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References

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, typically of four equal lengths, with either a 'short' or 'long' base, although an equilateral pentagon is possible, with two opposite right angles. The list is shown in three different ways, as according to various filters:

  1. A simple listing of attributed references i.e. mentioned in association with Cairo, e.g. 'is a favourite street-tiling in Cairo', arranged as according to chronology, with referral to the quote/diagram in question, in effect a simple bibliographic listing.
  2. A simple listing of non-attributed references i.e. not mentioned in association with Cairo, e.g. 'a pentagon tiling', arranged as according to chronology, with referral to the quote/diagram in question, in effect a simple bibliographic listing.
  3. A simple listing of attributed or not references of the Cairo tiling as a pentagon per se, of a 'standard model', with quotes and comments thereof where appropriate. In short, this listing combines the references of 1 and 2.

Can anyone add to this list, with either other references (the earlier the better) or of any additional further information?

Note that although most of these publications are in my possession, not all all; some were found on Google Books. In short, there is simply not time in the day to pursue 'all books', even as much as I would to! Some are simply beyond any practical use, say in metallurgy, and it is judged not worth the time and expense (and ever diminishing shelf space!) in obtaining. Where I have used Google books search facility, simply state 'Not seen, Google Books reference'.

On occasions, the numerical listing will be slightly out of sync, caused by a later entry. As may be imagined, especially for an 'early' entry, this will entail a lot of time consuming updating successive instances, and so with other additions at whim, I tend to update when there is 'obvious' need, broadly yearly. 



1. A simple listing of attributed references, mentioned in association with Cairo, e.g. 'is a favourite street-tiling 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. 366-369
…and is a favourite street-tiling in
Cairo.
A first-hand sighting, one of only two; also see Macmillan. Almost certainly the first account. 

2. 1975 Martin Gardner. Scientific American. Mathematical Games, July. ‘On tessellating the plane with convex polygon tiles’, pp.112-117 (p. 114 and 116 re Cairo aspect)
This beautiful tessellation is frequently seen as a street tiling in
Cairo, and occasionally on in the mosaics of Moorish buildings.
The second recorded attribution. Gardner’s account is interesting, in different ways. As such, this is not a first-hand sighting; although not stated as such, upon research this is based on James A. Dunn’s paper. The ‘Moorish buildings’ aspect is a reference to Richard K. Guy’s account of a supposed sighting now known to be supposedly at the Taj Mahal, almost certainly a mistaken account on his (Guy's) part (I asked him!).


3. 1978 Doris Schattschneider. Mathematics Magazine, January. ‘Tiling the Plane with Congruent Pentagons’, pp. 29-44 (p. 30 re
Cairo aspect).
It is said to appear as a street paving in
Cairo.
Likely referring to Martin Gardner or James Dunn’s quote; both authors are mentioned in the bibliography.

4. 1979 Robert H. Macmillan. Mathematical Gazette, 1979. ‘Pyramids and Pavements: some thoughts from Cairo’, pp. 251-255.
Many of the streets of
Cairo are paved with a traditional Islamic tessellation of pentagonal tiles, as shown in Fig. 4. The pentagons are all identical in size and shape, having four sides equal and two of their angles 90°…
A firsthand sighting, the second of only two, and so of the utmost significance; also see Dunn. No reference is made in the article itself or the references to any of the three above articles, and so this is likely an independent account, as a ‘discovery’. As such, this is a little surprising, in that Dunn’s article was also from the Mathematical Gazette! Of note is the reference to the tiles being coloured, or arranged of the same colour, ‘back to back’, this being the first recorded instance; indeed, the only one!

5. 1982 George E. Martin. Transformation Geometry: An Introduction to Symmetry, p. 119.
The beautiful
Cairo tessellation with a convex equilateral pentagon as its prototile is illustrated in Fig. 12.3. The tessellation is so named because such tiles were used for many streets in Cairo.
Likely referring to Martin Gardner or James Dunn’s quote.

6. 1984 William Blackwell. Geometry in Architecture, p. 54, Wiley 1984.
This unusual pattern, which is seen in street tiling in
Cairo and occasionally in the mosaic of Moorish buildings
Likely taking from the Gardner quote, as the latter part is almost word for word.

7. 1986 James McGregor and Alan Watt. The Art of Graphics for the IBM PC Addison Wesley 1986 pp. 196-197.

…is the well-known Cairo tile, so called because many of the streets of Cairo were paved in this pattern.

8. 1986 Ehud, Bar-On. ‘A programming approach to mathematics’. Computers & Education 10(4): pp. 393-401. December 1986. Elsevier.

… especially the Cairo tiling.


8. 1986 George E. Andrews. Percy Alexander MacMahon: Number theory, invariants, and applications. MIT Press, 1986 p. 196 Google Books

It is said to appear as street paving in Cairo (Purposefully re-quoting Schattschneider (1978))

9. 1989 W. K. Chorbachi. Computers and Mathematics with Applications. ‘In the Tower of Babel: Beyond Symmetry In Islamic Design’. Vol. 17, No. 4-6, pp. 751-789 (Cairo aspects pp. 783-794)
The pattern of a favorite street tiling in Cairo
Likely quoting from Dunn, as he is mentioned in the article (Note US spelling of favourite, note that Chorbachi also omits the dash between ‘favourite’ and ‘street’.

10. 1989 Pierre De La Harpe. Quelques Problèmes Non Résolus en Géométrie Plane. L’Enseignement Mathématique, t 35 (1989), pp. 227-243 (in French) Cairo tiling on p. 232
…dans les rues du Caire (…on the streets of
Cairo).
Likely taken from George Martin, given that the (‘unusual’) configuration of the diagram is the same.

11. 1990 Francis S. Hill. Computer Graphics. Macmillan Publishing Company, New York, p. 145.
An equilateral pentagon can tile the plane, as shown in Figure 5.4. This is called a
Cairo tiling because many streets in Cairo were paved with tiles using this pattern…
Likely quoting from McGregor and Watt, given that the text is very much alike, and their work is quoted and illustrations are used in the book.

12. 1991 Ann E. Fetter. et al. The Platonic Solids Activity Book. Key Curriculum Press/Visual Geometry Project. Backline Masters, pp. 21, 97
This pattern is seen in street tiling in
Cairo and in the mosaics of Moorish buildings.
Likely referring to the Gardner quote, both parts are almost word for word.

13. 1991 David Wells.The Penguin Dictionary of Curious and Interesting Geometry. Penguin Books, pp. 23, 61, 177.
So called because it often appears in the streets of
Cairo, and in Islamic decoration.
Likely referring to the Gardner quote, both parts are almost word for word.

14. 1993 Nenad Trinajstić. 'The Magic of the Number Five'. Croatia Chemica Acta 66 (1) pp. 227-254.

 ... 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 inter-professional: working together for health and welfare. Routledge; first edition 1994
In the Cairo tessellation (Wells 1991)…
Quotes the Wells reference. A very minor account. Note that this reference is only included for the sake ‘of everything’; the book is apparently of a non-mathematical nature, and is not illustrated with the tiling.

16. 1994 Carter Bays. Complex Systems Publications, Volume 8, Issue 2, pp. 127-150, Cairo aspect p. 148
'Cellular Automata in the Triangular Tessellation’.
… the Cairo tessellation (a tiling of identical equilateral pentagons)…
A cursory mention in passing.

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.
One particularly elegant tiling of the plane by pentagons is known as the
Cairo tessellation, because it can be seen as a street tiling in Cairo….

19. 1997 Michael Serra. Discovering Geometry: An Inductive Approach. Key Curriculum Press, p. 404
The
Cairo street tiling shown at right is a very beautiful tessellation that uses equilateral pentagons (the sides are congruent but not the angles).
Not seen, Google Books reference.

