maletsky is particularly excruciating. Note that this article was studied extensively of the day. In short, Teeters purports to set out one of his procedures for drawing Escher-like tessellations, with the subheading ‘a short, clear discussion showing how you or your students can create tessellation art’. If only! It’s not short, it’s not clear, and at the procedure is essentially unfathomable, or at least as given! The time I spent unravelling this unholy tangle was wholly disproportionate as to worth. As such, it is next to useless as a general procedure, as it applies to a specific tiling, of a staggered isosceles triangle, of which Teeters would better have presented as such without such convolutions. Teeters illustrates the article with three of his works, all oddly not captioned: Unicyclist, St. Bernard Dog and a Red Indian, of which only one (St Bernard) is referenced with a figure number, albeit discussed only in passing. Admittedly, all three are of interest as rare examples of such motifs. All three are of the same broad level of quality, neither particularly good nor bad. I believe that at the time of seeing they were one of the few non Escher-like instances available, and so assumed a position of greater status by default than they deserved. Termes, Richard A. ‘Hand with
Reflective Sphere to Six-Point Perspective Sphere’. In Doris Schattschneider
and Michele Emmer, Eds. Teuber, Marianne L. 'Sources of Ambiguity in the Prints of
Maurits C. Escher.' This article has generated considerable discussion, and of which in particular George Escher rebuts. Be that as it may, from Teuber’s premise, she quotes some psychology articles that supposedly Influenced Escher of which I now examine. She quotes, pp. 94 and 98:
and
Both assertions, as far as I am aware, are unproven.
Tennant, Raymond. ‘Islamic Constructions: The Geometry Needed by Craftsmen’. International Joint Conference of ISAMA, the International Society of the Arts, Mathematics, and Architecture, and BRIDGES, Mathematical Connections in Art Music, and Science, University of Granada, Spain, July, 2003 (5 November 2010)
Thiel, Anton. ‘M. C. Escher: Treppauf und Treppab’. (26 June 2011) There is much I am unsure of this article, in German, the references are a little unclear, hence the lack of bibliographic detail.
Thomas, B. G. and M. A. Hann. ‘Fundamental principles governing the patterning of polyhedra 2007’. IaSDR. (2010) The Cairo tessellation gets a mention (page 6), defined incorrectly as an equilateral pentagon.
————. ‘Patterned Polyhedra: Tiling the Platonic Solids’. In Bridges 2008. Again the Cairo tessellation is mentioned, with the same incorrect definition as above. This paper seems to be derived from the above.
Thomas, Dylan and
Doris Schattschneider. ‘Artist profile Dylan Thomas: Coast Salish artist’. Of interest mainly due to the Schattschneider collaboration, although this is limited to comments on the contents of the paper only. Quite why Schattschneider, with a primarily tiling and Escher interest chose to ‘promote’ this artist and this piece in particular is unclear; his work is undeserving, in both extent and quality. Although of a geometric nature, these are not inherently of tiling or Escher-like in the true sense. His one loosely defined tessellation artwork ‘Salmon Spirits’, is nothing special, despite a first mention in her 2008 Leeuwarden paper. Further, upon looking for him on the web, on a biography he lists the influences of three artists without even mentioning Escher!
Thomassen, Carsten. ‘Planarity
and Duality of Finite and Infinite Graphs’. From a reference in
Thompson, D’Arcy Wentworth. ‘On the Thirteen Semi-regular Solids of Archimedes, and on their development by the Transformation of certain Plane Configurations’. Proceedings of the Royal Society London Series A, 107 (1925) 181-188 (11 March 2013) From a reference in
Thro, E Broydic. ‘Distinguishing two classes of impossible
objects’. Numerous Escher references. Concave and Convex p. 738 and again p. 746. Tuning fork, Necker cubes.
Thompson, William. ‘On the Division of Space with Minimum
Partitional Area’.
