Academic

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

Includes MacMahon’s card.

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

Academic

Anonymous. ‘Life’s Rich Patterns and Art by Numbers’. Junior Education. January 1994 (January 1994)

Escher’s print page*

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

Pseudo Cairo tiling from India, picture 14 (2010) PDF

Anonymous. ‘The Yellow Book’ Vol. XI October 1896

From Andrew Crompton’s reference; only a single page is of interest!

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

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

Brief discussion of the Cairo tiling, page 30

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

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

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

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

Anonymous. H. S. M. Coxeter. Biography

Albright, Thomas. Commentary in Rolling Stone. 52. ‘Visuals’-Escher, page 40, February 21, 1970 (28 March 2011)

Appel, K and W. Haken. ‘The Four Color Proof Suffices’. The Mathematical Intelligencer Vol. 8, No. 1, 1986, 10-20 (9 December 2011) PDF

Academic nature throughout

Aljamali, Ahmad, M and Ebad Banissi. ‘Grid Method Classification of Islamic Geometric Patterns’. February 2003 (2010) PDF

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

Article on Quasicrystals, Penrose tiles.

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

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

Bagina, O. ‘Convex pentagons which tile the plane’. (4 June 2012)

Another largely theoretical paper, no diagrams

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

Academic

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

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

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

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

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

Discussion of polyhedra used in Escher’s prints

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

On Paul Thurán and Anthony Hill

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

Gathering for Gardner

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

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

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

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

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

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

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

Somewhat advanced, although there is the occasional diagram of interest

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

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

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

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

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

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!

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

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

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

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

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

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

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

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

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

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

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

Cairo diagram page 18, and much interesting discussion arising from this

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

General interest, academic

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

Not mentioned in any of Frederickson’s three books

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

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

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

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

(30 January 2012) PDF

‘Typical Coxeter’, too advanced.

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

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

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

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

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

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

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

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

Potted account of tile history.

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

Polyhedral lampshade

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

Curl, Robert F and Smalley, Richard E. Fullerenenes. These cagelike molecules constitute the third form of pure carbon (the other two are diamond and graphite). C60, the archetype, is the roundest molecule that can possibly exist. Scientific American October 1991. 32-41.

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

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

Danzer, Ludwig, Branko Grünbaum, G. C. Shephard. ‘Does Every Type of Polyhedron Tile Three-Space?’ Structural Topology 8, 3-11, 1983

Of no real interest

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

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

Composing Islamic patterns by means of lattices.

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

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’, page 554.

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

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

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

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

Academic

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

Academic

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

Of the utmost significance in regards to the Cairo tiling; the first reference to the pentagon tiling being associated with Cairo, and with a illustration, but not photograph

Some additional correspondence generated by the above article in The Mathematical Gazette, Letters by P. Nsanda Eba Vol. 56, No. 398 (December 1972), 332-335 and M. M. Risueno Vo. 56, No. 398 (December 1972) 332 PDF

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

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

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

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

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

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

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

Contains Kepler’s tilings, pages 126-128

Findeli, Alain. ‘Rhythm, Symmetry and Ornament’. Structural Topology 12, 37-56,1986

Symmetry groups illustrated with real like objects. Of very little use. Only obtained because I could…

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

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

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

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

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

Not really mathematics per se

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

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

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

Somewhat advanced

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

————. ‘The curious magic of anamorphic art’. Scientific American. January 1975. 110-116.

————. ‘On the remarkable Csaszar polyhedron and its applications in problem solving’. Scientific American May 1975. 102-105.

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

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

————. ‘Games of strategy for two players: star nim, meander and rex’. Scientific American June 1975. 106-111.

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

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

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

The most popular account of the Penrose tiles.

————. ‘The inspired geometrical symmetries of Scott Kim’. Scientific American. June 1981. 14-17.

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

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

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

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

Also see Reuben Hersch for a rebuttal of this

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

Gibbs, William. ‘Paper Patterns 3 With Circles’. Mathematics in School. September 1990 2-8

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

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

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

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

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

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

Not really mathematical, but of considerable interest nonetheless

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)

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

————. ‘The Emperor’s New Clothes: Full Regalia, G String, or Nothing?’ Mathematical Intelligencer Vol. 6, No.4, 1984 (24 November 2011) PDF

Accessible. Interestingly, has a type 13 Rice pentagon, of which I’ve been studying lately, but no connection is made with the Cairo tiling!

————. ‘Geometry Strikes Again’. Mathematics Magazine Vol. 58, No. 1 January 1985. 12-17. (18 September 1989)

As kindly supplied by Grünbaum, 18 September 1989

————. ‘Hypersymmetric Tiles’. Congressus Numerantium 50 (1985). 17-24. (18 September 1989)

As kindly supplied by Grünbaum following correspondence, 18 September 1989

————. Art and Science. ‘Mathematical Challenges in Escher's Geometry’, page (Supplement to article).19 September 1989.

Contains Koloman Moser’s tilings, MacMahon, Pólya, Delone, Heesch

————. ‘The Bilinski Dodecahedron, and Assorted Parallelohedra, Zonohedra, Monohedra, Isozonohedra, and Otherhedra’. Mathematical Intelligencer Vol 32, Number 4, 2010 (24 November 2011) PDF

Leaning heavily towards the academic. Polyhedra, no tiling as such

————. ‘A Relative of Napoleon’s Theorem’. Geombinatorics 10 (2001) 116-121(30 January 2012) PDF

Advanced.

————. ‘A star shaped polyhedron with no net’. Geombinatorics 11 (2001) 43-48 (30 January 2012) PDF

Advanced.

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

Advanced.

————. ‘Tilings with Congruent Tiles’. Bulletin (new Series) of the American mathematical Society Volume 3, Number 3, November 1980 (2010). PDF

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, pages 956-957

Grünbaum, Branko, and G. C. Shephard. ‘The Ninety-One types of Isogonal Tilings in the Plane’. Transactions of the American Mathematical Society. Volume 242, 335-353 August 1978. PDF (23 May 2011)

Academic. Mostly too technical for me, despite noticeable numerous diagrams.