20. 2000 M. Deza et al. 'Fullerenes as tilings of surfaces'. Journal of Chemical Information Computer and Modelling. ACS Publications, pp. 550-558
… is the Cairo tiling…

21. 2001 Edward Duffy, Greg Murty, Lorraine Mottershead. Connections Maths 7. Pascal Press, p. 83
Cairo streets have this Islamic pattern

22. 2003 Teacher’s Guide: Tessellations and Tile Patterns, p.30 (Cabri) Geometric investigations on the VoyageTM 200 with Cabri. Texas Instruments Incorporated
….Probably the most famous of these pentagonal patterns is the ‘
Cairo Tessellation’ named after the Islamic decorations found on the streets of Cairo
Begins by quoting David Wells’ book … Curious… and likely the text is based on his reference. However, the ‘Teacher’s Guide’ gives a different tiling, interestingly a ‘collinear pentagon.

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.
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…

24. 2003 Chris Pritchard. The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching, P. 421-427, Cairo aspect p. 421.
Is a favourite street tiling in
Cairo
This is an anthology, and simply repeats Dunn’s article, and follow-up correspondence. Nothing original is shown.
Not seen, Google Books reference.

25. 2004 Robert Parviainan. ‘Connectivity Properties of Archimedean and Laves Lattices’. Uppsala Dissertations in Mathematics 34. p. 9. 2004.
The lattice D (32. 4. 3. 4) is sometimes called the Cairo lattice, as the pattern occurs frequently as tilings on the streets of Cairo.

26. 2005 David Mitchell. Sticky note origami: 25 designs to make at your desk, Sterling Publication Company, pp. 58-61.
The Cairo Tessellation is an attractive and intriguing pattern of tiles named as a result of its frequent occurrence on the streets of Cairo and in other Islamic centers and sites

27. 2005 George McArtney Phillips. Mathematics Is Not a Spectator Sport. Springer, p. 193

This is called the Cairo tessellation.

28. 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.

29. 2005 Carter Bays. Complex Systems Publications, Volume 15, Issue 3, 245-252, Cairo aspect pp. 249-250
‘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. 165-169 pp. 167-169, in the context of a ‘Cairo Su Doku’.
Cairo tile So Doku with two overlapping hexagons.

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. 52-53, 70-71, 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. 2007 Kindt, Martin. ‘Wat te bewijzen is’ (in Dutch) (38) (translated ‘What is to be proved’). Nieuwe Wiskrant 27-1 September 2007
Article on Cairo tiling, 35-36, with initial reference to David Wells. The Nieuwe Wiskrant’, a Dutch journal for mathematics and computer science education, provided news of recent developments in these areas, and appeared quarterly between September 1981 and June 2013. Its focus was mainly on secondary education.

33. 2008 Anon. Key Curriculum Press. Chapter 7, Transformations and Tessellations, p. 396.
The beautiful
Cairo street tiling shown below uses equilateral pentagons.
Does anyone know of this book? I found it as a ‘part PDF’, without a title.

34. 2008 Merrilyn Goos et al. Teaching Secondary School Mathematics: Research and Practice for the 21st Century.
The particular tiling pattern of an irregular pentagon, shown in Figure 9.16, is called the
Cairo tessellation because it appears in a famous mosque in Cairo.
Not seen, Google Books reference. Very curious; the ‘famous mosque’ has evaded detection! Likely quoting, and extrapolating, from Gardner.

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’
There are, however, equilateral convex pentagons that do tessellate the plane, such as the well known
Cairo tessellation shown in Figure 1.
Also, other minor references essentially in passing.

36. 2008 B. G. Thomas and M. A. Hann. In Sarhangi, Reza (Ed). Bridges. Mathematical Connections in Art, Music, and Science, p. 101.
Probably the best known is the
Cairo tessellation.


37. 2008 Birgit Kaltenmorgen. Der mathematische Patchworker. (in German) Wagner, Gelnhausen;  first  edition  2008, pp. 82-83
Fünfeck beim Cairo-Tiling.

38. 2008 Robert Fathauer. Designing and Drawing Tessellations, 2008, p. 2.
A common street paving in
Cairo, Egypt is shown above left.

39. 2009 Craig S. Kaplan. Introductory Tiling Theory for Computer Graphics. Morgan & Claypool Publishers, p. 33.
The Laves tiling [32. 4. 3. 4] is sometimes known as the ‘
Cairo tiling’ because it is widely used there.

40. 2010 Claudi Alsina and Roger B. Nelsen. Charming Proofs: A Journey Into Elegant Mathematics. Dolciani Mathematical Expositions, p. 163.
This rather attractive monohedral pentagonal tiling is sometimes called the
Cairo tiling, for its reported use as a street paving design in that city.

41. 2010 Richard Elwes. Maths 1001: Absolutely Everything You Need to Know about Mathematics in 1001 Bite-Sized Explanations. Quercus, p. 109
it adorns the pavements of that city’ (
Cairo).

42. 2010 E. Ressouche et al. Magnetic frustration in an iron-based
Cairo pentagonal lattice. Physical Review Letters.
A famous one being the
Cairo tessellation whose name was given because it appears in the streets and in many Islamic decorations

43. 2011 M. Rojas et al. 'Frustrated Ising model on Cairo pentagonal lattice'Physical Review E 86(5-1):051116 November 2012
Cairo pentagonal lattice

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
Housing Exhibition in Vienna, Austria Project Description The Cairo Pods gave SPAN ... The Cairo Tessellation, known in mathematics also as an example of ...

45. 2011 Q. Ashton Acton (ed). Issues in General Physics Research: SchorlarlyAdditions, 2011. Google Books

Iron-Based Cairo Pentagonal Lattice


46. 2011 Bangura, Abdul Karim. African Mathematics: From Bones to Computers. University Press of America 2011. Not seen, Google Books

A basketweave tiling is topologically identical to the Cairo pentagonal tiling (not illustrated), p.113


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
… project uses a pentagonal Cairo tessellation pattern, flexibly aggregated to yield multiple overall arrangements. Each vertical layer of the cell was ...


48. 2013 Toshikazu Sunada. Topological Crystallography: With a View Towards Discrete Geometric Analysis. Springer, 2013. GB 

P. 132 Cairo pentagon (caption)

8.2 Cairo Pentagon Fig. 8.3 Merging two square lattices Figure 8.2 is a tiling of pentagons with picturesque properties that has become known as the Cairo pentagon.

Not seen, Google Books reference.


49. 2013 Gyynn, Ruairi and Bob Sheil (eds). Fabricate 2011: Making Digital Architecture. UCL Press, pp. 196-201 Riverside Architectural Press, 2013.  Joe MacDonald, ‘The Agency of Constraints’.

The Cairo hexagon (sic)... The streets of Cairo are paved with stones of this geometry.


49. 2014 Benölken, Ralf, Hans-Joachim Gorski and Susanne Müller-Philipp. Leitfaden Geometrie: Für Studierende der Lehrämter. Springer, 2014. In German. Translated: Guideline Geometry: For students of the teaching offices. p. 203

In abbildung 133 ist die parkettierung ‘Cairo tiling’ dargestellt

Translated: Figure 133 shows the parqueting 'Cairo tiling'


50. 2014 David E. Laughlin and ‎Kazuhiro Hono. Physical Metallurgy. P 76 5th edition. GB

The nets in … the Catalan Cairo pentagonal tiling V32.4.3.4

Not seen, Google Books reference.