Thurston, W. P. ‘Three Dimensional Manifolds, Kleinian
groups and Hyperbolic Geometry’. Reference in Grünbaum. Of an academic nature throughout; the tone of the paper is way beyond me. Of no practical use.
————. ‘Conway’s
Tiling Groups’. See fig. 5. 21, of three fused hexagons. Of an academic nature throughout; the tone of the paper is way beyond me. Of no practical use.
Tóth, Fejes, L. ‘What the Bees Know and What They Do Not
Know’. Both popular and academic in parts. Various aspects, noticeably on isoperimetric aspects.
————. ‘Symmetry Induced by Economy’. From the symmetry ‘special edition’ of the journal. Obscure.
————. ‘Tessellation of the Plane with Convex Polygons Having
a Constant Number of Neighbours’. From a reference in Has a Cairo tiling diagram on p. 274, with a possible later reference by Richard K. Guy to a sighting at the Taj Mahal! However, upon enquiring with Guy, he does not exactly recall this. Likely he was mistaken. Note that I have a whole host (15) of other papers by Toth, all of which are of an academic nature, of no practical use. Therefore, these are not listed here.
Trigg, Charles W. ‘What is Recreational Mathematics?’ General interest.
Trinajstić, Nenad. ‘The Magic of the Number Five’. Seemingly a conference paper, on aspects of symmetry concerning the number five. Of note is p. 234, of the Cairo tiling, of a relatively considered discussion with references, albeit with errors in fact, crediting Blackwell with Martin Gardner’s original line. Perhaps of most note is a reference to Lothar Collatz’s‘ use, of which I was unfamiliar with, and followed up. In general, a pleasing read. Has an extensive bibliography, of 151 references! Trinajstić, was a new name to me.
Trotter, Robert J. ‘Transcendental Meditation’. Although not a maths article
it is included here as it uses Escher's print
Tutte, W. T. ‘The dissection of equilateral triangles into
equilateral triangles’. From a reference in Grünbaum. Academic throughout, three figures only
————. ‘The Quest of a Perfect Square’. From a reference in Grünbaum. Academic throughout.
Tyler, Tom. ‘Benevolent Confraternity of Dissectologists’.
In On jigsaws. On the establishing of the BCD.
————. ‘1760s Jigsaw Puzzle Maps of Lady Charlotte Finch’. Analysis and speculation on the recent discovery of the Lady Charlotte Finch cabinet.
Usiskin, Zalman. ‘Enrichment Activities for Geometry’. Minor reference to convex pentagon problem; Marjorie Rice.
Vallete, G. and T. Zamfirscu. ‘Les
partages d’un polygone convexe en 4 polygones semblables au premier’. From a reference in Veldhuysen. Mark. ‘M.C. Escher
in Italy: The Trail Back’. In Coxeter, et al, Eds. Ver Sacrum. 1899. (4 November 2013) A collection of articles of the Art Nouveau period; quite where to categorise this is uncertain, as book or articles. A whole years’ worth of articles of 1899 is posted on-line, and although not in any way a mathematics book nonetheless contains Moser’s historic first true life-like tessellation, of fish, and of others of less importance, and so is of significance in that regard. On the off chance that there might be other instances from other contributors, I examined every page, but to no avail.
Vighi, Paola. ‘L’uso di mediatori artistici e informatici per l’insegnamento della Geometria’ Riv. Mat Univ. Parma 6 3 (2000) 183-197. (In Italian) (29 March 2016) An abstract in English which gives a meaning to this states: This work was inspired by a periodic drawing of M. C. Escher, based on a pentagonal tessellation. First we have reconstructed the basic pentagon by means of the CABRI software, then we have identified the basic grid and link from which the drawing derives. It is recognized that it falls under the C4 type in the 17-group classification of patterns design. This allows to treat the issue of geometry and its teaching, providing suggestions of pedagogical relevance. Finally we deal with the theme of the plane covering by pentagons and its fundamental points are illustrated. Ostensibly on pentagons. Has many instances of the Cairo tiling (although not stated as such) pp. 185-187, 193. Has an interesting pentagonal paving photo on p. 194.
