References

Or 

A chronicle‑style catalogue of all documented appearances of the Cairo tiling—explicit or implied—in print only (excluding web references), supported by quantitative analysis 

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 seven different ways, according to various categories and 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 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 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.

4.  "Cairo tiling" (Filtered from the above). The exact term, no variations, no matter how minor.

5.  "Cairo Pentagon" (Filtered from the above). The exact term, no variations, no matter how minor.

6.  Alphabetical listing of all entries of a "bare bones" bibliography, without annotations.

7.  Data Analysis

Rationale
As such, this is very much a continuous listing, of which assembling this, from different sources, some to hand and some not, of different times (years), with at times bibliography detail not fully available, is a task fraught with difficulty. As can be seen, not every entry is documented in full, although shortcomings are relatively minor, such as an omitted page number or other. And then there’s the time factor to take into consideration. As an example, the ‘Alphabetical, All’ section has taken a day and a half, for what, it must be admitted, is not exactly exciting fare. Add to that when I started reorganising the page on the website (perhaps with a day in mind, if that?), on 18 May 2026, this expanded, with a range from 18–22, 25–29 May; 1–3 June, of no less than thirteen days, I have had more than my fill of broad basic ‘documenting’, as important as ‘good housekeeping’ may be. 

Collating all this material consistently is a near‑impossible task, and even now it is still not finished. Once again, I underestimated the scale of the work—and it evolved as I went—but the result is at least better presented than before, with various expansions of text along the way. Even so, enough is enough. The plan now is to return at ad hoc intervals with whatever “found details” turn up and add them as they arise, rather than attempt another dedicated study of the kind described above.

Beyond a Standard Bibliography
Simply stated, this is far more than a “standard” bibliography. Although it is built upon a core list of references, many entries require explanation to make sense of their relevance, context, or nuances. Accordingly, each item is annotated. These annotations range from a brief one‑ or two‑line comment to, where necessary, a more substantial discussion—at times almost a miniature essay.

Research Constraints: Google Books, the Internet Archive, and Source Availability
Ideally, every entry would receive the same level of attention, but in practice this is not possible for several reasons. First, there is the sheer number of references—about one hundred in total. Even under ideal conditions, with every book and article to hand, this would be a formidable undertaking. In reality, I possess only around fifty of these sources. These are largely the “must‑have” references essential to the inquiry; others are less critical. Some items located through Google Books are simply too obscure or impractical to obtain—often appearing in specialist metallurgy or chemistry journals, with only a passing mention of the subject. In such cases, I note: “Not in possession; Google Books reference.”
Conspicuously absent (or at least recorded) was any use of the Internet Archive, an equally likely source. Quite when I first became aware of this outstanding archive—founded in 1996—is unclear; I cannot even estimate the decade with confidence. Tellingly, no reference to it appears at the time of study, and I may well have been unaware of it altogether, although I find this highly unlikely.
Whatever the case, I have now revisited the Internet Archive with this specific purpose in mind. Much to my surprise, searches for likely terms such as “Cairo tiling” and “Cairo pentagon” yielded remarkably sparse results: six entries for each (all distinct), with two repeats in both cases. I must admit I was taken aback by the paucity; I had expected considerably more. Be that as it may, the “new look” undertaken in May 2026 has uncovered a few additional references (about 10; an exact number is hard to give due to various factors in documenting), including some in recent publications, and I now take the opportunity to update the page accordingly.

“High-end” references - Chemistry and physics type papers
Aside from the obvious mathematical sources, another place where the Cairo tiling appears is in what I think of as “high‑end” chemistry and physics type papers. These references begin in 2009 and clearly mark a breakthrough moment in the chemistry/physics community: suddenly one encounters phrases such as “Frustrated Ising model” and “antiferromagnetic Heisenberg model on the Cairo pentagonal lattice”, followed by symbols and formalisms far beyond my popular‑science reach.