51. 2016 Walter Steurer and Julia Dshemuchadse. Intermetallics: Structures, Properties, and Statistics. OUP Oxford. p. 565. GB

... 34 Cairo pentagon tiling 496 

Not seen, Google Books reference.


52. 2017 Robert J. Lang. Twists, Tilings, and Tessellations: Mathematical Methods for Geometric Origami. CRC Press. GB Left: “Cairo Tessellation” (2011), a flagstone tessellation by Eric Gjerde, based on the Cairo tiling.

Not seen, Google Books reference.


53. 2017 Aakash Moncy. 'Mechanics of Cairo lattices'. Thesis

Numerous references throughout, too many to list.


53. 2017 Ponnadurai Ramasami (ed). Computational Sciences. De Gruyter, 2017. Google Books

P. 57 ...similar to Cairo pentagonal tiling.


53. 2017 Changzheng Wu (ed). Inorganic Two-dimensional Nanomaterials: Fundamental Understanding , Characterizations and Energy Applications (Smart Materials Series).  Royal Society of Chemistry; First Edition, 2017. Google Books

(Pentagraphene)… resembling the Cairo pentagonal tiling


54. 2019 Frank Morgan ‘My Undercover Mission to Find Cairo Tilings’. The Mathematical Intelligencer, September 2019, Volume 41, Issue 3, pp 19–22

References throughout. On his recent visit of the same year, following up the report on my page as to sightings.


55. 2019 Mircea Pitici (ed). The Best Writing on Mathematics. Princeton University Press, 2019. 114-116. Google Books

A portion of the Cairo tiling. 

N. J. Sloane on Chaim Goodman-Strauss’ ‘coloring book’ method.


55. 2019 Goodman-Strauss, Chaim and N. J. A. Sloane. ‘A Coloring Book Approach to Finding Coordination Sequences’, Acta Crystallographica Section A: Foundations and Advances, 2019, Volume A75, pp 121-134.

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



2. A simple listing of non-attributed 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. 178-179.

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 Goodman-Strauss, 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 ‘early-mid 20th century’.

3. 1909 Herbert C. Moore. ‘Tile’. United States Patents 928,320 and 928,321 of 20 July 1909
The first recorded instance, of a patent for a flooring. The tiling also appears in the second patent. As far as I am aware, no one has quoted his work in the Cairo tiling. Does anyone know of Moore at all? He is connected with Boston, Massachusetts, USA. Were the tiles actually made? Of note here is that Moore in principle shows a minimum and maximum deformations of the pentagon, to an implied rectangle and square, as outlined in more detail by Macmillan in his 1979 paper.

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), 567-578. Related discussion ibid pp. 578-582
Of note is that MacMahon refers to a ‘haystack’, meaning a Cairo tile, p. 573, after fig. 13. This term is also interestingly used by him in New Mathematical Pastimes. His nephew W. P. D. MacMahon also uses this word (haystack) in ‘The theory of closed repeating polygons…’, so confusion arises as to whom exactly determined the tile.

6. 1925 Friedrich Haag. 'Die pentagonale Anordung von sich berührenden Kriesen in der Ebene’. Zeitschrift fur Kristallographie 61  (1925), 339-340
Has Cairo tiling in the form of circle packing

7. 1926 Friedrich Haag. 'Die Symmetrie verhältnisse einer regelmässigen Planteilung’. Zeitschrift für mathematischen und naturwissenschaftlichen Unter-richt, Band 57 (1926), 262-263

Has Cairo tiling in the form of circle packing.


8. 1931 Fritz Laves. ‘Ebenenteilung in Wirkungsbereiche’. Zeitschrift für Kristallographie 76 (1931): 277-284. 


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 oft-quoted by Schattschneider, but surprisingly no one else.

P. 123 Cairo-like 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).
‘We have space for one of his; [MacMahon’s] it consists of equal-sided (but not regular) pentagons, but has the appearance of interlocking hexagons (Fig. 58)’
Cairo diagram (but not attributed) p. 63 (picture) and p. 65 (text). The diagram is derived from MacMahon’s book, as Cundy freely credits.

11. 1951 'Croton'. Cairo tiling used as a crossword puzzle, in The Listener, 13 December 1951, puzzle 1128 Hexa-Pentagonal I, by 'Croton'.
'Croton' is a pseudo-name; it's somewhat of a long shot given the time passed since the puzzle’s inception, but the does anyone know he is?

12. 1954 'Croton'. Cairo tiling used again as a crossword puzzle, in The Listener, 22 April 1954, puzzle 1251 Hexa-Pentagonal 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 non-stated ‘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.
Cairo diagram (but not attributed) on the cover of seemingly the second edition only. Interestingly, this is likely the first instance of using different coloured subsidiary hexagons to better feature the overlapping hexagon aspect.
The type of pentagon is not clear due to the nature of the drawing, with somewhat thick lines, but it would appear to be equilateral. An open question is does this appear on (or in) the first edition of 1947? I have not got the book to hand.

18. 1967 D. G. Wood. ‘Space Enclosure Systems’, Bulletin 203. Columbus, Ohio: Engineering Experiment Station, The Ohio State University. PP. 3-4, 30-31
Wood (a professor of industrial design rather than a mathematician) makes a curious observation as regards tilings with equal length sides, with the later to be known Cairo tiling being one of five such instances (the equilateral triangle, square, Cairo pentagon, hexagon, rhomb); as such, I do not recall seeing this simple observation elsewhere. Is this significant? Much of Wood’s work here, and elsewhere in the book, is in regards to prisms, of which he shows a ‘Cairo’ prism. Does anyone know of Wood? At the time of writing, he would be 99. Is he still alive? Did he do anything further with the tiling? He freely credits both MacMahon and Cundy and Rollett as the source of the pentagon per se, the observation of his appears to be his own.

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.
Cairo diagram (but not attributed) p. 49, but no text. This also has an interesting series of diagrams p. 83, best described as ‘variations’ with Cairo-like properties, with ‘par hexagon pentagons’ combined in tilings with regular hexagons.

20. 1970 Ernest R. Ranucci. Tessellation and Dissection. J. Weston Walch
Cairo-like diagram (but not attributed) p. 36 (picture and text).The inclusion of this Cairo of Ranucci’s is somewhat open to question, given that the diagram consists of two pentagons, rather than the given ‘standard model’ of one. Nonetheless, it is of interest due to the first example of this type.


21. 1970 H. S. M. Coxeter. 'Twisted Honeycombs' (CBMS Regional Conference Series in Mathematics), 1970, pp. 21-23.


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.18-19, 1974
Cairo diagram (but not attributed) p. 18. Interesting in that this credits MacMahon as the discoverer of the equilateral pentagon (p. 19), although this is not substantiated. Likely, reading from MacMahon’s book, he just assumed this.

24. 1975 John Parker. ‘Tessellations’, Topics, Mathematics Teaching 70, 1975, p. 34
Building on Clemens, immediately above, as noted by Parker. Loosely a Cairo diagram (but not attributed) 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 1st edition, 2nd edition 1992 Addison-Wesley Publishing Company. (Note that I have the 2nd edition, not the 1st)
While a regular pentagon will not tessellate the plane, it is interesting to note that there is a pentagon (see region A in Fig. 4.15) with all sides congruent [i.e. equilateral] (but with different size angles) that will tessellate the plane. A portion of this tessellation is shown in Fig. 4. 15. If four of these pentagonal regions are considered together (see Region B), an interesting hexagonal shape results that will tessellate the plane.
Cairo diagram (but not attributed) p. 95 (text continues to p. 96).

