Listing All Instances of the Cairo Tiling in Print
The criteria for the listing is of the Cairo tiling in text and/or in diagrammatic form, in which I list all instances of the Cairo tiling, as of the ‘standard model’ whose parameters can vary with basic features of a Cairo tiling, i.e. a symmetrical pentagon (not necessarily of equal lengths), with two opposite right angles. The list is shown in three different ways, as according to various filters:
Can anyone add to this list, with either other references (the earlier the better) or of any additional further information?
1. A simple listing of attributed references, mentioned in association with Cairo, e.g. 'is a favourite streettiling in Cairo' arranged as according to chronology, with Cairo quote.
1. 1971 James A. Dunn. ‘Tessellations with Pentagons’. The Mathematical Gazette, Vol. 55, No. 394 December, pp. 366369
2. 1975 Martin Gardner. Scientific American. Mathematical Games, July. ‘On tessellating the plane with convex polygon tiles’, pp.112117 (p. 114 and 116 re Cairo aspect)
4. 1979 Robert H. Macmillan. Mathematical Gazette, 1979. ‘Pyramids and Pavements: some thoughts from Cairo’, pp. 251255.
5. 1982 George E. Martin. Transformation Geometry: An Introduction to Symmetry, p. 119.
7. 1986 James McGregor and Alan Watt. The Art of Graphics for the IBM PC, Addison Wesley 1986 pp. 196197. …is the wellknown Cairo tile, so called because many of the streets of Cairo were paved in this pattern.
8. 1986 Ehud, BarOn. ‘A programming approach to mathematics’. Computers & Education 10(4): pp. 393401. December 1986. Elsevier. … especially the Cairo tiling.
9. 1989 W. K. Chorbachi. Computers and Mathematics with Applications. ‘In the Tower of Babel: Beyond Symmetry In Islamic Design’. Vol. 17, No. 46, pp. 751789 (Cairo aspects pp. 783794)
10. 1989 Pierre De La Harpe. Quelques Problèmes Non Résolus en Géométrie Plane. L’Enseignement Mathématique, t 35 (1989), pp. 227243 (in French) Cairo tiling on p. 232
11. 1990 Francis S. Hill. Computer Graphics. Macmillan Publishing Company, New York, p. 145.
12. 1991 Ann E. Fetter. et al. The Platonic Solids Activity Book. Key Curriculum Press/Visual Geometry Project. Backline Masters, pp. 21, 97
13. 1991 David Wells.The Penguin Dictionary of Curious and Interesting Geometry. Penguin Books, pp. 23, 61, 177.
14. 1993 Nenad Trinajstić. 'The Magic of the Number Five'. Croatia Chemica Acta 66 (1) pp. 227254. ... seen in street tiling in Cairo and occasionally in the mosaic of Moorish buildings. Seemingly quoting Blackwell, who in turn quotes Gardner.... 15. 1994 Audrey Leathard. Going interprofessional: working together for health and welfare. Routledge; first edition 1994
16. 1994 Carter Bays. Complex Systems Publications, Volume 8, Issue 2, pp. 127150, Cairo aspect p. 148
17. 1996 Michael O’Keefe and Bruce G. Hyde. Crystal Structures No. 1. Patterns & Symmetry. Mineralogical Society of America p. 207 The pattern is known as Cairo tiling, or MacMahon’s net and In Cairo (Egypt) the tiling is common for paved sidewalks…
18. 1998 David A. Singer. Geometry Plane and Fancy, 1998, pp. 34 and 37.
19. 1997 Michael Serra. Discovering Geometry: An Inductive Approach. Key Curriculum Press, p. 404
20. 2000 M. Deza et al. 'Fullerenes as tilings of surfaces'. Journal of Chemical Information Computer and Modelling. ACS Publications, pp. 550558
21. 2001 Edward Duffy, Greg Murty, Lorraine Mottershead. Connections Maths 7. Pascal Press, p. 83
22. 2003 Teacher’s Guide: Tessellations and Tile Patterns, p.30 (Cabri) Geometric investigations on the VoyageTM 200 with Cabri. Texas Instruments Incorporated
23. 2003 Catherine A. Gorini. The Facts on File Geometry Handbook. 2003, 2009 revised edition. Facts on File Inc, and imprint of Infobase publishing, p. 22.
24. 2003 Chris Pritchard. The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching, P. 421427, Cairo aspect p. 421.
25. 2004 Robert Parviainan. ‘Connectivity Properties of Archimedean and Laves Lattices’. Uppsala Dissertations in Mathematics 34. p. 9. 2004.
26. 2005 David Mitchell. Sticky note origami: 25 designs to make at your desk, Sterling Publication Company, pp. 5861.
27. 2005 George McArtney Phillips. Mathematics Is Not a Spectator Sport. Springer, p. 193 This is called the Cairo tessellation.
28. 2005 Sue JohnstonWilder and John Mason. Developing Thinking in Geometry, p. 182. … is often referred to as the Cairo tessellation as it appears in a mosque there.
