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 different ways, as according to various filters:
- 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.
- References of all instances of the Cairo tiling as a pentagon per se, of a 'standard model', Cairo attributed or not, with quotes and comments thereof where appropriate.
Due to the dual nature, there are on occasions overlaps in the text. 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 street-tiling in Cairo' arranged as according to chronology, with Cairo quote
1. 1971 Dunn, J. A. ‘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.
2. 1975 Gardner, M. 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 at the Taj Mahal, almost certainly a mistaken account on his part (I asked him!).
3. 1978 Schattschneider, D. 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 Macmillan, R.H. 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 Martin, George E. 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 Blackwell, William. 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 McGregor, James and Watt, J. The Art of Graphics for the IBM PC, p. 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. Computers and Education. Elsevier
… especially the Cairo tiling.
9. 1989 Chorbachi, W. K. 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 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 Harpe, P. De La. 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 Hill, Francis S. 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 Fetter, Ann E 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 Gartner quote, both parts are almost word for word.
13. 1991 Wells, David. 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 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.
14. 1994 Leathard, Audrey. Going inter-professional: working together for health and welfare
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.
15. 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
16. 1996 O’Keefe, Michael 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…
17. 1998 Singer, David A. 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….
18. 1997 Serra, Michael. 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.
19. 2000 Deza, M. et al. Fullerenes as tilings of surfaces. Journal of Chemical Information Computer and Modelling. ACS Publications, pp. 550-558
… is the Cairo tiling…
20. 2001 Edward Duffy, Greg Murty, Lorraine Mottershead. Connections Maths 7. Pascal Press, p. 83
Cairo streets have this Islamic pattern
21. 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.
22. 2003 Gorini, Catherine A. 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…
23. 2003 Pritchard, Chris. 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.
24. 2004 Parviainan, Robert. ‘Connectivity Properties of Archimedean and Laves Lattices’. Uppsala Dissertations in Mathematics 34. p. 9. 2004.
The lattice D (3^{2}. 4. 3. 4) is sometimes called the Cairo lattice, as the pattern occurs frequently as tilings on the streets of Cairo.
25. 2005 Mitchell, David. Sticky note origami: 25 designs to make at your desk, Sterling Publication Company, p. 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
26. 2005 Phillips, George McArtney. Mathematics Is Not a Spectator Sport. Springer, p. 193
This is called the Cairo tessellation.
27. 2005 Wilder, Johnston-Sue; John Mason, Developing Thinking in Geometry, p. 182.
… is often referred to as the Cairo tessellation as it appears in a mosque there.
28. 2005 Bays, Carter. 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…’
29. 2006 Sharp, John. ‘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
30. 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…
**2007 Ollerton, Mike. 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.
31. 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
32. 2008 Goos, Merrilyn et al. Teaching Secondary School Mathematics: Research and Practice for the 21^{st} 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.
33. 2008 Thomas, B. G. 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.
34. 2008 Thomas, B. G. 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
35. 2008 Kaltenmorgen, Birgit. Der mathematische Patchworker. (in German) Wagner, Gelnhausen; 1^{st} edition 2008, pp. 82-83
Fünfeck beim Cairo-Tiling
35. 2008 Fathauer, Robert. Designing and Drawing Tessellations, 2008, p. 2.
A common street paving in Cairo, Egypt is shown above left.
36. 2009 Kaplan, Craig. S. 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.
37. 2010 Alsina, Claudi 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.
38. 2010 Elwes, Richard. 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).
39. 2010 Ressouche, E. 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
40. 2011 Rojas, M. et al. Frustrated Ising model on Cairo pentagonal lattice
Cairo pentagonal lattice
41. 2011. Askew, Mike and Sheila Ebbutt. The Bedside Book of Geometry: From Pythagoras to the Space Race: The ABC of Geometry Murdoch Books Pty Limited
42. 2011 Goldemberg, Eric. 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 ...
43. 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
44. 2013 Iwamoto, Lisa. Digital Fabrications: Architectural and Material Techniques
… project uses a pentagonal Cairo tessellation pattern, flexibly aggregated to yield multiple overall arrangements. Each vertical layer of the cell was ...