27. 1977  Lorraine Mottershead. Sources of Mathematical Discovery. Basil Blackwell.
Cairo diagram (but not attributed) pp.106-107 on a chapter on tessellations, and a subset of irregular pentagons.
Of note is the use of the Cairo tiles as a letter puzzle; although this is not original with Mottershead, as perhaps might appear at first sight (as I did myself to 2012). Although titled ‘… by Croton’, no further detail of ‘Croton’ is given. This diagram has now been determined as to appearing in The Listener, as detailed above, see 5. 1951 and 6. 1954. Unfortunately, the determination as to which types of pentagon are here is fraught with difficulty due to such a small scale drawing and the accuracy of the drawing is also in question, of which I am not prepared to be categorical as to the type of pentagon here. They could be equilateral, or near.

28. 1977 Doris Schattschneider and Wallace Walker. M. C. Escher Kaleidocycles. Tarquin Publications. First edition, 1977; I have the ‘special edition’ of 1982.
One of Escher’s favourite geometric patterns was the tiling by pentagons shown (Figure 35). These pentagons are not regular since their angles are not all equal.
Cairo diagram (but not attributed) p. 26, also see p. 34, in the context of a dodecahedron tiling decoration and Escher’s ‘Flower’, PD 132.
The type of Cairo tiling is not explicitly stated; certainly, it is of a 4, 1 type, likely of the dual of the 3. 3. 4. 3. 4 type (90°, 120°), but Escher did not use this!

29. 1978 Peter Pearce and Susan Pearce. Polyhedra Primer. Dale Seymour Publications
Cairo diagram (but not attributed) on p. 35 and in the context of the dual tilings of the semiregular tilings, p. 39. Decidedly lightweight, no discussion as such.

30. 1978 Ernest H. Lockwood, and Robert H. Macmillan. Geometric symmetry. Cambridge University Press (and reprint 2008).
‘Indirect’ 
Cairo reference p. 88
… are patterns [semi regular] of congruent pentagons such as are often used for street paving in Moslem countries.
The inclusion of this book is somewhat of a moot point, in that
Cairo tiles are described very loosely here. However, as it is by Macmillan, this rather fragmentary account is worthy of note, and curiously it does not strictly tally with his later Mathematical Gazette article.

31. 1980 Michael O’Keefe and B. G. Hyde. ‘Plane Nets in Crystal Chemistry’. Philosophical Transactions Royal Society London. Series A, 295 1980, pp. 553-618

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. 194-205, 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 non-attributed Cairo tiling.


35. 1986 James McGregor and Jim Watt. The Art of Graphics for the IBM PC, pp 196-197

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 well-known 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.
A minor part of a chapter on tessellations. Diagram p. 197.

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 913-948

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. 1986 R. Mosseri and J. Sadoc. ‘Polytopes and Projection Method: An Approach to Complex Structures’. Journal de Physique Colloques, 1986, 47 (C3), pp. C3-281-C3-297.

Fig. 4: A tiling by pentagons with coordination 3 or 4 obtained with a squares tesselation (sic) decorated like in fig. 3b. Cairo tiling on p. C3-285.


39. 1987 Bob Burn. The Design of Tessellations. Cambridge University Press. Sheet 30. Equilateral pentagon.


40. 1987 Rudy v. B Rucker. Mind Tools: The Five Levels of Mathematical Reality. First Edition, Houghton Mifflin Company, Boston, 1987.

Fig. 39 Tessellation with irregular pentagons. And  If we give up the requirements that each tile be a regular polygon and that each corner look the same, a great many strange tessellations can be found. One very attractive one is made of irregular pentagons and is often used as a cobblestone pattern in Europe and the Near East.

Whether this quote is referring to the Cairo tiling is unclear.


40. 1989
 Dale Seymour and Jill Britton. Introduction to Tessellations. Dale Seymour Publications Cairo tiling (but not attributed) p. 39.

The exact pentagon not described, almost certainly the dual of the 3. 3. 4. 3. 4 (90°, 120° type).
Lightweight.

41. 1989 Piere De La Harpe. ‘Quelques Problèmes Non Résolus en Géométrie Plane’. L’Enseignement Mathématique, t 35 (1989), pp. 227-243 (in French)
Cairo tiling on p. 232, likely taken from George Martin, given that it is the same ‘unusual’ configuration

42. 1989. Marjorie Senechal. ‘Symmetry Revisited’. Computers and Mathematics with Applications. Vol 17, No. 1-3, pp. 1-12. 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. 783-794.
Not seen, Google Books reference.
 

44. 1991  Jay Kappraff. Connections The Geometric Bridge Between Art and Science. McGraw-Hill. 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. 96-98.

Minor instance of coloured Cairo tiling, p. 97, as devised by Rosemary Grazebrook.

46. 1999 Jinny Beyer. Designing Tessellations, Contemporary Books, p. 144.
Lightweight.


47. 2005 Paul Garcia. ‘The Mathematical Pastimes of Major Percy Alexander MacMahon. Part 2 triangles and beyond’. Mathematics in Schools, September 2005, pp. 20-22. 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. 165-169.

Cairo tiling on pp. 167-169, 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 well-known Cairo tessellation shown in Fig. 1.

Type of pentagon: Equilateral (p. 101).


51. 2014 Hans-Joachim Gorski  and Susanne Müller-Philipp. Leitfaden Geometrie: Für Studierende der Lehrämter, Springer p. 186.


52. 2016 Mark Neyrinck. ‘The Origami Cosmic Web’,  The Paper, No. 122, 26-27

How a 2D universe would fold up to form a [Cairo] pentagonal tiling of voids.


52. 2016 Simon Ray. Indian & Islamic Works of Art. Self Published, pp. 178-179.

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 Goodman-Strauss, 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!

53. 2017 Ed Southall and Vincent Pantaloni. Geometry Snacks. Tarquin, 2017, 90 pp. NOT SEEN Cairo tiling as an equilateral pentagon. Area problem p. 74, #47. Solution p. 82.

N.B. An apparent Cairo tiling in a 1923 paper by F. Haag, "Die regelmässigen Planteilungen und Punktsysteme." Zeitschrift fur Kristallographie 58 (1923): 478-488,

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. 178-179.

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 Goodman-Strauss, 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 ‘early-mid 20th century’.

3. 1909 Herbert C. Moore. ‘Tile’. United States Patents 928,320 and 928,321, of 20 July 1909.
The first recorded instance of a patent, for a flooring. The tiling also appears in the second patent. As far as I am aware, no one has quoted his work in the Cairo tiling. Does anyone know of Moore at all? He is connected with Boston, Massachusetts, USA. Were the tiles actually made? If so, there is no evidence. Of note here is that Moore in principle shows a minimum and maximum deformations of the pentagon, to an implied rectangle and square, as outlined in more detail by Macmillan in his 1979 paper.

4. 1921 Percy A. MacMahon. ANew Mathematical PastimesCambridge 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), 567-578. Related discussion ibid pp. 578-582
Of note is that MacMahon refers to a ‘haystack’, meaning a Cairo tile, p. 573, after fig. 13. This term is also interestingly used by him in New Mathematical Pastimes. His nephew W. P. D. MacMahon also uses this word (haystack) in ‘The theory of closed repeating polygons…’, so confusion arises as to whom exactly determined the tile.

6. 1925 Friedrich Haag. 'Die pentagonale Anordung von sich berührenden Kriesen in der Ebene’. Zeitschrift fur Kristallographie 61  (1925), pp. 339-340
Has Cairo tiling in the form of circle packing.