————. ‘Isohedral tilings of the plane by polygons’. Comment. Math. Helvetici 53 (1978) 542-571 (30 May 2012) PDF

Largely of an academic nature, but broadly accessible on occasions. A profusion of diagrams, has a Cairo tiling, and skew variation, page 568, the context of which requires examination. Interesting discussion on ‘straightening’ procedure, p. 556, that requires examination.

————. ‘Isotoxal Tilings’. Pacific Journal of Mathematics. Vol. 76, No. 2. 1978 407-430 (20 September 2012)

Largely academic throughout. Many ‘simple’ diagrams; however, all of this no practical use, being academic

————. ‘Is there an all-purpose tile?’ Amer. Math. Monthly 93 545-551 (1986) (18 September 1989)

As kindly supplied by Grünbaum, 18 September 1989. The premise is of tiling as according to each of the 17 wallpaper groups, using a triangular tile. Largely of academic interest only; certainly, it has not impacted on my studies.

————. Regular polyhedra – old and new. Aequationes mathematicae. 16 (1977) 1-20 (28 May (2012) PDF

Academic

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

Advanced.

————. ‘Tilings by Regular Polygons’. Math Magazine 50 (1977), 227-247 PDF

Broadly accessible, with reservation! A ‘follow up’, which I don’t have, is 51, (1978), 205-206)

————. ‘What Symmetry Groups are Present in the Alhambra?’ Notices of the AMS, Vol. 53, Number 6, 670-673, June/July 2006 (4 July 2011)

Popular account

————. ‘Unambiguous Polyhedral Graphs’. Journal unknown. 235-238.1963 (24 October 2012) PDF

Of a academic nature throughout. Of no practical use.

Grünbaum, B and Grünbaum, Z. ‘Symmetry in Moorish and other Ornaments’. Comp. & Maths. With Appls. Vol. 12B, Nos. 3/4. 641-653. (18 September 1989) PDF (30 April 2012)

As kindly supplied by Grünbaum, 18 September 1989. An examination of how many of the 17 wallpaper groups are present in the Alhambra, concluding that 13 are present, this being in contrast to the widely repeated claim that all 17 are to be found.

Gupta, Madhu S. ‘Escher’s Art, Smith Chart and Hyperbolic Geometry’. IEEE Microwave Magazine October 2006, pages 66-76 (11 October 2011) PDF

Haag, F. ‘Die regelmässigen Planteilungen’. Zeitschrift fur Kristallographie 49 (1911): 360-369.

(24 April 2012) PDF

Although this has what can be interpreted as ‘skewed Cairos’, there is not a standard Cairo tile here.

————. ‘Die regelmässigen Planteilungen und Punktsysteme.’ Zeitschrift fur Kristallographie 58 (1923): 478-488. (24 April 2012) PDF

This is the article Doris Schattschneider quoted me as a Cairo tiling, fig 13, page 487 in her tiling listserver response to my posting. However, after a translation was obtained, this is not so, Haag is referring to a quadrilateral tile, and not a pentagon; the pentagon ‘appears’ incidentally, upon a misinterpretation of the diagram.

————. ‘Die Planigone von Fedorow’ (Federov?) Zeitschrift fur Kristallographie 63 (1926): 179-186. (24 April 2012) PDF

Haak, Sheila. ‘Transformation Geometry and the artwork of M.C. Escher’. December 1976. 647-652. Mathematics Teacher (2010) PDF

Haake, A. ‘The Role of Symmetry in Javanese Batik Patterns’. Computers Math Applic. Vol. 17, No. 4-6, pp 815-826, 1989. PDF

Of limited interest.

Hankin, E. H. ‘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). PDF

Islamic style patterns, a ‘how it was done.’

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

Interview

————. ‘Lifelong Symmetry: A Conversation with H. S. M. Coxeter’. Mathematical Intelligencer Vol. 18 Number 4, 1996, 35-41 (28 November 2011) PDF

Interview with Coxeter. Minor Escher references, pages 36, 38-39

————. ‘Octagons Abound’. (Mathematical Tourist) Mathematical Intelligencer Vol. 17 Number 2, 1995, 52-54 (28 November 2011) PDF

Includes pavement

————. ‘Fullerene Geometry under the Lion's Paw’. (Mathematical Tourist) Mathematical Intelligencer Vol. 17 Number 3, 1995, 34-36 (28 November 2011) PDF

Chinese theme

————. ‘Quasicrystal Sculpture in Bad Ragaz’. (Mathematical Tourist) Mathematical Intelligencer Vol. 14 Number 3, 1992, 58-59 (14 December 2011) PDF

————. ‘Sacred Star Polyhedron’. (Mathematical Tourist) Mathematical Intelligencer Vol. 18 Number 3, 1996, 52-54 (14 December 2011) PDF

Hargittai, István and Magdolna Hargittai. ‘Symmetries of Opposites: Antisymmetry’ Mathematical Intelligencer Vol. 16 Number 2, 1994, 60-66 (28 November 2011) PDF

Occasional tessellation with Mamedov page 65, also in Symmetry book

Hargittai, Magdolna and István Hargittai. ‘Symmetry and Perception: Logos of Rotational Point-Groups Induce the Feeling of Motion’. Mathematical Intelligencer Vol. 19 Number 3, 1997, 55-58 (28 November 2011) PDF

Harrower, M. R. ‘Some Factors Determining Figure-Ground Articulation’. 407-424. (29 November 1993)

Hales, Thomas C. The Status of the Kepler Conjecture. The Mathematical Intelligencer Vol. 16, No. 3, 1994, 47-58 (15 December 2011) PDF

Academic

Heesch, H. Tiling the plane with congruent tiles. 115-117. 1935. Translation by John Berglund (2010)

Hemmings, Ray. Lobachevsky on a micro. Mathematics Teaching 111. June 1985. 23-27.

Hensley, Douglas. ‘Fibonacci Tiling and Hyperbolas’. Fibonacci Quarterly 16 (1978) 37-40

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

Henry, Richard. Pattern and Contemplation: Exploring the Geometric Art of Iran (2010) PDF

Public lecture given by Richard Henry at the Middle East Association on 27 April 2007. Published in the Journal of the Iran Society, September 2007.