Villiers, Michael D. ‘An Investigation of Some Properties of
the General Haag Polygon’. Contains a discussion of the Haag hexagon, of which I believe I have shown (upon looking at the paper in depth on the 13 March 2015) that Escher did indeed use this to at least E21 (Running Man), of which De Villiers himself, and John Rigby, who has done some work on this, left open-ended.
Vince, Andrew. ‘Replicating Tessellations’. Academic. Mentioned in Schattschneider’s bibliography, of no practical use.
————. ‘Rep-tiling Euclidean space’. Academic, of no practical use.
Vincent, Jill. ‘Shrine to University: Mathematics in the Constructed Environment’. 25-37. (2010) Penrose and pentagon paving tilings in situ in Australia. A picture this refers is on p. 35 , I believe, to a Rice type 13 pentagon.
Vincent, Jill and Claire Vincent. ‘Japanese temple
geometry’.
Vincent, Peter. ‘Tessellating into Algebra’. As the title suggest, largely algebra based; of no real interest. No diagrams.
Voderburg, H. ‘Zur Zerlegung der Umgebung eines ebenen Bereiches in kongruente’.
From a reference in
————. Zur Zerlegung der Ebene eines in kongruente’ Bereiche in Form einer Spirale.
From a reference in Vulihman, Valentin E.
‘Escher-Like Tessellations on Spherical Models’. In Doris Schattschneider and
Michele Emmer, Eds.
Waldman, Cye H. ‘Voderberg Deconstructed & Triangle Substitution Tiling’ 2014. No article (24 August 2015) Much of interest; spiral tilings. Both popular and academic.
Walker, Jearl. ‘What explains subjective-contour illusions,
those brightspots that are not really there?’
Walle, John Van de. ‘Concepts, Art, and Fun from simple
Tiling Patterns’. Children’s article.
Walter, Marion.
‘The day all the textbooks disappeared’.
Wang, David G. L. ‘Determining All Universal Tilers’. Of an academic nature throughout; the tone of the paper is way beyond me. Of no practical use.
Wang, Hao. ‘Proving Theorems by Pattern Recognition – II’.
From a reference in ————. ‘Games, Logic and Computers’. Scientific American, 231, No. 5, November 1965, 98-106. (4 May 2020. I recall seeing much earlier, c. late 1980s, but did not copy, judged not needed) From a reference in Tilings and Patterns. Popular, but advanced in concept, of no practical use. Of a ‘coloured domino’ tiling premise. A lot of Alan Turing references, way beyond me, and in general. ————. ‘Notes on a class of tiling problems’. From a reference in
Warner, Marina. ‘When People See My Drawings They Cannot Sleep, They Do Not Sleep’. In Escher interview by Marina
Walker, of 1968, with occasional use of Escher’s prints, in order of use,
Washburn, Dorothy K. ‘Pattern Symmetry and Coloured
Repetition in Cultural Contexts’. From the symmetry ‘special edition’ of the journal. Obscure.
Watson, R. ‘Semi-regular Tessellations’. From a reference in
Weaire, Denis. ‘C S Smith’s Development
of a Viewpoint’. Complex Ideas and their Demonstration in the 2D Soap Froth. One of a series of writings of the June 2006 dedicated issue paying tribute to C. S. Smith’s work. Of general interest regarding Smith and Weaire’s froth, but nothing specifically of tiling.