7. 1926 Friedrich Haag. 'Die Symmetrie verhältnisse einer regelmässigen Planteilung’. Zeitschrift für mathematischen und naturwissenschaftlichen Unter-richt, Band 57 (1926), pp. 262-263

Has Cairo tiling in the form of circle packing


8. 1931 Fritz Laves. ‘Ebenenteilung in Wirkungsbereiche’. Zeitschrift für Kristallographie 76 (1931): 277-284. 


9. 1933 Amos Day Bradley. The Geometry of Repeating Design and Geometry of Design for High Schools. Bureau of Publications Teachers CollegeColumbia UniversityNew York City, and 1972 reprint. Book as oft-quoted by Schattschneider, but surprisingly no one else.

P. 123 Cairo-like 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).
‘We have space for one of his; [MacMahon’s] it consists of equal-sided (but not regular) pentagons, but has the appearance of interlocking hexagons (Fig. 58)’
Cairo diagram (but not attributed) p. 63 (picture) and p. 65 (text). The diagram is derived from MacMahon’s book, as Cundy freely credits.


11. 1951 'Croton'. Cairo tiling used as a crossword puzzle, in The Listener, 13 December 1951, puzzle 1128 Hexa-Pentagonal I, by 'Croton'.
'Croton' is a pseudo-name; it's somewhat of a long shot given the time passed since the puzzle’s inception, but the does anyone know he is?


12. 1954 'Croton'. Cairo tiling used again as a crossword puzzle, in The Listener, 22 April 1954, puzzle 1251 Hexa-Pentagonal 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 non-stated ‘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.
Cairo diagram (but not attributed) on the cover of seemingly the second edition only. Interestingly, this is likely the first instance of using different coloured subsidiary hexagons to better feature the overlapping hexagon aspect.
The type of pentagon is not clear due to the nature of the drawing, with somewhat thick lines, but it would appear to be equilateral. An open question is does this appear on (or in) the first edition of 1947? I have not got the book to hand.

18. 1967 D. G. Wood. ‘Space Enclosure Systems’, Bulletin 203. ColumbusOhio: Engineering Experiment Station, The Ohio State University. PP. 3-4, 30-31
Wood (a professor of industrial design rather than a mathematician) makes a curious observation as regards tilings with equal length sides, with the later to be known Cairo tiling being one of five such instances (the equilateral triangle, square, Cairo pentagon, hexagon, rhomb); as such, I do not recall seeing this simple observation elsewhere. Is this significant? Much of Wood’s work here, and elsewhere in the book, is in regards to prisms, of which he shows a ‘Cairo’ prism. Does anyone know of Wood? At the time of writing, he would be 99. Is he still alive? Did he do anything further with the tiling? He freely credits both MacMahon and Cundy and Rollett as the source of the pentagon per se, the observation of his appears to be his own.


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.
Cairo diagram (but not attributed) p. 49, but no text. This also has an interesting series of diagrams p. 83, best described as ‘variations’ with Cairo-like properties, with ‘par hexagon pentagons’ combined in tilings with regular hexagons.


20. 1970 Ernest R. Ranucci. Tessellation and Dissection. J. Weston Walch
Cairo-like diagram (but not attributed) p. 36 (picture and text).The inclusion of this Cairo of Ranucci’s is somewhat open to question, given that the diagram consists of two pentagons, rather than the given ‘standard model’ of one. Nonetheless, it is of interest due to the first example of this type.


21. 1970 H. S. M. Coxeter. 'Twisted Honeycombs' (CBMS Regional Conference Series in Mathematics), 1970, pp. 21-23.


22. 1971 James A. Dunn. ‘Tessellations with Pentagons’. The Mathematical Gazette, Vol. 55, No. 394 (December, 366-369)

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 street-tiling in Cairo. The geometry of this basic pentagon is shown in Figure 7.
Of the utmost significance; the first recorded attribution.


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.
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.


24. 1974 Stanley R. Clemens. ‘Tessellations of Pentagons’. Mathematics Teaching, No. 67 (June), pp.18-19, 1974
Cairo diagram (but not attributed) p. 18. Interesting in that this credits MacMahon as the discoverer of the equilateral pentagon (p. 19), although this is not substantiated. Likely, reading from MacMahon’s book, he just assumed this.


25. 1975 John Parker. ‘Tessellations’, Topics, Mathematics Teaching 70, 1975, p. 34
Building on Clemens, immediately above, as noted by Parker. Loosely a Cairo diagram (but not attributed) p. 34.


26. 1975 Martin GardnerScientific American. Mathematical Games, July. ‘On tessellating the plane with convex polygon tiles’, pp. 112-117 (pp. 114, 116 re Cairo pentagon)
Gardner Quote Scientific American 1975 ‘On Tessellating the Plane with Convex Tiles’, pp. 112-117
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 1st edition, 2nd edition 1992 Addison-Wesley Publishing Company. (Note that I have the 2nd edition, not the 1st)
While a regular pentagon will not tessellate the plane, it is interesting to note that there is a pentagon (see region A in Fig. 4.15) with all sides congruent [i.e. equilateral] (but with different size angles) that will tessellate the plane. A portion of this tessellation is shown in Fig. 4. 15. If four of these pentagonal regions are considered together (see Region B), an interesting hexagonal shape results that will tessellate the plane.
Cairo diagram (but not attributed) p. 95 (text continues to p. 96).


29. 1977  Lorraine MottersheadSources of Mathematical Discovery. Basil Blackwell.
Cairo diagram (but not attributed) pp.106-107 on a chapter on tessellations, and a subset of irregular pentagons.
Of note is the use of the Cairo tiles as a letter puzzle; although this is not original with Mottershead, as perhaps might appear at first sight (as I did myself to 2012). Although titled ‘… by Croton’, no further detail of ‘Croton’ is given. This diagram has now been determined as to appearing in The Listener, as detailed above, see 5. 1951 and 6. 1954. Unfortunately, the determination as to which types of pentagon are here is fraught with difficulty due to such a small scale drawing and the accuracy of the drawing is also in question, of which I am not prepared to be categorical as to the type of pentagon here. They could be equilateral, or near.


30. 1977 Doris Schattschneider and Wallace Walker. M. C. Escher Kaleidocycles. Tarquin Publications. First edition, 1977; I have the ‘special edition’ of 1982.
One of Escher’s favourite geometric patterns was the tiling by pentagons shown (Figure 35). These pentagons are not regular since their angles are not all equal.
Cairo diagram (but not attributed) p. 26, also see p. 34, in the context of a dodecahedron tiling decoration and Escher’s ‘Flower’, PD 132.
The type of Cairo tiling is not explicitly stated; certainly, it is of a 4, 1 type, likely of the dual of the 3. 3. 4. 3. 4 type (90°, 120°), but Escher did not use this!


31. 1978 Doris Schattschneider. Tiling the Plane with Congruent Pentagons’ Mathematics Magazine. ‘Vol.1, 51, No.1 January 1978. 29-44.
P. 3
Three of the oldest known pentagonal tilings are shown in FIGURE 1. As Martin Gardner observed in [5], they possess ‘unusual symmetry’. This symmetry is no accident, for these three tilings are the duals of the only three Archimedean whose vertices are valence 5. The underlying Archimedean tilings are shown in dotted outline. Tiling (3) (dual of the 3. 3. 4. 3. 4) of FIGURE 1 has special aesthetic appeal. It is said to appear as a street paving in Cairo [likely referring to Martin Gardner or James Dunn’s quote; both authors are mentioned in the bibliography]; it is the cover illustration for Coxeter’s Regular Complex Polytopes [apparently equilateral], and was a favorite pattern of the Dutch artist, M.C. Escher [square based intersections]. Escher’s sketchbooks reveal that this tiling is the unobtrusive geometric network which underlies his beautiful; ‘shells and starfish’ pattern. He also chose this pentagonal tiling as the bold network of a periodic design which appears as a fragment in his 700 cm. Long print ‘Metamorphosis II’.