At first, only a handful of such papers turned up through ordinary searching, and it seemed practical to list “most”, if not “all”, of them. A later sweep through Google Scholar, however, revealed many more —enough that attempting to document every instance would be unrealistic. Because the Cairo‑tiling reference in these papers is usually brief and embedded in material I cannot meaningfully interpret, there is little value in spending days (or weeks) compiling an exhaustive list. Instead, I am content with the representative examples I have gathered and with establishing when this wave of advanced scientific references first appeared. Typically, the model chosen is the dual, although whether this is by “convention”, with the researchers copying after the earliest references or is somehow intrinsic to the “high-end” model, I am unsure.
The earliest instance I have, with the caveats above, is “Magnetic frustration in an iron-based Cairo pentagonal lattice” by E Ressouche et al., 2009. 

Foreign References

Of interest are foreign references, that is, outside English-speaking countries. These are conspicuous by their absence, although whether this is down to my own deficiencies in searching foreign archives, or there is genuinely little interest is uncertain. However, I would be amazed if the earliest account was foreign. For what it is worth, the first French reference I have is 1989 (De la Harpe) and German (Benölken et al) 2014.

Any native of the above, I would be delighted to hear from you. So far as I know, there is no reference to the tiling in any Egyptian publication (excluding my two publications)!

References in Print Only (i.e. no web references) 

For this compilation, I have deliberately restricted myself to references in print and have therefore excluded web‑based material. In principle, I would like to include online sources as well, but doing so would risk overwhelming the project. By nature I am thorough, and even a preliminary survey revealed such an abundance of online material that any attempt at a comprehensive listing would be daunting—potentially a task of many weeks, if not months. Experience has taught me to refrain from embarking on such open‑ended undertakings.

A selective approach might seem a reasonable compromise: include only the more substantial online sources, such as Wikipedia or Wolfram (albeit they perpetuate falsehoods). Yet this quickly becomes arbitrary. Lesser‑known but equally deserving references would then demand inclusion, and before long the list would grow unwieldy and inconsistent.

There is also the matter of quality. Web references vary widely, and many lack the editorial oversight typically found in print publications, where some degree of third‑party review is the norm. For all these reasons—scope, manageability, consistency, and standards—I have chosen to omit web references and confine the compilation to print sources alone.

Bias
There is also an intended bias. Because this study is primarily historical, the earliest references are examined in greater depth, whereas more recent material—published twenty or thirty years later—typically receives a lighter treatment.

Colour Coding
It will be seen that colour coding is used throughout the text. Although not essential, it greatly aids the quick identification of terms of interest, especially in the more detailed annotations.
The scheme is as follows:

• Red marks any Cairo association, e.g. “Cairo tiling”, "Cairo Pentagon", "Cairo tessellation".

• Blue marks references to street‑paving or related observations, e.g. street tiling in Cairo.

• Green marks the (erroneous) claim that the pattern is common in Islamic decoration, mosques, and related contexts, e.g. traditional Islamic tessellation. 

Outlets

ArXiv - “high-end”

Core - “high-end”

Google Books - Good

Haithi Trust - Poor

Internet Archive - Good

JSTOR - Poor

Newspapers (any) Nothing!

Open Alex - “high-end”

Open Library - I don’t see any difference from its “source”, Internet Archive

ResearchGate - Good for “high-end” references (although infuriating to use with “security checks” required each time)

Semantic Scholar - Good for “high-end” references

zbMATH - Poor

For finding "standard” references, there are really only two archives worth searching, namely Google Books and the Internet Archive. Some what I describe as “high-end” references are so numerous and trivial that it is exhausting, and so I effectively stopped documenting after a little while.

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

1. A simple listing of attributed references, mentioned in association with Cairo, e.g. 'is a favourite street-tiling in Cairo', arranged according to chronology, with Cairo quote.

1971 (1)

James A. Dunn. ‘Tessellations with Pentagons’. The Mathematical Gazette, Vol. 55, No. 394, December, pp. 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. [An equilateral Cairo-like tile is depicted]

A first‑hand sighting, one of only two; see also Macmillan. Almost certainly the earliest account. The Cairo reference appears almost as an aside, placed at the end, within a broader study of pentagonal tilings.