————. ‘A Philomorph looks at Foam’. A reference in the article above. Of mild interest. Smith discussion. Wegner, G. ‘Bewegungsstabile Packungen Konstanter Nachbarnzahl’. Studia Scientiarum Math Hungaricae 6, 1971, pp. 431–438 (24 May 2021). Translated: Movement-stable packs consistent number of neighbors From a reference in Tilings and Patterns. Academic, of no practical use; no ‘normal’ tiling diagrams as such. Wells, A. F. ‘The Geometrical Basis of Crystal Chemistry IX Some properties of Plane Nets’. Acta Chry v. B24, pp. 50-57. WANTED Welsh, T. ‘Designing and Coloring of Scotch Tweeds’, Posselt’s Textile Journal, October 12, 1912, pp. 90-91 (29 April 2019) Of houndstooth interest. Reference to Sir Walter Scott and black and white checked trousers, with reference is made to ‘shepherd’s plaid’, presumably equating to shepherd’s check. From Wikipedia: Emanuel Anthony Posselt (1858–1921) was an authority on Jacquard looms and weaving. His book on the Jacquard machine is considered to be a classic…. Posselt's Textile Journal that ran between 1907 and 1923. Wenninger, Magnus J. ‘Artistic
Tessellation Patterns the Spherical Surface’. No Escher-like tessellation used; the article is more on polyhedra rather than tessellations. Wheeler, G. E. ‘Cell face correlations and geometrical necessity’. American Journal of Botany, 45, 1958, 439-449 (4 May 2020) From a reference in Tilings and Patterns. Academic, of no practical use. ‘Biological tiling’.
Wheeler, Harvey. ‘The Politics of Ecology’. Use of Escher’s
Wilkie, H.C. ‘On non-Euclidean
crystallographic groups’. From a reference in
Williams, Anne D. ‘Perplexity
Puzzles’. Jigsaw puzzle interest. On Margaret Richardson’s ‘Perplexity’ Puzzles. Contains some new detail.
————. ‘Not Spilsbury, Not Finch, But Who? Jill Shefrin’s
Discoveries’. Jigsaw puzzle interest. On Shefrin’s investigations as to the originator of the jigsaw puzzle.
Anon. ‘The Perplexity Puzzle: The Fad of the Year’, brochure, circa 1908
Pauline Wixon Derick Library Dennis, Massachusetts “Emily Evans” Margaret Richardson”
Willcocks, T. H. ‘A note on some
perfect squared squares’. From a reference in
Wieting, Thomas. ‘Capturing Infinity’. Reed, 21-29 (March 2010) A layman’s guide to constructing hyperbolic tessellations using compass and straight edge.
Wilker, J. B. ‘Topologically Equivalent Two-Dimensional
Isometries’. From a reference in
————. ‘Open Disk Packings of a Disk’. From a reference in
Wilkie, Ken. ‘The Weird World of Escher the Impossible’.
Magazine of the Netherlands
Delightful! Popular essay on Escher by Wilkie with many interesting sub tales, much of which is new. Perhaps of most note concerns the background to the Mick Jagger-Escher correspondence.
Willcut, Bob. ‘Triangular Tiles for Your Patio?’ Simple tilings. The title is a little misleading, in that the implication is on ‘practical patio concerns’, but this is put in the context of a hypothetical situation.
Willson, John Scott. ‘Tessellated Designs in my Op Art
Paintings’. This must be the same Willson
of
Wood, Donald G. ‘Space Enclosure Systems’. Cairo tiling interest, from a
footnote in Wood makes a curious observation as regards tilings with equal length sides, with the Cairo tiling being one of five such instances; as such, I do not recall seeing this simple observation elsewhere. Much of his work here, and elsewhere in the book, is in regards to prisms.
Wood, Elizabeth A. ‘The 80 Diperiodic Groups in Three
Dimensions’. From a reference in Grünbaum. Wholly academic.
Woods, Jimmy C. ‘Let the Computer Draw the Tessellations
That You Design’. A little dated, due to the computer program of the day used. Only one tessellation is shown, of a geometric bird, of a reasonable standard.
Woods, H. J. ‘The Geometrical Basis of Pattern Design. Part
I: Point and Line Symmetry in Simple Figures and Borders’. From a reference in Grünbaum, and elsewhere. A four-part work: (i) Point and Line Symmetry in Simple Figures and Borders (ii) Nets and Sateens (iii) Geometrical Symmetry in Plane Patterns (iv) Counterchange Symmetry in Plane Patterns. Of no practical use. Of a crystallographic viewpoint. Of interest in the early usage of the term ‘counterchange’.