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.
The third recorded attribution, but not of a firsthand sighting. Of note is Schattschneider’s care as to attribution, stating ‘it is said to appear as a street tiling..’, likely as she had not seen an in situ picture, and so did not state so categorically that it was a street tiling.


32. 1978 Peter Pearce and Susan Pearce. Polyhedra Primer. Dale Seymour Publications
Cairo diagram (but not attributed) on p. 35 and in the context of the dual tilings of the semiregular tilings, p. 39. Decidedly lightweight, no discussion as such.


33. 1978 Ernest H. Lockwood, and Robert H. Macmillan. Geometric symmetry. Cambridge University Press (and reprint 2008).
‘Indirect’
Cairo reference p. 88
… are patterns [semi regular] of congruent pentagons such as are often used for street paving in Moslem countries.
The inclusion of this book is somewhat of a moot point, in that 
Cairo tiles are described very loosely here. However, as it is by Macmillan, this rather fragmentary account is worthy of note, and curiously it does not strictly tally with his later Mathematical Gazette article.


34. 1979 Robert H. Macmillan. Mathematical Gazette, 1979. ‘Pyramids and Pavements: some thoughts from Cairo’, pp. 251-255
On a recent visit to 
Cairo I was struck by two matters [concerning the pyramids and pentagon tiling]
and
P. 253
A pentagonal tessellation
Many of the streets of Cairo are paved with a traditional Islamic tessellation of pentagonal tiles, as shown in Fig. 4. The pentagons are all identical in size and shape, having four sides equal and two of their angles 90°, as shown in Fig. 5, where angles (* and *) and lengths (a and b) are marked. The tiles are often in two colours, as in Fig. 4, and their pattern can then be classified as belonging to the plane dichromatic symmetry group p4’ g’m. By making all those tiles with a particular orientation of a single colour a polychromatic symmetry pattern, of group p4(4), would be achieved; by an alternative colouring it would be also be possible to produce a symmetry of group p4(4)mg (4), but I have never seen either of these actually used. (See [1], p.89, Fig. 13.12.)

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:…..
P. 255
(iv) If * is such that, in Fig. 4, AB and CD are collinear, the tessellation is particularly pleasing to the eye, and this is in fact the proportion (108. ) often adopted in Cairo…
The fourth recorded attribution. Of note as to the depth of detail Macmillan gives. Notably, he describes an in situ pentagon possessing of collinearity properties. A firsthand sighting, the second of only two, and so of the utmost significance; also see Dunn. No reference is made in the article itself or the references to any of the three above articles, and so this is likely an independent account, as a ‘discovery’. As such, this is a little surprising, in that Dunn’s article was also from the Mathematical Gazette! Of note is the reference to the tiles being coloured, or arranged of the same colour, ‘back to back’, this being the first recorded instance; indeed, the only one!


35. 1980 Michael O’Keefe and B. G. Hyde. ‘Plane Nets in Crystal Chemistry’. Philosophical Transactions Royal Society London. Series A, 295 1980, pp. 553-618

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 GeometryAn Introduction to Symmetry, p. 119
The beautiful 
Cairo tessellation with a convex equilateral pentagon as its prototile is illustrated in Fig. 12.3. The tessellation is so named because such tiles were used for many streets in Cairo.
Gives the ‘45°’ construction.


37. 1982 Patrick Murphy. Modern Mathematics Made Simple. Heinemann London Tessellations, Chapter 10, pp. 194-205, 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 non-attributed Cairo tiling


40. 1984
 William Blackwell. Geometry in Architecture, Wiley 1984, p. 54

...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.
The latter part of the quote is taken from Martin Gardner, word for word.


41. 1986 James McGregor and AlanWatt. The Art of Graphics for the IBM PC, pp 196-197
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 well-known 
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.
A minor part of a chapter on tessellations. Diagram p. 197.


42. 1986 Ehud, Bar-On. ‘A Programming Approach to Mathematics’. ‘A programming approach to mathematics’. Computers & Education 10(4): pp. 393-401. December 1986. Elsevier.

 … then the possible ways of tiling with pentagons are explored, especially the Cairo tiling.
Inconsequential reference. No diagrams are shown.


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 913-948

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. 1986 R. Mosseri and J. Sadoc. ‘Polytopes and Projection Method: An Approach to Complex Structures’. Journal de Physique Colloques, 1986, 47 (C3), pp. C3-281-C3-297.

Fig. 4: A tiling by pentagons with coordination 3 or 4 obtained with a squares tesselation (sic) decorated like in fig. 3b. Cairo tiling on p. C3-285.


47. 1986 George E. Andrews. Percy Alexander MacMahon: Number theory, invariants, and applications. MIT Press, 1986 p. 196 Google Books

It is said to appear as street paving in Cairo (Purposefully re-quoting Schattschneider (1978))


47. 1987 Branko Grünbaum and Geoffrey C. Shephard. Tilings and Patterns. W. H. Freeman and Company
For an account of a street tiling with pentagonal tiles common in Cairo (Egypt) see Macmillan [1979]
P. 5, no discussion, just a reference to Macmillan’s article.


48. 1987 Bob Burn. The Design of TessellationsCambridge University Press. Sheet 30. Equilateral pentagon.


49. 1987 Rudy v. B Rucker. Mind Tools: The Five Levels of Mathematical Reality. First Edition, Houghton Mifflin Company, Boston, 1987.

Fig. 39 Tessellation with irregular pentagons. And If we give up the requirements that each tile be a regular polygon and that each corner look the same, a great many strange tessellations can be found. One very attractive one is made of irregular pentagons and is often used as a cobblestone pattern in Europe and the Near East.

Whether this quote is referring to the Cairo tiling is unclear.


49. 1989 Dale Seymour and Jill Britton. Introduction to Tessellations. Dale Seymour Publications Cairo tiling (but not attributed) p. 39.
The exact pentagon not described, almost certainly the dual of the 3. 3. 4. 3. 4 (90°, 120° type).
Lightweight.


50. 1989 Piere De La Harpe. ‘Quelques Problèmes Non Résolus en Géométrie Plane’. L’Enseignement Mathématique, t 35 (1989), pp. 227-243 (in French)
Cairo tiling on p. 232, likely taken from George Martin, given that it is the same ‘unusual’ configuration.


51. 1989. Marjorie Senechal. ‘Symmetry Revisited’. Computers and Mathematics with Applications. Vol 17, No. 1-3, pp1-12. 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. 4-6, pp 751-789 (Cairo aspects 783-794), 1989 (reprinted in I. Hargittai, ed. Symmetry 2: Unifying Human Understanding, Pergamon, New York, 1989.
The pattern of a favorite street tiling in 
Cairo (US spelling of favourite, note that Chorbachi also omits the dash between favourite and street)
Fig. 19.16c 2-3. Two different semiregular pentagons are drawn at the bottom of the page. On the right side is the Islamic pentagon, where * is the critical value in the design. On the left is the Western one given by J. A. Dunn in an article on ‘Tessellations with pentagons’ [30]. Dunn’s pentagon has an isosceles pentagon triangle that has a critical length * for the two equal sides while the third side is a or any given length. This tiling (Fig. 19.16c 1) is referred to as the ‘favorite street tiling in Cairo’. In it, the tessellation is considered hexagonal, each hexagon being a combination of four semi regular pentagons. However, this tessellation is based on the 4-fold rotation of the semi regular pentagon, with sides equal to two units and two opposite right angles. The latter combination permits the 4-fold rotation of symmetry group 244 or p4g 
Has interesting Cairo tiling references, pp. 783-784, and quotes James Dunn’s 1971 article, and beyond any reasonable doubt the quote given by Chorbachi is taken from him as well. Equilateral pentagons.
Has references to ‘semi regular pentagons’ which is surely the wrong terminology; I had a web search for this, but I couldn't find references.