29. 2005 Carter Bays. Complex Systems Publications, Volume 15, Issue 3, 245252, Cairo aspect pp. 249250 ‘A Note on the Game of Life in Hexagonal and Pentagonal Tessellations’. ‘Here we have chosen the Cairo tiling…’ 30. 2006 John Sharp. ‘Beyond Su Doku’. Mathematics Teaching in the Middle Years. Vol. 12, No. 3 October 2006 pp. 165169 pp. 167169, in the context of a ‘Cairo Su Doku’. 31. 2007 B. G. Thomas and M. A. Hann. In Sarhangi, Reza (Ed). Bridges. Mathematical Connections in Art, Music, and Science. (Ninth) Conference Proceedings 2007. Donostia, Spain. Patterned Polyhedra: Tiling the Platonic Solids There are however various equilateral pentagons that can tessellate the plane. Probably the best known is the Cairo tessellation… 32. 2007 B. G. Thomas and M. A. Hann. Patterns in the Plane and Beyond: Symmetry in Two and Three Dimensions. 2007. The University of Leeds and the authors. Ars Textrina, No. 37 Cairo tiling pp. 5253, 7071, 79. Stated as equilateral. 32. 2007 Mike Ollerton. 100+ Ideas for Teaching Mathematics p. 66 This tessellation not only begs interesting questions about angle sizes and side ... The Cairo tessellation... A The challenge is to use this tile to fill 2D space.
33. 2008 Anon. Key Curriculum Press. Chapter 7, Transformations and Tessellations, p. 396.
35. 2008 B. G. Thomas and M. A. Hann in Sarhangi, Reza (Ed). Bridges. Mathematical Connections in Art, Music, and Science. (Tenth) Conference Proceedings 2008. Leeuwarden, Netherlands p. 102 in ‘Patterning by Projection: Tiling the Dodecahedron and other Solids’ 37. 2008 Birgit Kaltenmorgen. Der mathematische Patchworker. (in German) Wagner, Gelnhausen; first edition 2008, pp. 8283
38. 2008 Robert Fathauer. Designing and Drawing Tessellations, 2008, p. 2.
39. 2009 Craig S. Kaplan. Introductory Tiling Theory for Computer Graphics. Morgan & Claypool Publishers, p. 33.
43. 2011 M. Rojas et al. 'Frustrated Ising model on Cairo pentagonal lattice'. Physical Review E 86(51):051116 November 2012 Seemingly one of the first of 'high level physics' of the Ising model, from this date there are many others. 44. 2011 Mike Askew and Sheila Ebbutt. The Bedside Book of Geometry: From Pythagoras to the Space Race: The ABC of Geometry. Murdoch Books Pty Limited
45. 2011 Eric Goldemberg. Pulsation in Architecture p. 338. J Ross Publishing, 2011 46. 2012 Edited by Christoph Gengnagel, A. Kilian, Norbert Palz, Fabian Scheurer. Computational Design Modeling: Proceedings of the Design Modeling Symposium. Springer, p. 229... …on the mathematical configuration of a Cairo tessellation 47. 2013 Lisa Iwamoto. Digital Fabrications: Architectural and Material Techniques. Princeton Architectural Press; first edition 2009 48. 2014 HansJoachim Gorski and Susanne MüllerPhilipp. Leitfaden Geometrie: Für Studierende der Lehrämter, Springer p. 186
2. A simple listing of nonattributed references of the Cairo tiling, in chronological order, with quotes and comments where considered appropriate 1. 17th Century? Simon Ray. Indian & Islamic Works of Art. Self Published, 2016, pp. 178179. The first recorded instance in whatever capacity. As such, the 17th century dating here, of an Indian jali, is largely taken on trust. The entry in the catalogue is rather sparse, and so naturally I attempted to contact Simon Ray (a dealer in Indian and Islamic Works of Art, in London, UK) for more detail. However, despite two emails from myself, and then at my behest two others from interested parties, namely professors Gregg De Young and Chaim GoodmanStrauss, Ray once more chose not to reply! Hence the date given is thus on trust. In the (unlikely) event of you reading this Simon, do by all means redeem yourself here!A reference to a jali, albeit matters of provenance are a little understated. See p. 178 in the Indian and Islamic Art catalogue, 2016. 2. Early to Mid 20th century. Not Published The first recorded instance of a flooring, at a room in Heidelberg Castle, Germany. As such, although there is (so far), no evidence of this sighting appearing in print, I nonetheless include for the sake of an inclusive listing, of which by its strict omission would thus be lost. This is of a floor, apparently of white marble. The date is not entirely clear, beyond being ‘earlymid 20th century’.
3. 1909 Herbert C. Moore. ‘Tile’. United States Patents 928,320 and 928,321 of 20 July 1909 4. 1921 Percy A. MacMahon. A. New Mathematical Pastimes. Cambridge University Press 1921 and 1930. (Reprinted by Tarquin Books 2004) Cairo diagram p. 101.
5. 1922 Percy A. MacMahon. ‘The design of repeating patterns for decorative work’. Journal of the Royal Society Arts 70 (1922), 567578. Related discussion ibid pp. 578582
6. 1925 Friedrich Haag. 'Die pentagonale Anordung von sich berührenden Kriesen in der Ebene’. Zeitschrift fur Kristallographie 61 (1925), 339340 7. 1926 Friedrich Haag. 'Die Symmetrie verhältnisse einer regelmässigen Planteilung’. Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht, Band 57 (1926), 262263 Has Cairo tiling in the form of circle packing. 8. 1931 Fritz Laves. ‘Ebenenteilung in Wirkungsbereiche’. Zeitschrift für Kristallographie 76 (1931): 277284. 9. 1933 Amos Day Bradley. The Geometry of Repeating Design and Geometry of Design for High Schools. Bureau of Publications Teachers College, Columbia University, New York City, and 1972 reprint. Book as oftquoted by Schattschneider, but surprisingly no one else. P. 123 Cairolike diagram, dual of the 3. 3. 4. 3. 4. Possibly based on the work of Haag, of which the diagram resembles, and of whom articles he quotes
10. 1951 H. Martyn Cundy and A. P. Rollett. Mathematical Models. Oxford University Press (I have the second edition, of 1961).