And here, at the very first mention, the misattributions of the in‑situ model begin! For background on this, see Dunn’s entry on the “pentagon confusion”:

https://www.magnificent-tessellation.com/cairo-tiling/mathematics/pentagon-confusion

1975 (1)

Martin Gardner. "On tessellating the plane with convex polygon tiles". Scientific American, July, 1975, pp. 112–117 (pp. 114 and 116 re Cairo aspect).

The most remarkable of all the pentagonal patterns is a tessellation of equilateral pentagons. It belongs to types 2* and 4. Observe how quadruplates of these pentagons can be grouped into oblong hexagons in two different ways, each set tessellating the plane at right angles to the other. This beautiful tessellation is frequently seen as a street tiling in Cairo and occasionally in the mosaics of Moorish buildings.

In his discussion of tiling with convex polygonal tiles, Gardner first considers convex hexagonal tiles, drawing on the work of Karl Reinhardt. He then turns to convex pentagonal tiles, following the work of Richard Kershner, who in turn built on Reinhardt’s results. From there, Gardner highlights three illustrated pentagons of “extra” interest, one of which is an equilateral, Cairo‑like tiling.
This is the second recorded attribution. Gardner’s account is interesting in several ways. It is not a first‑hand sighting; although not stated explicitly, it is, on inspection, based on James Dunn’s 1971 paper. The reference to “Moorish buildings” derives from Richard K. Guy’s account of a supposed sighting—now understood to have been at the Taj Mahal, and almost certainly a mistaken recollection on Guy’s part (I asked him).

* Gardner elsewhere corrects to type 4 only.

For background on this, again see Gardner’s entry on the type of “pentagon confusion”, link above.

1978 (1)
Doris Schattschneider. ‘Tiling the Plane with Congruent Pentagons’, Mathematics Magazine, January, 1978, pp. 29–44 (p. 30 re Cairo aspect).
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 tilings whose vertices are of valence 5. The underlying Archimedean tilings are shown in dotted outline. Tiling (3) of FIGURE 1 has special aesthetic appeal. It is said to appear as street paving in Cairo; it is the cover illustration for Coxeter's Regular Complex Polytopes, and was a favorite pattern of the Dutch artist, M. C. Escher. 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 fragment in his 700 cm. long print "Metamorphosis II." 

[Figure]

The three pentagonal tilings which are duals of Archimedean tilings. The underlying Archimedean tiling is shown in dotted outline.

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 tilings are superimposed at right angles to each other. 

The reference is likely to Gardner or Dunn, or perhaps to both authors, as each is included in the bibliography and gives a comparable account. 

In an article on pentagonal tiling, Schattschneider discusses the Cairo tiling within the context of the duality of the three Archimedean tilings consisting of pentagons. I take issue with the Cairo tiling or Cairo-like being included as one of “the oldest known pentagonal tilings”. I presume this is based on the duality feature, but there is no evidence of this as this being “old”, as in centuries, say.

Curiously, she bases her model on the dual of the 3.3.4.3.4 tiling, not given by Dunn or Gardner. 

1979 (1)
Robert H. Macmillan. ‘Pyramids and pavements: some thoughts from Cairo’, Mathematical Gazette, 1979, pp. 251–255. See pp. 253–255.
A pentagonal tessellation

On a recent visit to Cairo I was struck by two matters which may be of interest to other members and their pupils…[pyramid discussion, which I omit, and paving]

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'g1m. By making all those tiles with a particular orientation of a single colour a polychromatic symmetry pattern, of group *) would be achieved; by an alternative colouring it would also be possible to produce a symmetry of group *. but I have never seen either of these actually used. (See Ill, 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 other dimensions are altered suitably. The geometric conditions to be satisfied are seen from Fig. 5 to be as follows:

[Equations]