Herda, Hans. ‘Tiling the Plane With Incongruent Regular Polygons’. Fibonacci Quarterly 19 (1981) 437-439

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

Hersh, Reuben. Reply to Martin Gardner. Opinions Column. Mathematical Intelligencer

An open letter to Martin Gardner (25 November 2011) PDF Vol. 23, No. 2, 2001 3-5

Hilton, Peter, and Jean Pedersen. ‘Comments on Grünbaum’s Article’. Mathematical Intelligencer Vol. 6, No.4 1984 (24 November 2011) PDF

Hirschhorn, M. D. and Hunt, D. C. ‘Equilateral Convex Pentagons Which Tile the Plane’. Journal of Combinatorial Theory, Series A. Vol. 39, No.1, May 1985. (22 May 1996)

Somewhat technical in places, but still of considerable interest in regard of Cairo-type tiling

Hirschhorn, M. D. ‘Tessellations with Convex Equilateral Pentagons’. Parabola 2-5, 18, 23, 36. Vol.13, No.1 February/March 1977 (22 May 1996)

Considerable Cairo-esque type pentagons

————. ‘More Tessellations with Convex Equilateral Pentagons’. Parabola 20-22, 36. Vol.13, No.2 May/June 1977 (22 May 1996)

Considerable Cairo-esque type pentagons

————. Parabola 20-22, 36. Vol.13, No.3 August/September 1977 (22 May 1996)

Limited to a single pentagon patch tiling, pages 14 and 17, by Hirschhorn junior!

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

William Huff’s student-inspired parquet deformations

Holden, Herbert L. ‘Fibonacci Tiles’. Fibonacci Quarterly 13 (1975), 45-49 (26 October 2012) PDF

Of an academic nature throughout, albeit , with the occasional diagram ‘understandable! Of no practical use. From a reference in Tilings and Patterns.

Huson, H. Daniel. ‘The Generation and Classification of Tile-k-Transitive Tilings Of The Euclidean Plane, the Sphere And The Hyperbolic Plane’. Geometriae Dedicata 47: 269-296, 1993. (9 September 2010). PDF

Highly technical, of limited interest

Jansen, René. ‘Polycairos in Disguise’. Newsletter Nederlandske Cube Club’, CFF 63, March 2004 (June 2011)

Of note is the Jansen has a Cairo tiling article in the form of Polycairos, and with a request for an in situ tiling picture.

Jaworski. ‘A Mathematician’s Guide to the Alhambra’. Second revised edition 2006 (25 October 2012) PDF

Jones, Christopher B. ‘Periodic tilings with vertices of species number 3’. Structural Topology 20, 1993

A whole host of ‘demi-regular’ tilings of squares and triangles

Juhel, Alain. ‘Prince of Samarqand Stars’. Mathematical Intelligencer (The Mathematical Tourist) Volume 29, Number 4, 44-50, 2007 (30 April 2012) PDF

On Ulugh-Beg

Kaplan, C. S. ‘Computer Generated Islamic Star Patterns’. In Bridges 2000 105-112

————. ‘Escherization’. In proceedings of Siggraph 2000

Most informative, with a considered approach to life-like tiling

————. ‘Islamic Star Patterns from Polygons in Contact’

Building upon Hankin’s ‘polygons in contact’ method

————. ‘The trouble with five’. December 2007.

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

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

————. ‘Patterns on Surfaces’

Various tilings applied to an arbitrary 3D model, here a rabbit. Includes Escher’s Shells and Starfish drawing, no. 42

Kaplan, C. S. and Salesin, David H. ‘Islamic Star Patterns in Absolute Geometry’. ACM Transactions on Graphics, Vol. 23, No. 2 April 2004, pages 97-119

Keeton, Greg. 'The Artist who Aims to Tease.' Readers Digest (March 1981): 37-41.

This is, I believe to the best of my dim and distant recollection (but still clear enough to plainly recall), my first encounter with Escher’s work, in c. 1983, but I didn’t do anything about it at the time. I inscribed on the front cover, likely in 1990 ‘saw first prob(bably) (19)83, rediscovered January (19)90’. Uses Escher’s prints: Hand with Reflecting Globe,37; Three Worlds, 38; Bond of Union, 38; Day and Night, 39, Belvedere, 40; Mobius Strip II, 41. Sent whole journal to Jeffrey Price, 16 April 2010

Kershner, R. B. ‘On Paving the Plane’. The American Mathematical Monthly. Vol. 75, No. 8 (October 1968) 839-844. PDF (12 December 2010)

Of significance. Gives eight of the convex pentagon types then known

Klarner, David A. ‘Some Results Concerning Polyominoes’. Fibonacci Quarterly 3 (1965), 9-20 (26 October 2012)

Of an largely academic nature throughout, with occasional diagram ‘understandable! That said, still of no practical use. From a reference in Tilings and Patterns.

Koizumi, Hiroshi and Kokichi Sugihara. ‘Maximum Eigenvalue Problem for Escherization’.

The authors’ own Escherization program. PDF (2010)

Krizic, Michal, Jakub Solz, Alena Solkova. ‘Is There a Crystal Lattice Possessing Five-Fold Symmetry?’ Notices of the AMS Vol. 59 No. 1 22-30

References to Kepler and Penrose

Kulpa, Zenon. ‘Are Impossible Figures Possible?’ Signal Processing 5 (1983) 201-220. (17 July 1993)

From a reference in The Eye Beguiled

Kvern, Olav, M. ‘Eschersketch – An Adventure in the World of Tessellations’. Desktop Science. Adobe magazine 43-46 Winter 1998. PDF

Tessellation tutorial

Le, San. ‘The Art of Space Filling in Penrose tilings and Fractals’. On-line article, pending print

The title is somewhat misleading, in that other, non Penrose tilings feature. Escher is prominently mentioned. Le makes uses of what I term as ‘placements’.