11. 1951 'Croton'. Cairo tiling used as a crossword puzzle, in The Listener, 13 December 1951, puzzle 1128 HexaPentagonal I, by 'Croton'.
12. 1954 'Croton'. Cairo tiling used again as a crossword puzzle, in The Listener, 22 April 1954, puzzle 1251 HexaPentagonal II, by 'Croton'.
13. 1954 Cyril Stanley Smith. “The Shape of Things.” Scientific American, vol. 190, no. 1, 1954, pp. 58–65. In a general discussion on tiling. 14. 1955 A. P. Rollett. ‘A Pentagonal Tessellation’. The Mathematical Gazette, Vol 39, No. 329 (Sep. 1955) note 2530 p. 209. Rollett states 'My colleague Mr. R. C. Lyness noticed this [Cairo tiling] pattern on the floor of a school in Germany. It has also appeared in a crossword puzzle in The Listener'. The detail given is infuriatingly sparse to try and locate this sighting. Does anyone know about Lyness's connection to Germany, and if so where is it? By 'school' does he mean university? 'The Listener' reference has been found; see 1951 and 1954 entries above. 15. 1956 C. Dudley Langford. ‘Correspondence’. The Mathematical Gazette, Vol. 40, No. 332 May 1956 p. 97. Drawing readers' attention to MacMahon’s Cairo tiling picture in New Mathematical Pastimes. Of importance, due to Cairo tiling reference, referring to Rollett’s piece in the Gazette (Note 2530). Also of note in that Langford gives a different construction to MacMahon’s. Also see T. Bakos, which completes a nonstated ‘trilogy’ of writings of the day.
16. 1958 T. Bakos. ‘2801 On Note 2530’ (Correspondence on C. Dudley Langford's 'Cairo' tile reference)’. The Mathematical Gazette, Vol 42, No. 342 December 1958, p. 294 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. 17. 1963 H. S. M. Coxeter. Regular Complex Polytopes. Second edition. Dover Publications Inc., New York.
18. 1967 D. G. Wood. ‘Space Enclosure Systems’, Bulletin 203. Columbus, Ohio: Engineering Experiment Station, The Ohio State University. PP. 34, 3031
19. 1969? Keith Critchlow. Order in Space. A Design Source Book. Thames & Hudson. A date of 1969 is given in the book but it is unclear if this was when it was first published. The published date is apparently given as 1987. 2000 reprint.
20. 1970 Ernest R. Ranucci. Tessellation and Dissection. J. Weston Walch 21. 1970 H. S. M. Coxeter. 'Twisted Honeycombs' (CBMS Regional Conference Series in Mathematics), 1970, pp. 2123. 22. 1972 Robert Williams. The Geometrical Foundation of Natural Structure. A Source Book of Design. Dover Publications, Inc. 1979. Another edition, of another name, was of 1972. Cairo diagram (but not attributed) p. 38 in the context of the Laves tilings. This is also interesting in that it shows ‘minimum and maximum’ values of the tiling, of a square, and two rectangles (basketweave). The source of the D.G. Wood reference.
23. 1974 Stanley R. Clemens. ‘Tessellations of Pentagons’. Mathematics Teaching, No. 67 (June), pp.1819, 1974
24. 1975 John Parker. ‘Tessellations’, Topics, Mathematics Teaching 70, 1975, p. 34 25. 1976 Marc G. Odier. ‘Puzzle with Irregular Pentagonal Pieces’. United States Patent 3,981,505 21 September 1976 Cairo tile diagram Fig. 3, and various patches of tiles formed with the pentagons.
26. 1976 Phares O’Daffer. G; Clemens, Stanley R. Geometry. An Investigative Approach 1^{st} edition, 2^{nd} edition 1992 AddisonWesley Publishing Company. (Note that I have the 2^{nd }edition, not the 1^{st})
27. 1977 Lorraine Mottershead. Sources of Mathematical Discovery. Basil Blackwell.
28. 1977 Doris Schattschneider and Wallace Walker. M. C. Escher Kaleidocycles. Tarquin Publications. First edition, 1977; I have the ‘special edition’ of 1982.
29. 1978 Peter Pearce and Susan Pearce. Polyhedra Primer. Dale Seymour Publications
30. 1978 Ernest H. Lockwood, and Robert H. Macmillan. Geometric symmetry. Cambridge University Press (and reprint 2008).
31. 1980 Michael O’Keefe and B. G. Hyde. ‘Plane Nets in Crystal Chemistry’. Philosophical Transactions Royal Society London. Series A, 295 1980, pp. 553618 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. 32. 1982 Patrick Murphy. Modern Mathematics Made Simple. Heinemann London Tessellations, Chapter 10, pp. 194205, 262. Cairo diagram, of equilateral pentagons (but not attributed) p. 200. 33. 1983 John Willson. Mosaic and Tessellated Patterns. How to Create Them. Dover Publications, Inc. 1983. Plate 3 Cairo tiling plate 3. (Neglected until 7 May 2013!) 34 1983 Cyril Stanley Smith. A Search for Structure. The MIT Press, 1983 Has nonattributed Cairo tiling. 35. 1986 James McGregor and Jim Watt. The Art of Graphics for the IBM PC, pp 196197
The plane cannot be tesselated (sic) by regular pentagons, but there are an a number of irregular pentagons that will tessellate the plane. An example of a pentagon that will tesselate (sic) is the wellknown Cairo tile, so called because many of the streets of Cairo were paved in this pattern (Fig. 5.2): The Cairo tile is equilateral but not regular because its angles are not all the same.