53. 1989 Istvan Hargittai. Symmetry 2, Unifying Human Understanding. Volume 2, Source of Chorbachi article, see above pp. 783-794.
Not seen, Google Books reference.

54. 1990 Francis S. Hill. Jr. Computer Graphics. Macmillan Publishing Company, New York, P. 145.
An equilateral pentagon can tile the plane, as shown in Figure 5.4. This is called a Cairo tiling because many streets in Cairo were paved with tiles using this pattern. Note that this figure can also be generated by drawing an arrangement of overlapping (irregular) hexagons.
Likely quoting from McGregor and Watt, given that the text is very much alike, and their work is quoted and illustrations are used in the book.

55. 1991 Ann E. Fetter et al. The Platonic Solids Activity Book. Key Curriculum Press/Visual Geometry Project. Backline Masters.
Regular pentagons don’t tile, but many equilateral (though not equiangular) pentagons do. [A Cairo tiling diagram is then shown.] This pattern is seen in street tiling in Cairo and in the mosaics of Moorish buildings. A similar tiling can be obtained of the dual of a semi regular tiling (see exercise 8)
Cairo tiling pp. 21 and 97 (the latter of which repeats, as student activities)
Almost certainly this quote is taken from Gardner, as detailed above.


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. McGraw-Hill. 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) 227-254

 ... seen in street tiling in Cairo and occasionally in the mosaic of Moorish buildings.

Seemingly quoting Blackwell.

 

59. 1994 Audrey Leathard. Going inter-professional: working together for health and welfare
In the Cairo tessellation (Wells 1991), dual tessellations are formed by overlaying a second grid rotated 90 degrees to the first…P. 45:
Not seen, Google Books reference. Note that this reference is only included for the sake ‘of everything’; the book is apparently of a non-mathematical nature, and is not illustrated with the tiling. Quotes the Wells reference.


60. 1994 Carter Bays. Complex Systems Publications, Volume 8, Issue 2, 127-150, Cairo aspect p. 148
‘Cellular Automata in the Triangular Tessellation’
… the Cairo tessellation (a tiling of identical equilateral pentagons)…
Cursory mention in passing.


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
The 
Cairo street tiling shown at right is a very beautiful tessellation that uses equilateral pentagons (the sides are congruent but not the angles). The pentagon is shown below right, with angle measures that will help you draw your ...
Not seen, Google Books reference.


63. 1998 David A. Singer. Geometry Plane and Fancy, 1998, p.34. Springer-Verlag
One particularly elegant tiling of the plane by pentagons is known as the Cairo tessellation, because it can be seen as a street tiling in Cairo. The pentagon used for this tiling can be constructed using straight edge and compass… although it is not regular, it is equilateral
Not seen, Google Books reference.


64. 1999 Ian Stewart. ‘The Art of Elegant Tiling’. Scientific American. July 1999, pp. 96-98
Minor instance of coloured Cairo tiling, p. 97, as devised by Rosemary Grazebrook.


65. 1999 Jinny Beyer. Designing Tessellations, Contemporary Books, p. 144.
Lightweight.


66. 2001 Edward Duffy, Greg MurtyLorraine Mottershead. Connections Maths 7. Pascal Press, p. 83
Cairo streets have this Islamic pattern
Not seen, Google Books reference.


67. 2003 Teacher’s Guide: Tessellations and Tile Patterns, p. 30 (Cabri) Geometric investigations on the VoyageTM 200 with Cabri. Texas Instruments Incorporated
….Probably the most famous of these pentagonal patterns is the ‘Cairo Tessellation’ named after the Islamic decorations found on the streets of Cairo…Begins by quoting David Wells’ book … Curious… and likely the text is based on his reference. However, the ‘Teacher’s Guide’ gives a different tiling, interestingly a ‘collinear pentagon.


68. 2003  Catherine A GoriniThe 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
Is a favourite street tiling in Cairo
pp. 421-427. This is an anthology, and simply repeats Dunn’s article and follow-up correspondence. Nothing original is shown. 
Not seen, Google Books reference.


70. 2003 Eric W. Weisstein. CRC concise encyclopedia of mathematics, p. 313
A tessellation appearing in the streets of 
Cairo and in many Islamic decorations. Its tiles are obtained by projection of a dodecahedron, and it is the dual tessellation of the semiregular tessellation of squares and equilateral triangles.

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.
The lattice D (32. 4. 3. 4) is sometimes called the Cairo lattice, as the pattern occurs frequently as tilings on the streets of Cairo.
A fleeting mention in the context of a study on Laves tilings.


72. 2005 David Mitchell. Sticky Note Origami: 25 Designs to make at your desk, Sterling Publication Company
The 
Cairo Tessellation is an attractive and intriguing pattern of tiles named as a result of its frequent occurrence on the streets of Cairo and in other Islamic centers and sites. Cairo tiles are a special kind of pentagon that unlike ordinary regular pentagons will fit together without leaving gaps between them. Four of these slightly squashed pentagonal tiles will from a stretched hexagon in the final pattern, stretched hexagons laid in a vertical direction intersect other stretched hexagons laid horizontally across and through them. If you make the tiles in four different colours the resulting pattern is particularly interesting and attractive.
Mitchell doesn’t state exactly what type of Cairo tiling he is referring to. However, upon checking his diagram, p. 58 it would appear to be equilateral. However, due to the small scale nature, this is not categorically so.
Not seen, Google Books reference.


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 tessellationObserve that it has a pentagonal motif that has four sides of one length and one shorter side
Not seen, Google Books reference.


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.
 Although the diagram is too small in scale to measure with certainty, it appears to be of the dual of the 3. 3. 4. 3. 4 (90°, 120° type).
Not seen, Google Books reference.


75. 2005 Carter Bays. Complex Systems Publications, Volume 15, Issue 3, 245-252, Cairo aspect p. 249-250
‘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, 20-22. 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. 165-169

Cairo tiling on pp. 167-169, 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
…without gap or overlap. There are however various equilateral pentagons that can 
tessellate the plane. Probably the best known is the Cairo tessellation, formed…


80. 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.


81. 2007 Kindt, Martin. ‘Wat te bewijzen is’ (in Dutch) (38) (translated ‘What is to be proved’). Nieuwe Wiskrant 27-1 September 2007

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


81. 2008 Anon. Key Curriculum Press. Chapter 7 Transformations and Tessellations, p. 396

The beautiful Cairo street tiling shown below uses equilateral pentagons.

This also gives a construction, of the well known ‘45° type’.


82. 2008 Merrilyn Goos et al. Teaching Secondary School Mathematics: Research and Practice for the 21st Century.
The particular tiling pattern of an irregular pentagon, shown in Figure 9.16, is called the Cairo tessellation because it appears in a famous mosque in Cairo.
Not seen, Google Books reference.


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. LeeuwardenNetherlands
There are, however, equilateral convex pentagons that do tessellate the plane, such as the well known 
Cairo tessellation shown in Figure 1.
Also, other minor references essentially in passing.
Cairo reference and diagram p. 102 in ‘Patterning by Projection: Tiling the Dodecahedron and other Solids’ gives an equilateral pentagon.