Lee, A. J. ‘Islamic Star Patterns’. Muqarnas 182-197 (2010) PDF

Loe, Brian J. ‘Penrose Tiling in Northfield, Minnesota’. (Mathematical Tourist) Mathematical Intelligencer Vol. 17 Number 2, 1995, 54 (28 November 2011) PDF

Loeb, A. L. ‘On my Meetings and Correspondence with M.C. Escher’. Leonardo, Vol. 15, No. 1 (Winter, 1982) pp. 23-27 (9 September 2010) PDF

Macaulay, W. H. ‘The Dissection of Rectilinear Figures’. Messenger of Mathematics, Volume 48, 1919a. 159-165

————. ‘The Dissection of Rectilinear Figures (continued)’ Messenger of Mathematics, Volume 49, 1919b. 111-121

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

Macmillan, R. H. ‘Pyramids and Pavements: some thoughts from Cairo’. Mathematical Gazette 1979 (10 August 2010) PDF

Highly significant as regards the Cairo tiling, Cairo pentagon discussion, in depth, the third reference (1979), after Dunn (1971) and Gardner (1975).

Mackay, Alan L. ‘Bending the Rules. Crystallography, Art and Design’ (lecture 1997/98) (2010) PDF.

Also see his review of Visions of Symmetry.

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

Mentioned in Grünbaum bibliography

Makovicky, E. ‘Ornamental Brickwork. Theoretical and applied symmetrology and classification of patterns’. Computers Math. Applic. Vol. 17, No. 4-6, pp 955-999, 1989. (30 April 2012) PDF

Of limited, but still general interest. Of note is an interesting tessellation, with many stackings, page 962

MacMahon, P. A. ‘The design of repeating patterns for decorative work’. Journal of the Royal Society Arts 70 (1922) Friday, June, 567-578. Related discussion ibid pp. 578-582 (3 May 2012) PDF

Of note is that (PA) MacMahon refers to a ‘haystack’, meaning a Cairo tile, page 573, after fig. 13. This term is also interestingly used by him in New Mathematical Pastimes. WPD also uses this word in ‘The theory of closed repeating polygons…’ . So who devised it?!

MacMahon, P. A. and W. P. D. MacMahon. ‘The Design of Repeating Patterns’. Part I N.B. There is no Part II) Proc. Royal Society London 101 (1922), 80-94 (30 April 2012) PDF

MacMahon, W. P. D. ‘The theory of closed repeating polygons in Euclidean space of two dimensions’. Proc. London Math Society (2) 23 (1925) 75-93 (30 April 2012) PDF

Of note is that (WPD) MacMahon refers to a ‘haystack’, meaning a Cairo tile, page 89, after fig. 6. As this term is also used in New Mathematical Pastimes, by PA, which of the MacMahons devised this is unclear

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

Maletsky, Evan, M. Activities: ‘Designs with Tessellations’. The Mathematics Teacher, April 1974, Volume 67, Number 4, 335-338 and continued 360. (Confusingly, inside the book this is also titled as ‘Mathematics Teacher’ (18 June 2011)

‘Special edition’ on tessellations, specifically concerning three Escher inspired tessellation articles

Mallinson, Philip R. ‘Geometry and its Applications. Tessellations’. 1-74. PDF (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.

Martin, G. E. ‘Polytaxic Polygons’. Structural Topology No.12 (1986) 5-10. (2 February 1998) PDF (Also see Fontaine, A; Martin, G. E. Polymorphic Polyominoes)

Moore, Calvin, C. ‘Mathematical Sciences Research Institute Berkeley, California’. The Mathematical Intelligencer Vol. 6, No. 1, 1984, 59-64 (9 December 2011) PDF

General interest

Mozes, Shahar. ‘Aperiodic tilings. Inventiones Mathematicae. 128, 603-611 (1997). (24 October 2012) PDF

Largely of a academic nature throughout, with not a single diagram! Of no practical use..

Necefoğlu, Hacali. ‘Turkish Crystallographic Patterns: From Ancient to Present’

Includes Escher-like tessellations from Imameddin Amiraslan and Khudu S. Mamedov

O’Beirne, T. H. ‘Puzzles and Paradoxes’ (?): New Scientist. (No.258) 26 October 1961. 261 (7 February 1998)

————. ‘Puzzles and Paradoxes 44: Pentominoes and Hexiamonds’. New Scientist (No.259) 2 November 1961. 316-317 (7 February 1998)

————. ‘Puzzles and Paradoxes 45: Some Hexiamond solutions: and an introduction to a set of 25 remarkable points’. New Scientist. (No.260) 9 November 1961. 379 (7 February 1998)

————. ‘Puzzles and Paradoxes 50: Thirty-six triangles make six hexiamonds make one triangle’. New Scientist (No.265) 14 December 1961. 706-707 (7 February 1998)

————. ‘Puzzles and Paradoxes 51: Christmas Puzzles and Paradoxes’. New Scientist (No.266) 21 December 1961. 752-753 (7 February 1998)

————. ‘Puzzles and Paradoxes 55: Some tetrabolical difficulties’. New Scientist. (No. 270) 18 January 1962. 158-159. (27 March 1993)

Orton, Tony. ‘From Tessellations to Fractals’. Mathematics in Schools. March 1991 30-31

Orton, T. and S. M. Flower. ‘Analysis of an ancient tessellation’. The Mathematical Gazette 297-298 (2010) PDF

The ‘Hammerhead’ tessellation as described by the authors

Osborn, J. A. L. ‘Amphography: The Art of Figurative Tiling’. Leonardo, Vol. 26 No. 4 pp 289-291, 1993 (9 September 2010) PDF

Ostromoukhov, Victor. ‘Multi-Color and Artistic Dithering’. Siggraph 1999. PDF (2010)

Of no interest beyond a Cairo tile reference that cannot be seen! A sample image produced using a threshold matrix inspired by the Cairo tessellation. Figures (a)-(e) show the building process of the threshold matrix’.