36. 1986 A. L. Loeb. 'Symmetry and Modularity'. Computers and Mathematics with Applications, Elsevier 37. 1986 Jay Kappraff. ‘A Course in the Mathematics of Design’. Computers and Mathematics with Applications Vol. 12B, Nos. 3/4, pp 913948 Cairo tiling in the context of the set of 11 Laves tiling; p. 923 but as such, inconsequential.
38 1986 Lothar Collatz. Geometrische Oranamente (in German) Cairo tiling diagram in context of 43433 classification. 39. 1987 Bob Burn. The Design of Tessellations. Cambridge University Press. Sheet 30. Equilateral pentagon.
40. 1989 Dale Seymour and Jill Britton. Introduction to Tessellations. Dale Seymour Publications Cairo tiling (but not attributed) p. 39.
41. 1989 Piere De La Harpe. ‘Quelques Problèmes Non Résolus en Géométrie Plane’. L’Enseignement Mathématique, t 35 (1989), pp. 227243 (in French)
42. 1989. Marjorie Senechal. ‘Symmetry Revisited’. Computers and Mathematics with Applications. Vol 17, No. 13, pp. 112. 1989. Cairo diagram in the context of the set of 11 Laves diagrams, p. 9; as such per se, inconsequential.
43. 1989 Istvan Hargittai. Symmetry 2, Unifying Human Understanding. Volume 2, Source of Chorbachi article, see above pp. 783794. 44. 1991 Jay Kappraff. Connections The Geometric Bridge Between Art and Science. McGrawHill. p. 181. Shown as the dual of 3. 3. 4. 3. 4 tiling. Poorly executed diagram, with four different pentagons! However, the intention, due to an accompanying diagram, is indeed clear. 45. 1999 Ian Stewart. ‘The Art of Elegant Tiling’. Scientific American. July 1999, pp. 9698. Minor instance of coloured Cairo tiling, p. 97, as devised by Rosemary Grazebrook.
46. 1999 Jinny Beyer. Designing Tessellations, Contemporary Books, p. 144. 47. 2005 Paul Garcia. ‘The Mathematical Pastimes of Major Percy Alexander MacMahon. Part 2 triangles and beyond’. Mathematics in Schools, September 2005, pp. 2022. PDF Contains a Cairo tiling 'of sorts', p. 22
48. 2006 John Sharp. ‘Beyond Su Doku’. Mathematics Teaching in the Middle Years. Vol. 12, No. 3 October 2006 pp. 165169. Cairo tiling on pp. 167169, in the context of a ‘Cairo Su Doku’.
49. 2006 Mark Eberhart. Excerpts selected by Mark Eberhart in Resonance from C. S. Smith's A Search for Structure, of which p. 87 contains a Cairo tiling 50. 2008 B. G. Thomas, B. G. and M. A. Hann. In Bridges. Mathematical Connections in Art, Music, and Science. There are, however, equilateral convex pentagons that do tessellate the plane, such as the wellknown Cairo tessellation shown in Fig. 1. Type of pentagon: Equilateral (p. 101). 51. 2014 HansJoachim Gorski and Susanne MüllerPhilipp. Leitfaden Geometrie: Für Studierende der Lehrämter, Springer p. 186. 52. 2016. Simon Ray. Indian & Islamic Works of Art. Self Published, pp. 178179. As such, the 17th century dating here, of an Indian jali, is largely taken on trust. The entry in the catalogue is rather sparse, and so naturally I attempted to contact Simon Ray (a dealer in Indian and Islamic Works of Art, in London, UK) for more detail. However, despite two emails from myself, and then at my behest two others from interested parties, namely professors Gregg De Young and Chaim GoodmanStrauss, Ray once more chose not to reply! Hence the date given is thus on trust. In the (unlikely) event of you reading this Simon, do by all means redeem yourself here! N.B. An apparent Cairo tiling in a 1923 paper by F. Haag, "Die regelmässigen Planteilungen und Punktsysteme." Zeitschrift fur Kristallographie 58 (1923): 478488, Figure 13, in that it is frequently quoted as a pentagonal tiling is misleading; it's not a pentagon, but rather a quadrilateral. 2. References of all instances of the Cairo tiling, attributed or not, in chronological order, with quotes and comments where appropriate 1. 