84. 2008 Robert Fathauer. Designing and Drawing Tessellations, p. 2.
A common street paving in 
CairoEgypt is shown above left. It is notable for the interesting tessellation formed by pentagons, four of which form larger hexagons, with hexagon patterns running in two different directions
Type of pentagon: Equilateral. Has a brief discussion on tessellations in the 'real world', p. 2, with many photos of brickwork and paving stone tessellations, all except for the ‘Cairo Pentagon’ tiling, where although this is discussed, he shows a line drawing. Presumably, the reason for this is that he was unable to locate a photo.


85. 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 well-known 
Cairo tessellation shown in Fig. 1.

Type of pentagon: Equilateral (p. 101).


86. 2008 Birgit Kaltenmorgen. Der mathematische Patchworker. (in German) Wagner, Gelnhausen; 1st edition pp. 82-83
Fünfeck beim Cairo-Tiling


87. 2009 Craig S. Kaplan. Introductory Tiling Theory for Computer Graphics. Morgan & Claypool Publishers, p. 33
The Laves tiling [32. 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
Another pentagonal tiling can be created by overlaying two non-regular hexagonal tilings illustrated in Figure 10.6. This rather attractive monohedral pentagonal tiling is sometimes called the 
Cairo tiling, for its reported use as a street paving design in that city.
Cairo diagram p. 163. The type of pentagon is not detailed; unfortunately, the diagram is too small a scale to measure with certainty.
Not seen, Google Books reference.


90. 2011 Richard Elwes. Maths 1001: Absolutely Everything You Need to Know about Mathematics in 1001 Bite-Sized Explanations. Quercus, p. 109
it adorns the pavements of that city’ (
Cairo).
Although it would appear likely that a single pentagon is intended, this shows two different, but roughly alike pentagons, of which I assume that it just a careless drawing. Given that the type of pentagon Elwes is referring to is unclear; no assessment as to type is made.


91. 2011 Abdul Karim BanguraAfrican Mathematics: From Bones to Computers University Press of America, 2011
A basketweave tessellation is topologically equivalent to the Cairo pentagonal tiling…
Not seen, Google Books reference. Cursory mention in passing.


92. 2011 Eric Goldemberg. Pulsation in Architecture  p. 338
Housing Exhibition in Vienna, Austria Project Description The Cairo Pods gave SPAN ... The Cairo Tessellation, known in mathematics also as an example of ...


92. 2011 Q. Ashton Acton (ed). Issues in General Physics Research: SchorlarlyAdditions, 2011. Google Books

Iron-Based Cairo Pentagonal Lattice


93. 2012 Christoph A. Kilian (ed), Norbert Palz, Fabian Scheurer
Computational Design Modeling: Proceedings of the Design Modeling Symposium. Springer, p. 229...
…on the mathematical configuration of a Cairo tessellation


94. 2013 Lisa Iwamoto. Digital Fabrications: Architectural and Material Techniques. Princeton Architectural Press; first edition 2009
… project uses a pentagonal Cairo tessellation pattern, flexibly aggregated to yield multiple overall arrangements. Each vertical layer of the cell was ...


95. 2013 Toshikazu Sunada. Topological Crystallography: With a View Towards Discrete Geometric Analysis. Springer, 2013. GB 

P. 132 Cairo pentagon (caption)

8.2 Cairo Pentagon Fig. 8.3 Merging two square lattices Figure 8.2 is a tiling of pentagons with picturesque properties that has become known as the Cairo pentagon.

Not seen, Google Books reference.


96. 2013 Gyynn, Ruairi and Bob Sheil (eds). Fabricate 2011: Making Digital Architecture. UCL Press, pp. 196-201 Riverside Architectural Press, 2013.  Joe MacDonald, ‘The Agency of Constraints’.

The Cairo hexagon (sic)... The streets of Cairo are paved with stones of this geometry.


96. 2014 Benölken, Ralf, Hans-Joachim Gorski and Susanne Müller-Philipp. Leitfaden Geometrie: Für Studierende der Lehrämter. Springer, 2014. In German. Translated: Guideline Geometry: For students of the teaching offices. p. 203, ‘Cairo tiling’

In abbildung 133 ist die parkettierung ‘Cairo tiling’ dargestellt. Translated: Figure 133 shows the parqueting 'Cairo tiling'


97. 2014 David E. Laughlin and ‎Kazuhiro Hono. Physical Metallurgy. P 76 5th edition. GB

The nets in … the Catalan Cairo pentagonal tiling V32.4.3.4

Not seen, Google Books reference.


98. 2016 Walter Steurer and Julia Dshemuchadse. Intermetallics: Structures, Properties, and Statistics. OUP Oxford. p. 565. GB

... 34 Cairo pentagon tiling 496 

Not seen, Google Books reference.


99. 2016 Mark Neyrinck. ‘The Origami Cosmic Web’, The Paper, No. 122, 26-27

How a 2D universe would fold up to form a [Cairo] pentagonal tiling of voids.


99. 2017 Robert J. Lang. Twists, Tilings, and Tessellations: Mathematical Methods for Geometric Origami. CRC Press. GB Left: “Cairo Tessellation” (2011), a flagstone tessellation by Eric Gjerde, based on the Cairo tiling.

Not seen, Google Books reference.


100. 2017 Ed Southall and Vincent Pantaloni. Geometry Snacks. Tarquin, 2017, 90 pp. NOT SEEN Cairo tiling as an equilateral pentagon. Area problem p. 74, #47. Solution p. 82.


101. 2017 Aakash Moncy. 'Mechanics of Cairo lattices'. Thesis

Numerous references throughout, too many to list.


101. 2017 Ponnadurai Ramasami (ed). Computational Sciences. De Gruyter, 2017. Google Books

P. 57 ...similar to Cairo pentagonal tiling.


101. 2017 Changzheng Wu (ed). Inorganic Two-dimensional Nanomaterials: Fundamental Understanding , Characterizations and Energy Applications (Smart Materials Series).  Royal Society of Chemistry; First Edition, 2017. Google Books

(Pentagraphene)… resembling the Cairo pentagonal tiling


102. 2019 Frank Morgan. ‘My Undercover Mission to Find Cairo Tilings’. The Mathematical Intelligencer, September 2019, Volume 41, Issue 3, pp 19–22

A dedicated piece, with references throughout, too many to list here. On his visit of the same year, following up the report on my page as to sightings.


102. 2019 Mircea Pitici (ed). The Best Writing on Mathematics. Princeton University Press, 2019

pp. 114-116. Google Books

A portion of the Cairo tiling. 

N. J. Sloane on Chaim Goodman-Strauss’ ‘coloring book’ method.


103. 2019 Chaim Goodman-Strauss and N. J. A. Sloane. ‘A Coloring Book Approach to Finding Coordination Sequences’, Acta Crystallographica Section A: Foundations and Advances, 2019, Volume A75, pp .121-134.

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


N.B. An apparent Cairo tiling in a 1923 paper by F. Haag, "Die regelmässigen Planteilungen und Punktsysteme." Zeitschrift fur Kristallographie 58 (1923): 478-488,

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
For the sake of accuracy, I restrict the listings here to a few prime mathematics sites:

Wolfram MathWorld
A tessellation appearing in the streets of
Cairo and in many Islamic decorations. Its tiles are obtained by projection of a dodecahedron, and it is the dual tessellation of the semiregular tessellation of squares and equilateral triangles.

Wikipedia
In geometry, a pentagon tiling is a tiling of the plane by pentagons. A regular pentagonal tiling on ... Tiling Dual Semiregular V3-3-4-3-4 Cairo Pentagonal.svg ...



Update History

N.B. This partial, or as 'of note' in some way, as I don't see the need to detail every new entry.


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 tidy-up.


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 non-attributed 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.

Created on: 9 September 2011. 

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