Özdural, Alpay. ‘On Interlocking Similar or Corresponding Figures and Ornamental Patterns of Cubic Equations’. 191-211 (2010). PDF

No citation for this article

————. ‘Mathematics and Arts: Connections between Theory and Practice in the Medieval Islamic World’. Historia Mathematica 27 (2000), 171-201. (30 April 2012) PDF

————. ‘The Use of Cubic Equations in Islamic Art and Architecture’. ? (30 April 2012) PDF

Pargeter, A. R. ‘Plaited Polyhedra’. The Mathematical Gazette, Vol. 43, No. 344 (May, 1959) 88-101

As quoted in Mathematical Models by H. Martyn Cundy and A. P. Rollett, page 152

Parker, John. ‘Tessellations’, Topics, Mathematics Teaching 70, 1975, page 34 (3 May 2012) PDF

Cairo-esque pentagon, page 34. Parker states that this is a footnote to the article by Clemens of the same journal

Penrose, L. S. and Penrose R. ‘Impossible Objects: A Special Type of Visual Illusion’. Brit. Journal of Psychology. Vol.49, No.1, (1958). 31-33

(Reprinted in The Eye Beguiled by Bruno Ernst, page 72-73).

Penrose, R. ‘On the Cohomology of Impossible Figures’. Structural Topology 11-16.

Impossible triangles

————. ‘Pentaplexity. A Class of Non-Periodic Tilings of the Plane’. Mathematical Intelligencer Vol ? Number ?, 1?, 4-11 (25 November 2011) PDF

Penrose tiles, a popular account. The article is a reprint from the Archimedeans’ of Cambridge University, which first appeared in Eureka No. 39.

Perigal, H. ‘On Geometric Dissections and Transformations’. Messenger of Mathematics, Volume 2, 1873. 103-106

————. ‘Geometrical Dissections and Transformations. No. II’. Messenger of Mathematics, Volume 4, 1875. 103-104

Petersen, Mark A. ‘The Geometry of Piero della Francesca’. The Mathematical Intelligencer. Vol. 19, No. 3, 1997, 33-40 (2 December 2011) PDF

General and academic

Ranucci, Ernest R. ‘Master of Tessellations: M.C. Escher, 1898-1972’. The Mathematics Teacher, April 1974, Volume 67, Number 4, 299-306. (Confusingly, inside the book this is also titled plainly as ‘Mathematics Teacher’ (18 June 2011)

‘Special edition’ on tessellations, specifically concerning three Escher inspired tessellation articles, by Ranucci, Teeters, and Maletsky

Redondo Buitrago, Antonia and Reyes Iglesias, Encarnación. ‘The Geometry of the Cordovan Polygons’, Visual Mathematics 2008b 10, 4. http://www.mi.sanu.ac.rs/vismath/redondo2009/cordovan.pdf

Cairo-like tiles page 12, based on the ‘Cordovan proportion’. Also see page 14

Rigby, J. F. ‘Napoleon, Escher, and Tessellations’. Structural Topology 19, 1991 17-23. Also appeared in Mathematics Magazine 64 (1991) 242-246

————. ‘Precise Colourings of Regular Triangular Tilings’. Mathematical Intelligencer Vol 20, Number 1, 1998, 4-11 (25 November 2011) PDF

Richert, Michael and Franz Gähler. ‘Cluster Models of Decagonal Tilings’. (2010)

Penrose-like material, somewhat advanced, but of some interesting diagrams 2003. PDF

Roberts, Siobhan, and Asia Ivić Weiss. ‘Donald in Wonderland: The Many Faceted Life of H. S. M. Coxeter’. Mathematical Intelligencer Vol. 26 Number 3, 2004 (25 November 2011) PDF

Robinson, J. O. and Wilson, J. A. ‘The Impossible Colonnade and Other Variations of a Well-Known Figure’. Brit. Journal of Psychology 64, 3 (1973) 363-365. (10 August 1993)

Robinson, Raphael M. ‘Undecidability and Nonperiodicity for Tilings of the Plane’. Inventiones Mathematicae. 12, 177-209 (1971). (24 October 2012 PDF

Of a academic nature, of which some parts are broadly readable, but of no practical use. The premise is of the Hao Wang conjecture. From a reference in Tilings and Patterns.

————. ‘Multiple Tilings on n-Dimensional Space by Unit Cubes’. Mathematische Zeitschrift 166, 225-264 (1979) (24 October 2012) PDF

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

————. ‘Undecidable Tiling Problems in the Hyperbolic Plane. Inventiones Mathematicae. 44, 259-264 (1978). (24 October 2012) PDF

Largely of a academic nature throughout, with not a single diagram! Minor recreational references to Gardner, Penrose and Ammann, but of no practical use. From a reference in Tilings and Patterns.

Robinson, Sara M.C Escher: ‘More Mathematics Than Meets the Eye’. SIAM news, Volume 35, Number 8, 1-4, 2002 (11 July 2011)

Examination of ‘Print Gallery’ type effect, with H. Lenstra quoted

Rollett, A. P. ‘A Pentagonal Tessellation’. The Mathematical Gazette, Vol 39, No. 329 (Sep. 1955) page 209, note 2530 (2 April 2012).

Cairo-like diagram page 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, and The Listener

Rowe, David E. ‘Coxeter on People and Polytopes’. (In ‘Years Ago’ column). Mathematical Intelligencer Vol. 26, Number 3, 2004, 26-30 (22 December 2011) PDF

Minor Escher reference page 30

————. ‘Herman Weyl, the Reluctant Revolutionary’. (In ‘Years Ago’ column). Mathematical Intelligencer Vol. 25, Number 1, 2003, 61-70 (22 December 2011) PDF

General interest

————. ‘Puzzles and Paradoxes and Their (Sometimes) Profounder Implications’. Mathematical Intelligencer Vol. 33, Number 1, 2011, 55-? (In ‘Years Ago’ column). (29 December 2011) PDF

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). Mathematical Intelligencer Vol. 25, Number 2, 2003, 61-70 (22 December 2011) PDF