17th Century? Simon Ray. Indian & Islamic Works of Art. Self Published, 2016, pp. 178179. The first recorded instance in whatever capacity. As such, the 17th century dating here, of an Indian jali, is largely taken on trust. The entry in the catalogue is rather sparse, and so naturally I attempted to contact Simon Ray (a dealer in Indian and Islamic Works of Art, in London, UK) for more detail. However, despite two emails from myself, and then at my behest two others from interested parties, namely professors Gregg De Young and Chaim GoodmanStrauss, Ray once more chose not to reply! Hence the date given is thus on trust. In the (unlikely) event of you reading this Simon, do by all means redeem yourself here!A reference to a jali, albeit matters of provenance are a little understated. See p. 178 in the Indian and Islamic Art catalogue, 2016. 2. Early to Mid 20th century. Not Published The first recorded instance of a flooring, at a room in Heidelberg Castle, Germany. As such, although there is (so far), no evidence of this sighting appearing in print, I nonetheless include for the sake of an inclusive listing, of which by its strict omission would thus be lost. This is of a floor, apparently of white marble. The date is not entirely clear, beyond being ‘earlymid 20th century’. 3. 1909 Herbert C. Moore. ‘Tile’. United States Patents 928,320 and 928,321, of 20 July 1909. 4. 1921 Percy A. MacMahon. A. New Mathematical Pastimes. Cambridge University Press 1921 and 1930. (Reprinted by Tarquin Books 2004) Cairo diagram p. 101. 5. 1922 Percy A. MacMahon. ‘The design of repeating patterns for decorative work’. Journal of the Royal Society Arts 70 (1922), 567578. Related discussion ibid pp. 578582 6. 1925 Friedrich Haag. 'Die pentagonale Anordung von sich berührenden Kriesen in der Ebene’. Zeitschrift fur Kristallographie 61 (1925), pp. 339340 7. 1926 Friedrich Haag. 'Die Symmetrie verhältnisse einer regelmässigen Planteilung’. Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht, Band 57 (1926), pp. 262263 Has Cairo tiling in the form of circle packing 8. 1931 Fritz Laves. ‘Ebenenteilung in Wirkungsbereiche’. Zeitschrift für Kristallographie 76 (1931): 277284. 9. 1933 Amos Day Bradley. The Geometry of Repeating Design and Geometry of Design for High Schools. Bureau of Publications Teachers College, Columbia University, New York City, and 1972 reprint. Book as oftquoted by Schattschneider, but surprisingly no one else. P. 123 Cairolike diagram, dual of the 3. 3. 4. 3. 4. Possibly based on the work of Haag, of which the diagram resembles, and of whom articles he quotes. 10. 1951 H. Martyn Cundy and A. P. Rollett. Mathematical Models. Oxford University Press (I have the second edition, of 1961). 11. 1951 'Croton'. Cairo tiling used as a crossword puzzle, in The Listener, 13 December 1951, puzzle 1128 HexaPentagonal I, by 'Croton'. 12. 1954 'Croton'. Cairo tiling used again as a crossword puzzle, in The Listener, 22 April 1954, puzzle 1251 HexaPentagonal II, by 'Croton'. 13. 1954 Cyril Stanley Smith. “The Shape of Things.” Scientific American, vol. 190, no. 1, 1954, pp. 58–65. In a general discussion on tiling. 14. 1955 A. P. Rollett. ‘A Pentagonal Tessellation’. The Mathematical Gazette, Vol 39, No. 329 (Sep. 1955) note 2530 p. 209. Rollett states 'My colleague Mr. R. C. Lyness noticed this [Cairo tiling] pattern on the floor of a school in Germany. It has also appeared in a crossword puzzle in The Listener'. The detail given is infuriatingly sparse to try and locate this sighting. Does anyone know about Lyness's connection to Germany, and if so where is it? By 'school' does he mean university? 'The Listener' reference has been found; see 1951 and 1954 entries above. 15. 1956 C. Dudley Langford. ‘Correspondence’. The Mathematical Gazette, Vol. 40, No. 332 May 1956 p. 97. Drawing readers' attention to MacMahon’s Cairo tiling picture in New Mathematical Pastimes. Of importance, due to Cairo tiling reference, referring to Rollett’s piece in the Gazette (Note 2530). Also of note in that Langford gives a different construction to MacMahon’s. Also see T. Bakos, which completes a nonstated ‘trilogy’ of writings of the day.