45. 2014 Gorski, Hans-Joachim and Susanne Müller-Philipp. Leitfaden Geometrie: Für Studierende der Lehrämter, Springer p. 186
2. References of all instances of the Cairo tiling, attributed or not, in chronological order, with quotes and comments where appropriate
1. 17th Century. Ray, Simon. p
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 20th century. A sighting at Heidelberg Castle, Germany
3. 1909 Moore, Herbert C. ‘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 MacMahon, P. A. New Mathematical Pastimes. Cambridge University Press 1921 and 1930. (Reprinted by Tarquin Books 2004)
Cairo diagram p. 101
5. 1922 MacMahon, P. A. ‘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 Haag, F. '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 Haag, F. '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 Laves, F. ‘Ebenenteilung in Wirkungsbereiche’. Zeitschrift für Kristallographie 76 (1931): 277-284.
9. 1933 Bradley, Amos Day. 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 Cundy H. Martyn; Rollett, A.P. 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 Smith, Cyril Stanley. “The Shape of Things.” Scientific American, vol. 190, no. 1, 1954, pp. 58–65.
In a general discussion on tiling.
14. 1955 Rollett, A. P. ‘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 Langford, C. Dudley. ‘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 Bakos, T. ‘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 Coxeter, H. S. M. Regular Complex Polytopes. Second edition. Dover Publications Inc., New York.
Cairo diagram (but not attributed) on the cover. 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 Wood, D. G. ‘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? Critchlow, Keith. 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 Ranucci, Ernest R. 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 Coxeter, H. S. M. 'Twisted Honeycombs' (CBMS Regional Conference Series in Mathematics), 1970, pp. 21-23.
22. 1971 Dunn, J. A. ‘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 Williams, Robert. 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 Clemens, Stanley R. ‘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 Parker, John. ‘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 Gardner, M. Scientific 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.
23. 1976 Odier, Marc G. ‘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.
24. 1976 O’Daffer, Phares. G; Clemens, Stanley R. Geometry. An Investigative Approach 1^{st} edition, 2^{nd} edition 1992 Addison-Wesley Publishing Company. (Note that I have the 2^{nd }edition, not the 1^{st})
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).
25. 1977 Mottershead, Lorraine. 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.
26. 1977 Schattschneider, Doris; Walker, W. 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!
27. 1978 Schattschneider, D. 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.
28. 1978 Pearce, Peter and Pearce, Susan. 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.
29. 1978 Lockwood, E.H; R.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.
30. 1979 Macmillan, R.H. 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!
31. 1982 Martin, George E. 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.
Gives the ‘45°’ construction.
32. 1982 Murphy, Patrick. 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 Willson, John. Mosaic and Tessellated Patterns. How to Create Them. Dover Publications, Inc. 1983. Plate 3Cairo tiling plate 3. (Neglected until 7 May 2013!)
* 1983 Smith, Cyril Stanley. A Search for Structure. The MIT Press, 1983
Has non-attributed Cairo tiling
34. 1984 Blackwell, William, 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.
35. 1986 McGregor, James and Watt, J. 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 Ehud, Bar-On. ‘A Programming Approach to Mathematics’. Computers and Education. Elsevier. … then the possible ways of tiling with pentagons are explored, especially the Cairo tiling
Inconsequential reference. No diagrams are shown
37. 1986 Loeb. A. L. 'Symmetry and Modularity'. Computers and Mathematics with Applications, Elsevier
38. 1986 Kappraff, Jay. ‘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.
* 1986 Collatz, Lothar. Geometrische Oranamente (in German)
Cairo tiling diagram in context of 43433 classification.
39. 1987 Grünbaum, Branko; Shephard, G. C. 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.
40. 1987 Burn, Bob. The Design of Tessellations. Cambridge University Press. Sheet 30. Equilateral pentagon.
41. 1989 Seymour, D; Britton, J. 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.
42. 1989 Harpe, P. De La. ‘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
43. 1989. Senechal, M. ‘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.
44. 1989 Chorbachi, W. K. ‘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.