General interest

————. ‘On Projecting the Future and Assessing the Past – the 1946 Princeton Bicentennial Conference’ Mathematical Intelligencer Vol. 25, Number 4, 2003, 8-15 (31 December 2011) PDF

limited interest

————. Euclidean Geometry and Physical Space. Mathematical Intelligencer Vol. 28, Number 2, 2006, 51-59 (31 December 2011) PDF

————. Felix Klein, Adolf Hurwitz, and the Jewish Question in German Academia Mathematical Intelligencer Vol. 29, Number 2, 2007, 18-? (31 December 2011) PDF

December 2011) In ‘Years Ago’ column). PDF

General interest

Ruane, P.N. ‘The curious rectangles of Rollett and Rees’. (2010) PDF

Of limited interest

Rush, Jean C. ‘On the Appeal of M. C. Escher’s Pictures’. Leonardo, Vol. 12, No. 1 (Winter, 1979), pp.48-50 (9 September 2010). PDF

Sallows, Lee. ‘The Lost Theorem’. The Mathematical Intelligencer Vol. 19, No. 4, 1997, 51-54 (3 January 2012) PDF

Magic Squares

Schattschneider, Doris. ‘Tiling the Plane with Congruent Pentagons’. Mathematics Magazine Vol.1, 51, No.1 January 1978. 29-44. (13 February 1996 and PDF, 2010)

Of fundamental importance, and full of interest, all largely accessible. ‘Cairo tiling’ as a term is mentioned, as an Archimedean dual, page 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’. Amer. Math. Monthly June-July 1978. 439-450. (3 October 1996)

Largely of an academic nature, and the subject itself is of limited interest. Two uses of Escher's tessellations, page 440. Tilings in the form of Chinese lattice designs, pages 444-445

————. ‘The Fascination of Tiling’. Leonardo, Vol. 25, No. 3/4 (Winter, 1979), pp.341-348 (15 September 2010). PDF

Full of interest; various aspects; Escher, Rice, pentagons, Penrose, kites and darts, rep-tiles

————. ‘Will it Tile? Try the Conway Criterion!’ Mathematics Magazine. 53, No. 4, 224-233. 1980 (13 February 1996)

Of both academic and popular nature. Rightly or wrongly, it has had no practical application in my studies. Figure 6 is an obvious fish, of which I haven’t found the time to compose. Needs a re-read

————. ‘In Black And White: How To Create Perfectly Colored Symmetric Patterns’. Comp. & Maths. with Appls. Vol. 12B, Nos. 3/4. 673-695, 1986. (9 September 2010) PDF

As such, of limited interest as regards tessellation. This borders on the popular and academic, and in relation to tessellation per se is of little value.

————. ‘The Pólya-Escher Connection’. Mathematics Magazine Vol. 60, No. 5. (Dec. 1987) pp 292-298 (17 March 2010) PDF

Contains a previously unpublished page from Escher's sketchbook, which is of some significance, in that it shows how Escher formed his Eagle motif (PD 17), by fusing two tiles. This formation had previously escaped me. This is all the more galling, in that the information was available from as far back as 1987 with this article, but it took me 23 years to find it! Much of the material here later appears in Visions. The background to the creation of the Eagle motif is discussed in Visions, page 289, of which I was aware of previously, but here the fusing is not mentioned, and so I couldn’t understand Schattsneider’s belief at the time as to the accreditation. This article reveals it.

————. ‘Escher: A Mathematician In Spite Of Himself’. Structural Topology No.15 1988. 9-22. PDF

This largely features aspects arising from Escher’s notebooks of 1941-1942, in which Schattschneider examines his mathematics.

————. ‘Escher’s Metaphors’. Scientific American 271, No. 5 66-71 November 1994 (13 June 2011) PDF

Somewhat curious; the premise here is unclear, and there is nothing that has not been discussed before in Visions.

————. ‘Math and Art in the Mountains’. Mathematical Intelligencer (Mathematical Communities) Vol. 28 Number 3, 2006 (24 November 2011) PDF

Talking about the Banff Bridges Conference of 2005

————. ‘The Mathematical Side of M. C. Escher’. Notices of the AMS, Vol. 57, Number 6, June/July 2010. 706-718 (2010) PDF

Although full of interest, this largely covers ground already discussed in Visions, but new is Escher analysis of Coxeter’s diagram, and occasional snippets of interest, such as with Speiser.

————. Afterword. On pentagon article. source, date not recorded (24 November 2009)

————. ‘Escher’s Combinatorial Patterns’. The Electronic Journal of Combinatorics 4 (NO.2) (1997), #R17. ‘Potato printing game’ of Escher updated. (29 May 2007)

http://www.combinatorics.org/Volume_4/wilftoc.html

An examination of Escher’s ‘combinatory tile’ problem. Of very little interest in itself; it’s really an instance of personal study to the person concerned.

————. ‘Mathematics and Art’. Math Awareness Month – April 2003. Web. (6 November 2007)

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

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

Senechal, Marjorie. ‘The Algebraic Escher’. Structural Topology No.15 31-42. PDF

————. ‘Parallel Worlds: Escher and Mathematics, Revisited’. Mathematical Intelligencer Vol. 21 Number 1, 1999, 13-19 (24 November 2011)

PDF

Note that this is reprinted in M.C. Escher’s Legacy

————. ‘The Mysterious Mr. Ammann’. (Mathematical Communities) Mathematical Intelligencer Vol. 26 Number 4, 2004, 10-21 (25 November 2011) PDF

————. Martin Gardner tribute (1914-2010). Mathematical Intelligencer Vol. 33 Number 1, 2011, 51-54 (25 November 2011) PDF

————. ‘Coxeter and Friends’. Mathematical Intelligencer Vol. ? Number ?, 2004, 16 ‘Mathematical Communities’ column. (28 November 2011) PDF

————. ‘Tiling the Torus and Other Space Forms’. Discrete Comput Geom 3:55-72 (1988)

Academic

Senechal, Marjorie and Jean Taylor. ‘Quasicrystals: The View from Les Houches’. Mathematical Intelligencer Vol. 12 Number 2, 1990, 54-64 (28 November 2011)

————. ‘Tilings, quasicrystals, and Hilbert’s 18^th problem’ (lower case as in article). Structural Topology No.20 7-26. 1993. PDF.