16. 1958 T. Bakos. ‘2801 On Note 2530’ (Correspondence on C. Dudley Langford's 'Cairo' tile reference)’. The Mathematical Gazette, Vol 42, No. 342 December 1958, p. 294 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. 17. 1963 H. S. M. Coxeter. Regular Complex Polytopes. Second edition. Dover Publications Inc., New York. 18. 1967 D. G. Wood. ‘Space Enclosure Systems’, Bulletin 203. Columbus, Ohio: Engineering Experiment Station, The Ohio State University. PP. 34, 3031 19. 1969? Keith Critchlow. Order in Space. A Design Source Book. Thames & Hudson. A date of 1969 is given in the book but it is unclear if this was when it was first published. The published date is apparently given as 1987. 2000 reprint. 20. 1970 Ernest R. Ranucci. Tessellation and Dissection. J. Weston Walch 21. 1970 H. S. M. Coxeter. 'Twisted Honeycombs' (CBMS Regional Conference Series in Mathematics), 1970, pp. 2123. 22. 1971 James A. Dunn. ‘Tessellations with Pentagons’. The Mathematical Gazette, Vol. 55, No. 394 (December, 366369) Finally, if the sides are all equal and x = x’ =90°, the tessellation in Figure 5 becomes Figure 6 which is shown in Cundy and Rollett and is a favourite streettiling in Cairo. The geometry of this basic pentagon is shown in Figure 7. 23. 1972 Robert Williams. The Geometrical Foundation of Natural Structure. A Source Book of Design. Dover Publications, Inc. 1979. Another edition, of another name, was of 1972. 24. 1974 Stanley R. Clemens. ‘Tessellations of Pentagons’. Mathematics Teaching, No. 67 (June), pp.1819, 1974 25. 1975 John Parker. ‘Tessellations’, Topics, Mathematics Teaching 70, 1975, p. 34 26. 1975 Martin Gardner. Scientific American. Mathematical Games, July. ‘On tessellating the plane with convex polygon tiles’, pp. 112117 (pp. 114, 116 re Cairo pentagon) Gardner Quote Scientific American 1975 ‘On Tessellating the Plane with Convex Tiles’, pp. 112117 P. 114: The most remarkable of all the pentagonal patterns is a tessellation of equilateral pentagons [‘c’]. It belongs only to Type 1*. Observe how quadruplets of these pentagons can be grouped into oblong hexagons, each set tessellating the plane at right angles to the other. This beautiful tessellation [of equilateral pentagons] is frequently seen as a street tiling in Cairo, and occasionally on in the mosaics of Moorish buildings. *errata (September 1975?) corrects this to Types 2 and 4 Gardner then gives the construction: The equilateral pentagon is readily constructed with a compass and straightedge…. (What I refer to as the ‘45° construction’) The second recorded attribution, based upon Dunn's account. 27. 1976 Marc G. Odier. ‘Puzzle with Irregular Pentagonal Pieces’. United States Patent 3,981,505 21 September 1976 Cairo tile diagram Fig. 3, and various patches of tiles formed with the pentagons. 28. 1976 Phares O’Daffer. G; Clemens, Stanley R. Geometry. An Investigative Approach 1^{st} edition, 2^{nd} edition 1992 AddisonWesley Publishing Company. (Note that I have the 2^{nd }edition, not the 1^{st}) 29. 1977 Lorraine Mottershead. Sources of Mathematical Discovery. Basil Blackwell. 30. 1977 Doris Schattschneider and Wallace Walker. M. C. Escher Kaleidocycles. Tarquin Publications. First edition, 1977; I have the ‘special edition’ of 1982. 31. 1978 Doris Schattschneider. Tiling the Plane with Congruent Pentagons’ Mathematics Magazine. ‘Vol.1, 51, No.1 January 1978. 2944. Tiling (3) can also be obtained in several other ways. Perhaps most obviously it is a grid of pentagons which is formed when two hexagonal tiles are superimposed at right angles to each other. F. Haag noted that this tiling can also be obtained by joining points of tangency in a circle packing of the plane [12]. It can also be obtained by dissecting a square into four congruent quadrilaterals and then joining the dissected squares together [26]. The importance of these observations is that by generalising these techniques, other pentagonal tiles can be discovered. 32. 1978 Peter Pearce and Susan Pearce. Polyhedra Primer. Dale Seymour Publications 33. 1978 Ernest H. Lockwood, and Robert H. Macmillan. Geometric symmetry. Cambridge University Press (and reprint 2008). 34. 1979 Robert H. Macmillan. Mathematical Gazette, 1979. ‘Pyramids and Pavements: some thoughts from Cairo’, pp. 251255 It will be seen that the pattern formed by the tile edges can also be taken as two interlinked and identical meshes. The question of interest is what may be the possible variations in the shape of these pentagons and hexagons. We can see that the slope of line CD in Fig. 4 can be varied, provided that the other dimensions are altered suitably. The geometric conditions to be satisfied are seen from Fig. 5 to be as follows:….. 35. 1980 Michael O’Keefe and B. G. Hyde. ‘Plane Nets in Crystal Chemistry’. Philosophical Transactions Royal Society London. Series A, 295 1980, pp. 553618 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. 36. 1982 George E. Martin. Transformation Geometry: An Introduction to Symmetry, p. 119 37. 1982 Patrick Murphy. Modern Mathematics Made Simple. Heinemann London Tessellations, Chapter 10, pp. 194205, 262. Cairo diagram, of equilateral pentagons (but not attributed) p. 200. 38. 1983 John Willson. Mosaic and Tessellated Patterns. How to Create Them. Dover Publications, Inc. 1983. Plate 3 Cairo tiling plate 3. (Neglected until 7 May 2013!) 39. 1983 Cyril Stanley Smith. A Search for Structure. The MIT Press, 1983 Has nonattributed Cairo tiling