45. 1989 Hargittai, Istvan. Symmetry 2, Unifying Human Understanding. Volume 2, Source of Chorbachi article, see above pp. 783-794.
Not seen, Google Books reference.
46. 1990 Hill, Francis S. 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.
47. 1991 Fetter, Ann E 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.
48. 1991 Wells, David. 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.
49. 1991 Kappraff, Jay. 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.
*. 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.
50. 1994 Leathard, Audrey. 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.
51. 1994 Bays, Carter. 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.
52. 1997 Serra, Michael. 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.
53. 1998 Singer, David A. 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.
54. 1999 Stewart, Ian. ‘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.
55. 1999 Beyer, Jinny. Designing Tessellations, Contemporary Books, p. 144.
Lightweight.
56. 2001 Duffy, Edward, Greg Murty, Lorraine Mottershead. Connections Maths 7. Pascal Press, p. 83
Cairo streets have this Islamic pattern
Not seen, Google Books reference.
57. 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.
58. 2003 Gorini, Catherine A. 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…
59. 2003 Pritchard, Chris. 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.
60. 2003 Weisstein, Eric W. 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.
61. 2004 Parviainan, Robert. ‘Connectivity Properties of Archimedean and Laves Lattices’. Uppsala Dissertations in Mathematics 34. p. 9. 2004.
The lattice D (3^{2}. 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.
62. 2005 Mitchell, David. 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.
63. 2005 Phillips, George McArtney. 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
Not seen, Google Books reference.
64. 2005 Wilder, Johnston-Sue; 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.
65. 2005 Bays, Carter. 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.
66. 2005 Garcia, Paul. ‘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
67. 2006 Sharp, John. ‘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’.
* 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
68. 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…
** 2007 Ollerton, Mike. 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.
69. 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’.
70. 2008 Goos, Merrilyn et al. Teaching Secondary School Mathematics: Research and Practice for the 21^{st} 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.
71. 2008 Thomas, B. G. and M. A. Hann. In Sarhangi, Reza (Ed). Bridges. Mathematical Connections in Art, Music, and Science. (Tenth) Conference Proceedings 2008. Leeuwarden, Netherlands
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.
72. 2008 Fathauer, Robert. Designing and Drawing Tessellations, p. 2.
A common street paving in Cairo, Egypt 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.
73. 2008 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).
**. 2008 Kaltenmorgen, Birgit. Der mathematische Patchworker. (in German) Wagner, Gelnhausen; 1^{st} edition pp. 82-83
Fünfeck beim Cairo-Tiling
74. 2009 Kaplan, Craig. S. 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.
75. 2009. Ollerton, Mike. The mathematics teacher's handbook, p. 148
… use four different colours to make the 'Cairo' tiling design.
Not seen, Google Books reference.
76. 2010 Alsina, Claudi 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.
77. 2011 Elwes, Richard. 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.
78. 2011 Abdul Karim Bangura. African 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.
79. 2011 Goldemberg, Eric. 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 ...
80. 2012 Gengnagel, Christoph (ed), 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
81 2013 Iwamoto, Lisa. Digital Fabrications: Architectural and Material Techniques
… project uses a pentagonal Cairo tessellation pattern, flexibly aggregated to yield multiple overall arrangements. Each vertical layer of the cell was ...
82. 2014 Gorski, Hans-Joachim and Susanne Müller-Philipp. Leitfaden Geometrie: Für Studierende der Lehrämter, Springer p. 186
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 ...
Created on: 9 September 2011. Revised and enlarged: 22 September, 26 October, 1 November 2011; 21 April, 12 May, 3 December (D. G. Wood), 10 December (R. Parviainan) 2012. Wholesale revision 14 December 2012, with quotes added to Section 1, and colour coding removed from Sections 1 and 2, which was a little contrived, and did not make for easy reading. Also, a general tidy-up. 4 October 2013. 10 July 2014: Moore and Odier references added. 7 January 2015: Haag (2) and Laves references added, inexplicably omitted previously). 30 October 2018: Smith (2), Trinajstic, Collatz, Eberhart entries added.