No Cairo tiles. Escher tiling E128 (ghosts) page 9

————. ‘What is… a Quasicrystal?’ Notices of the AMS September 2006, 886-887. (2010) PDF

Somewhat advanced

Senechal, Marjorie and R. V. Galiulan. ‘An Introduction to the Theory of Figures: the Geometry of E. S. Federov’. Structural Topology No. 10 1984 5-20 (2010) PDF

Simpson, R. ‘Locally equiangular triangulations’. The Computer Journal. 21 (1978) 243-245. (31 October 2012) PDF

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

Situngkic, Hokky. ‘What is the relatedness of mathematics and art and why should we care?’ (Escher page 5) PDF (2010)

Socolar, Joshua E. S. ‘Hexagonal Parquet Tilings k-Isohedral Monotiles with Arbitrarily Large k’ The Mathematical Intelligencer Vol. 29 No.2 2007. 1-6. PDF (2010)

Somewhat advanced. N.B. this is not parquet deformation per se!

Socolar, Joshua E. S and Joan M. Taylor. ‘An aperiodic hexagonal tile’ 1-21 (2010) PDF

Stein, Sherman. ‘Tiling, Packing, and Covering by Clusters’. Rocky Mountain Journal of Mathematics Vol 16, Number 2, Spring 1986 (20 September 2012) PDF

Academic throughout. Bare minimum of diagrams; much theory, all of no practical use.

Stewart, Ian. ‘Rep-Tiling the Plane’. Scientific American. May 2000. 84-85. (5 May 2000)

————. ‘The Art of Elegant Tiling’. Scientific American. July 1999. 96-98. (30 July 2012)

Minor instance of Cairo tiling, page 97, as devised by Rosemary Grazebrook

Sugimoto, T. ‘Classification of Convex Pentagons That Can Generate Edge-to-edge

Monohedral Tilings of The Plane’ 2012? PDF (25 May 2012)

Sugimoto, T and Tohru Ogawa. ‘Tiling Properties of tilings by Convex Pentagon’. Forma 21, 113-128 (26 November 2009) PDF

Set of 14 Convex pentagons

Sugimoto, T and Tohru Ogawa. ‘Convex Pentagonal Tiling Problem and Properties of Nodes in Pentagonal Tilings’, 452-455 Form and Symmetry Art and Science Buenos Aires Conference, 2007 (26 November 2009) PDF

Sugimoto, T. and OGAWA, T. (2000a) ‘Tiling Problem of Convex Pentagon’, Forma, 15, 75–79, 2000 PDF.

Sugimoto, Teruhisa and Tohru Ogawa. ‘Systematic Study of Convex Pentagonal Tilings’, I: Case of Convex Pentagons with Four Equal-length Edges. Forma 20, 1-18, 2005 (26 November 2009) PDF

Tapson, Frank. ‘Filling in Space’. Mathematics in School. March 1989?

Taylor, H.M. ‘On some geometrical dissections’ Messenger of Mathematics, Volume 35, 81-101

Taylor, M.V. ‘The Roman Tessellated Pavement at Stonesfield, Oxon’. Oxoniesia Vol. VI 1941. PDF (2010)

Historical account, of limited interest.

Teeters, Joseph L. ‘How to Draw Tessellations of the Escher Type’. The Mathematics Teacher, April 1974, Volume 67, Number 4, 307-310. (Confusingly, inside the book this is also titled as ‘Mathematics Teacher’ (18 June 2011)

‘Special edition’ on tessellations, specifically concerning three Escher-inspired tessellation articles; (i) Ernest R. Ranucci, Master of Tessellations M.C. Escher; (ii) Joseph L. Teeters How to draw tessellations of the Escher Type, and (iii) Evan M. Maletsky Activities: Designs with Tessellations. That by Maletsky is particularly excruciating,

Teuber, Marianne L. 'Sources of Ambiguity in the Prints of Maurits C. Escher.' The fascinating graphic inventions of the late Dutch artist reflect a strong mathematical and crystallographic influence. Their original inspiration, however, came from experiments on visual perception. Scientific American 231 no. 1 (July 1974): 90-104. (1987)

Thomas, B.G and M. A. Hann. ‘Fundamental principles governing the patterning of polyhedra 2007’. IaSDR. (2010) PDF

The Cairo tessellation gets a mention (page 6), defined as a equilateral pentagon

————. ‘Patterned Polyhedra: Tiling the Platonic Solids’. In Bridges 2008. PDF

Again the Cairo tessellation is mentioned, with the same definition as above. This paper seems to be derived from the above.

Vince, Andrew. ‘Replicating Tessellations’. Siam Journal of Discrete Math, Vol. 6 No. 3, pp 501-521, August 1993. (2010) PDF

Academic. Mentioned in Schattschneider’s bibliography.

————. ‘Rep-tiling Euclidean space’. Aequationes Mathematicae 50 (1995) 191-213 (30 May 2012) PDF

Academic

Vincent, Jill. ‘Shrine to University: Mathematics in the Constructed Environment’. 25-37. PDF (2010)

Penrose and pentagon tilings in situ in Australia.

Walker, Jearl. ‘What explains subjective-contour illusions, those brightspots that are not really there?’ Scientific American 84-87. (1988)

Walter, Marion. ‘The day all the textbooks disappeared’. Mathematics Teaching 112. September 1985. 8-11.

Wieting, Thomas. ‘Capturing Infinity’. Reed, March 2010 21-29

A layman’s guide to constructing hyperbolic tessellations using compass and straight edge

Wilkie, Ken. ‘The Weird World of Escher the Impossible’. Magazine of the Netherlands Holland Herald Volume 9 Number 1 1974 3, 20-43 (28 March 2011)

Wollny, Wolfgang. ‘Contributions to Hilbert’s Eighteenth Problem’. Pacific Journal of Mathematics Vol 112, No 2, 1984 (20 September 2012)

Academic throughout. First diagram appears on p.468, followed by a profusion of diagrams; however, all of this no practical use, being academic

Zeeman, Christopher and Ian Stewart. ‘Mathematics for Young People: The Royal Institution Masterclasses’. The Mathematical Intelligencer. Vol. 7, No. 3, 1985, 59-64 (9 December 2011) PDF

General interest

PAMPHLETS

Unknown Author. Perplexing Poultry Instructional Manual. Pentaplex Limited. 1983.