...appearance of interlocking hexagons but consists of identical equal sided (but not equal angular) pentagons. The hexagonal patterns cross at right angles and the while pattern can be fit into a square or subdivided into modular squares. This unusual pattern, which is seen in street tiling in Cairo and occasionally in the mosaic of Moorish buildings. 41. 1986 James McGregor and AlanWatt. The Art of Graphics for the IBM PC, pp 196197 42. 1986 Ehud, BarOn. ‘A Programming Approach to Mathematics’. ‘A programming approach to mathematics’. Computers & Education 10(4): pp. 393401. December 1986. Elsevier. … then the possible ways of tiling with pentagons are explored, especially the Cairo tiling. 43. 1986 A. L. Loeb. 'Symmetry and Modularity'. Computers and Mathematics with Applications, Elsevier 44. 1986 Jay Kappraff. ‘A Course in the Mathematics of Design’. Computers and Mathematics with Applications Vol. 12B, Nos. 3/4, pp 913948 Cairo tiling in the context of the set of 11 Laves tiling; p. 923 but as such, inconsequential. 45. 1986 Lothar Collatz. Geometrische Oranamente (in German) Cairo tiling diagram in context of 43433 classification. 46. 1996 Michael O’Keefe and Bruce G. Hyde. Crystal Structures No. 1. Patterns & Symmetry. Mineralogical Society of America p. 207 The pattern is known as Cairo tiling, or MacMahon’s net and In Cairo (Egypt) the tiling is common for paved sidewalks… 47. 1987 Branko Grünbaum and Geoffrey C. Shephard. Tilings and Patterns. W. H. Freeman and Company 48. 1987 Bob Burn. The Design of Tessellations. Cambridge University Press. Sheet 30. Equilateral pentagon. 49. 1989 Dale Seymour and Jill Britton. Introduction to Tessellations. Dale Seymour Publications Cairo tiling (but not attributed) p. 39. 50. 1989 Piere De La Harpe. ‘Quelques Problèmes Non Résolus en Géométrie Plane’. L’Enseignement Mathématique, t 35 (1989), pp. 227243 (in French) 51. 1989. Marjorie Senechal. ‘Symmetry Revisited’. Computers and Mathematics with Applications. Vol 17, No. 13, pp112. 1989 Cairo diagram in the context of the set of 11 Laves diagrams, p. 9; as such per se, inconsequential. 52. 1989 W. K. Chorbachi. ‘In the Tower of Babel: Beyond Symmetry In Islamic Design’. Computers and Mathematics with Applications. Vol. 17, No. 46, pp 751789 (Cairo aspects 783794), 1989 (reprinted in I. Hargittai, ed. Symmetry 2: Unifying Human Understanding, Pergamon, New York, 1989. 54. 1990 Francis S. Hill. Jr. Computer Graphics. Macmillan Publishing Company, New York, P. 145. 56. 1991 David Wells. The Penguin Dictionary of Curious and Interesting Geometry. Penguin Books P. 23: So called because it often appears in the streets of Cairo, and in Islamic decoration. It can be seen in many ways, for example as cross pieces rotated about the vertices of a square grid, their free ends joined by short segments, or as two identical tessellations of elongated hexagons, overlapping at right angles. Its dual tessellation, formed by joining the centre of each tile to the centre of every adjacent tile, is a semiregular tessellation of square and equilateral triangles. P. 61: …Thus the dual of the tessellation of squares and equilateral triangles is the Cairo tessellation. P. 177: The regular pentagon will not tessellate. Less regular pentagons may, as in the Cairo tessellation…. The first line of p. 23 bears resemblance to Gardner's quote. 57. 1991 Jay Kappraff. Connections The Geometric Bridge Between Art and Science. McGrawHill. p. 181 Shown as the dual of 3. 3. 4. 3. 4 tiling. Poorly executed diagram, with four different pentagons! However, the intention, due to an accompanying diagram, is indeed clear. 58. 1993 Nenad Trinajstic. The Magic of the Number Five. Croatia Chemica Acta 66 (1) 227254 ... seen in street tiling in Cairo and occasionally in the mosaic of Moorish buildings. Seemingly quoting Blackwell. 59. 1994 Audrey Leathard. Going interprofessional: working together for health and welfare 60. 1994 Carter Bays. Complex Systems Publications, Volume 8, Issue 2, 127150, Cairo aspect p. 148 61. 1996 Michael O’Keefe and Bruce G. Hyde. Crystal Structures. 1. Patterns & Symmetry. Mineralogical Society of America p. 207 The pattern is known as Cairo tiling, or MacMahon’s net and In Cairo (Egypt) the tiling is common for paved sidewalks… Not fully seen, Google Books reference? 62. 1997 Michael Serra. Discovering Geometry: An Inductive Approach. Key Curriculum Press, p. 404 63. 1998 David A. Singer. Geometry Plane and Fancy, 1998, p.34. SpringerVerlag 64. 1999 Ian Stewart. ‘The Art of Elegant Tiling’. Scientific American. July 1999, pp. 9698 65. 1999 Jinny Beyer. Designing Tessellations, Contemporary Books, p. 144. 66. 2001 Edward Duffy, Greg Murty, Lorraine Mottershead. Connections Maths 7. Pascal Press, p. 83 67. 2003 Teacher’s Guide: Tessellations and Tile Patterns, p. 30 (Cabri) Geometric investigations on the VoyageTM 200 with Cabri. Texas Instruments Incorporated 68. 2003 Catherine A Gorini. The Facts on File Geometry Handbook. 2003, 2009 revised edition. Facts on File Inc, and imprint of Infobase publishing Cairo tiling illustrated p. 22, equilateral. Gives the following definition: Cairo tessellation: A tessellation of the plane by congruent convex equilateral pentagons that have two nonadjacent right angles; so called because it can be found on streets in Cairo. Oddly, Gorini shows an accompanying picture of a pentagon that is not equilateral, a 4, 1 type… 69. 