————.‘The Geometrical Basis of pattern Design Part IV: Counterchange
Symmetry in Plane Patterns’.
Wollny, Wolfgang. ‘Contributions to Hilbert’s Eighteenth
Problem’. From a reference in ## Wright, Aaron Sidney. ‘The origins of Penrose diagrams in Physics, Art, and the Psychology of perception’, 1958–62. Endeavour, Volume 37, Issue 3, September 2013, pp. 133-139. WANTEDWunderlich, Walter. ‘Starre, kippende, wackelige und
begwegliche Achtfläche’. From a reference in
Jie, Xu and Craig S. Kaplan. ‘Calligraphic Packing’.
In Although not strictly mathematics, included for the sake of general interest. All most clever, as is of all of Kaplan’s work. Broadly, taking words and fitting them into a connected object.
Yazar, Tuğral. ‘Revisting Parquet Deformations from a
computational perspective: A novel method for design and analysis’. Broadly, written from an architecture viewpoint. Tuğral, who I have corresponded with previously on parquet deformations, and has studied these extensively, and here presents his (first?) paper on them. However, I don’t quite know what to make of this, as the writing and explanations are somewhat technical. It is essentially a study of Huff student-inspired works and in particular ‘Trifoliate’ by Glen Paris. Other Huff-related works include ‘Crossover’, by Richard Long and ‘I at the Center’ by David Oleson. Further, his own students works are included as well. Extensive use is made of Grasshopper, much beyond my understanding, or indeed interest, as good as it may be in the right hands. Mentions of myself p. 254 and in the acknowledgements, p. 265. Gives a good bibliography, although many of the references are peripheral.
Yen, Jane and Carlo Séquin. ‘Escher Sphere Construction
Kit’. In Partly on free-forming Escher's lizards; a little technical in places. Yet another Beyer reference in the paper… Yen-Lin Chen, Ergun Akleman, Jianer Chen and Qing Xing. ‘Designing Biaxial Textile Weaving Patterns’ Texas A&M University, (19 February 2019)
Yoneyama, K. ‘Theory of continuous set of points’. From a reference in
Zahn,
Markus. ‘The Contributions of Arthur Robert von Hippel to Electrical Insulation
Research’. Upon rereading Cyndie Campbell’s book ‘M. C. Escher Letters to Canada, 1958-1972’, I noticed a reference to von Hippel, page 65, whose name I was unfamiliar with. Upon looking on the web for him, I found various papers, with this, containing the background to Escher’s ‘Man with Cuboid’ print, of which the background, with the von Hippel connection, was unknown to me. Page 798 titles this as ‘The Thinker’. For more on von Hippel, see the article by Frank N. von Hippel.
Zeeman, Christopher and Ian Stewart. ‘Mathematics for Young People:
The Royal Institution Masterclasses’. General interest.
Zeitler H. ‘Über Netze aus regularen Polygonen in der
hyperbolischen Geometrie’. From a reference in
Zitronenbaum, A. C. editor Queries. Reply to Query 23 913 (20 March 2013) Dudeney utility reference re its history.
Zongker, Douglas. ‘Creation of Overlay Tilings Through
Dualization of Regular Networks’. From a reference in Craig Kaplan’s thesis. As an aside, the first ISAMA conference. Somewhat advanced as to concept in places. The premise, although simple to state, is not fully understood. Use is made of the Laves tilings. Be that as it may, the resulting four tilings so produced are impressive. However, I have no plans to re-read or adopt the methodology.
Zucker, Andrew A. ‘Student Projects in Geometry’. Brief reference to Escher-like tessellation, inconsequential.
Zurstadt, Betty K. ‘Tessellations and the Art of M. C.
Escher’. Child-oriented tessellation guide. Very poor indeed, typical teacher lack of understanding. Also, poor presentation; she cannot even present a print of Escher’s in full! |