Isenberg, Cyril. Soap Film Experiments with Kubic Bubbles. 1974.

JOURNALS

Journal of the Mathematics and Arts. Volume 1 Number 4 ed. Greenfield, Gary, R. December 2007

Mathematics Teaching number 50 spring 1970 (13 August 2000).

No particular reason for obtaining; a chance purchase at a bargain price. Peano curves 13-21

Mathematics Teaching number 51 summer 1970 (13 August 2000).

No particular reason for obtaining; a chance purchase at a bargain price.

Mathematics in School. 1979: September, November

Mathematics in School. 1980: January, May, September, November

Mathematics in School. 1981: January

N.B. None of the MIS above have anything in particular about tessellation

Scientific American. 1961: Front cover, Escher’s birds (14 January 1988)

Scientific American. 1963: May. Solomon Golomb, reptiles

Scientific American. 1965: July, polyominoes, 24 heptiamonds twisted cord illusion, page 102-103, page 102

Scientific American. 1965: August, proof that V-heptiamond will not tile heptiamonds

Scientific American. 1966: April-August; October-December

Scientific American. 1975: January. George A. Escher’s reply (letter, pages 8-9) to Marianne Teuber’s article, and Teuber’s immediate response

Scientific American. 1975: September. Penrose loaded wheelbarrow solution to last month’s problem, page 180.

Scientific American. 1977: August; Minor tiling, dividing polygons as equal bisections, page 122

Scientific American. 1986: January-August; October-December

Undated: Richard E. James III pentagonal tessellation, page 118

PATENTS

Fisher, Adrian. Tessellation Set 2001 (2010) PDF

Penrose, Roger

Lalvani, Heresh. Crescent shaped polygonal tiles, of 4 November 1986 (4 July 2011)

WEBSITES

www.euler.slu.edu/escher

www.mathfroum.org/sketchup Escher Square tiles, In Google SketchUp. Tutorial in creating Escher-like tilings, 12 pages

Kaplan, Craig. Taprats – Islamic geometric designs: Design notes, architecture, Gallery, User’s manual

Maze design

THESES

Chavey, Darrah, P. ‘Periodic Tilings and Tilings by Regular Polygons’. 1984 (PDF) (18 May 2011)

Harriss, E. O. ‘On Canonical Substitution Tilings’. 2004 (PDF)

Kaplan, C. S. ‘Computer Graphics and Geometric Ornament’. 2002 (PDF)

Reinhardt, K. ‘Über die Zerlegung der Ebene in Polygone’. Universität Frankfurt 1918 (PDF) (24 May 2012)

This is interesting in many ways, and specifically in context with the Cairo tiling, in that it shows what I term as a ‘skew’ Cairo tiling, pages 80-81, the first instance I have of this. It’s also curiously noticeable for the almost total exclusion of diagrams, despite it being a paper on tiling!

Calendars

All these maths calendars are from John Bibby. Various oddments of mathematics, still relevant

The Mathematics Calendar 1992

The Fun Maths Calendar 1993 (9 April 1993)

Fun Maths [calendar] 1994 (30 April 1994)

Mathematics Education Calendar 1994 (30 April 1994). Historical figures for each month

Exhibition Catalogues

Escher: Pattern & Paradox. Catalogue of the exhibition held at Lowe Museum, Miami, USA, October to November 1984 (28 March 2011)

Catalogue of the exhibition held at Artists’ Market, Norwalk, Connecticut, USA, March-April 1987 (28 March 2011)

Structural Topology 15 1988 (Escher Special, articles by M. Emmer, D. Schattschneider, H. S. M. Coxeter, M. Senechal, G. C. Shephard)

Structural Topology 17 1991 (‘continuation’ of Escher Special: articles by M. Emmer, Roger Penrose, J. F. Rigby, A. Dress and D. H. Huson)

O Mundo Mágico de Escher Brazil exhibit catalogue, in Brasilia 12October-26 December 2010, Rio de Janeiro 17 January-27 March 2011, Sao Paulo 18 April-17 July 2011. Curated by Pieter Tjabbes (10 October 2012) PDF

Reviews

Pickover, Cliff. Tribute to a Mathemagician. Edited by Barry Cipra, Erik D. Demaine, Martin L. Demaine and Tom Rodgers. Review in Mathematical Intelligencer Vol. 29, No. 3 2007, 71

Martin Gardner tribute (25 November 2011)

Mackay, Alan L. Stimulating patterns (sic). Review in Nature Vol. 349 7 February 1991, 471-472. Visions of Symmetry, Notebooks Periodic Drawings, and Related Works of M.C. Escher. (15 June 2011) PDF

Sigmund, Karl. Kepler’s Conjecture: How Some of the Greatest Minds in History Helped Solve One of the Oldest Math Problems in the World by George G. Szpiro. Review in Mathematical Intelligencer Vol. 26, No. 1 2004, 66-67

Miscellaneous

The New Yorker. July 5 2010. Escher's Sky and Water II appropriated/adapted for cover design by Bob Staake ‘Gulf Sky and Water’.

Escher Notebook

Other items seen, but not deemed worthy of purchasing, but may be referred to in others work

Dresser, Christopher. Design.

Library Books

Callender, Jane. 2000 Patterns Combinations. Batsford 2011 (7 April 2012)

20 demi-regular howler.

Bellos, Alex. Alex’s Adventures in Numberland. Dispatches from the Wonderful World of Mathematics. Bloomsbury Publishing Ltd, 2010. Library.

Occasional references to Escher, page 244, 392 hyperbolic geometry, with Circle Limit IV

Mankiewicz, Richard. The Story of Mathematics. Cassell & Co 2000 (Grimsby Library).

Escher, 6, 125 Circle Limit IV, 129 Mobius Strip II