2003 Chris Pritchard. The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching 70. 2003 Eric W. Weisstein. CRC concise encyclopedia of mathematics, p. 313 Interesting in that Weisstein defines this as the ‘projection of a dodecahedron’ before the dual. 71. 2004 Robert Parviainan. ‘Connectivity Properties of Archimedean and Laves Lattices’. Uppsala Dissertations in Mathematics 34. p. 9. 2004. 72. 2005 David Mitchell. Sticky Note Origami: 25 Designs to make at your desk, Sterling Publication Company 73. 2005 George McArtney Phillips. Mathematics Is Not a Spectator Sport. P. 193. Springer Problem 6. 5. 3 Construct a dual of the 3. 3. 4. 3. 4 tessellation by joining the centers of adjacent polygons. This is called the Cairo tessellation. Observe that it has a pentagonal motif that has four sides of one length and one shorter side 74. 2005 Sue Johnston Wilder and John Mason. Developing Thinking in Geometry, P. 182. … is often referred to as the Cairo tessellation as it appears in a mosque there. 75. 2005 Carter Bays. Complex Systems Publications, Volume 15, Issue 3, 245252, Cairo aspect p. 249250 ‘A Note on the Game of Life in Hexagonal and Pentagonal Tessellations’ ‘Here we have chosen the Cairo tiling…’ A curiosity, with the Cairo tiling acting as a backdrop on the Game of Life. 76. 2005 Paul Garcia. ‘The Mathematical Pastimes of Major Percy Alexander MacMahon. Part 2 triangles and beyond’. Mathematics in Schools, September 2005, 2022. PDF Contains a Cairo tiling 'of sorts', p. 22
77. 2006 John Sharp. ‘Beyond Su Doku’. Mathematics Teaching in the Middle Years. Vol. 12, No. 3 October 2006 pp. 165169 Cairo tiling on pp. 167169, in the context of a ‘Cairo Su Doku’.78. 2006 Mark Eberhart. Excerpts selected by Mark Eberhart in Resonance from C. S. Smith's A Search for Structure, of which p. 87 contains a Cairo tiling 79. 2007 B. G. Thomas and M. A. Hann. in Sarhangi, Reza (Ed). Bridges. Mathematical Connections in Art, Music, and Science. (Ninth) Conference Proceedings 2007. Donostia, Spain. Patterned Polyhedra: Tiling the Platonic Solids 80. 2007 Mike Ollerton. 100+ Ideas for Teaching Mathematics p. 66 81. 2008 Anon. Key Curriculum Press. Chapter 7 Transformations and Tessellations, p. 396 82. 2008 Merrilyn Goos et al. Teaching Secondary School Mathematics: Research and Practice for the 21^{st} Century. 83. 2008 B. G. Thomas and M. A. Hann. In Sarhangi, Reza (Ed). Bridges. Mathematical Connections in Art, Music, and Science. (Tenth) Conference Proceedings 2008. Leeuwarden, Netherlands 84. 2008 Robert Fathauer. Designing and Drawing Tessellations, p. 2. 85. 2008 B. G. Thomas, B. G. and M. A. Hann. In Bridges. Mathematical Connections in Art, Music, and Science. Type of pentagon: Equilateral (p. 101). 86. 2008 Birgit Kaltenmorgen. Der mathematische Patchworker. (in German) Wagner, Gelnhausen; 1^{st} edition pp. 8283 87. 2009 Craig S. Kaplan. Introductory Tiling Theory for Computer Graphics. Morgan & Claypool Publishers, p. 33 The Laves tiling [3^{2}. 4. 3. 4] is sometimes known as the ‘Cairo tiling’ because it is widely used there. p. 103 Not seen, Google Books reference. 88. 2009 Mike Ollerton. The mathematics teacher's handbook, p. 148 … use four different colours to make the 'Cairo' tiling design. Not seen, Google Books reference. 89. 2010 Claudi Alsina and Roger B. Nelsen. Charming Proofs: A Journey Into Elegant Mathematics. Dolciani Mathematical Expositions 90. 2011 Richard Elwes. Maths 1001: Absolutely Everything You Need to Know about Mathematics in 1001 BiteSized Explanations. Quercus, p. 109 91. 2011 Abdul Karim Bangura. African Mathematics: From Bones to Computers University Press of America, 2011 92. 2011 Eric Goldemberg. Pulsation in Architecture p.338 93. 2012 Christoph A. Kilian (ed), Norbert Palz, Fabian Scheurer 94. 2013 Lisa Iwamoto. Digital Fabrications: Architectural and Material Techniques. Princeton Architectural Press; first edition 2009
N.B. An apparent Cairo tiling in a 1923 paper by F. Haag, "Die regelmässigen Planteilungen und Punktsysteme." Zeitschrift fur Kristallographie 58 (1923): 478488, Figure 13, in that it is frequently quoted as a pentagonal tiling is misleading; it's not a pentagon, but rather a quadrilateral.
Web References
Wolfram MathWorld
Wikipedia
Created on: 9 September 2011. 2011 22 September, 26 October, 1 November 2011; 21 April, 12 May, 3 December (D. G. Wood), 10 December (R. Parviainan). All revised and enlarged. 2012 14 December 2012. Wholesale revision, with quotes added to Section 1, and colour coding removed from Sections 1 and 2, of which although well intentioned, was a little contrived, and did not make for easy reading. 2013 4 October 2013. General tidyup. 2014 10 July 2014. Moore and Odier references added. 2015 7 January 2015. Haag (2) and Laves references added, inexplicably omitted previously.
2018 30 October 2018. Smith (2), Trinajstic, Collatz, Eberhart entries added. 2019 21 June 2019. Added a new major section, of nonattributed references (rather late in the day!), excised from the combined listing. This now better 'balances' the different sections. Detail added to Simon Ray and Heidelberg Castle entries, replacing mere one liners. Also a general tidying up.

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