It is now of interest to consider a number of special cases which satisfy the two conditions above...[dual, the equilateral, collinear feature, basket weave degeneration]
In an article on a shared Egyptian theme—pyramids and paving—the depth of discussion is far more extensive than in the earlier accounts by Dunn, Gardner, and Schattschneider. He identifies variants such as the dual, the equilateral form, the collinearity feature of the in situ model, and even the degeneration to a basketweave tiling and its associated colouring. The first two are mentioned only in isolation, while the third, fourth, and fifth are not mentioned at all.
This constitutes a firsthand sighting—the second of only two—and is therefore of the utmost significance; see also Dunn. No reference is made in the article or its bibliography to any of the three earlier papers, suggesting that this is likely an independent account, effectively a rediscovery. This is somewhat surprising, given that Dunn’s article also appeared in the Mathematical Gazette. Of particular note is the reference to the tiles being coloured, or arranged in the same colour “back to back”: this is the first recorded instance, and indeed the only one.
Oddly, he refers to the Cairo tiling as a “traditional Islamic tessellation”, for which there is no evidence. The text is not an easy read for the enthusiast, and what can be understood is open to interpretation. Whether by design or not, the diagram he provides is not the in situ model, although he does appear to infer awareness of it through his mention of the collinearity feature.

1982 (1)
George E. Martin. Transformation Geometry: An Introduction to Symmetry, New York: Springer-Verlag, 1982, 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 a Dunn, Gardner, Schattschneider, or Macmillan quote.
https://archive.org/details/transformationge0000mart (search only)

1984 (1)

William Blackwell. Geometry in Architecture. John Wiley & Sons, New York, 1984, pp. 54–55.

The tile pattern of Figure 5.12 has the appearance of interlocking hexagons but consists of identical equal sided (but not equal angular) pentagons. The hexagonal patterns cross at right angles and the whole 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 mosaics of Moorish buildings, combines elements of four-, five-, and six-sided polygons and is another of the shapes in geometry in which, mathematically, the square root of seven plays a part. (The diagonal dimension of the rectangle enclosing the equilateral triangle includes the square root of seven.) 

Like all other tile patterns, this pattern can provide the basis for a three-dimensional assembly of prisms. The photograph of a model (Figure 5.13) shows such an assembly with the pentagonal prisms proportioned to minimize the total surface area of each piece. With this proportion, the height as well as the dimensions of the base embodies the square root of seven.

[Caption] A pentagonal tile pattern with pieces equilateral but not equiangular. The pattern retains a right-angular relationship to the walls of a rectangular room and can be subdivided in several ways.

Likely taking his lead from the Gardner quote, as the latter part is almost word-for-word. In so many words, leaning on Gardner, states that the in situ paving is equilateral. Gives a nice diagram showing different unit cells for a repeating square matrix.

Shows the Donald Wood pentagonal prisms.

Snippet view in Google Books, showing what appears to be an equilateral pentagon

https://archive.org/details/geometryinarchit0000blac

Search only, but there is a time-consuming workaround for single pages of interest.

1985 (1)

Steven Maddock. ‘Shape Up With Tesselations’ (sic). Acorn User Magazine, December 1985, pp. 82–85.

…The plane cannot be tesselated by regular pentagons, but there are a number of irregular pentagons that will. Such an example is the well-known Cairo tile, shown opposite, so-called because many of the streets of Cairo were paved in this pattern. The Cairo tile is equilateral but not regular because its angles are not all the same.

A discussion/promotion on McGregor and Watt's forthcoming new book, The Art of Microcomputer Graphics for the BBC Micron/Electron. Some text repeats from the book (as in the quote above). Interesting in that as early as 1985, the Cairo tile was described as “well-known”! 

Also, a nice witty title!

1986 (3)
James McGregor and Alan Watt. The Art of Microcomputer Graphics for the BBCMicron/Electron. Addison-Wesley, 1986, pp. 196–197.

The plane cannot be tesselated by regular pentagons, but there are a number of irregular pentagons that will. Such an example is the well-known Cairo tile, so called because many of the streets of Cairo were paved in this pattern.The Cairo tile is equilateral but not regular because its angles are not all the same.

A popular discussion within a chapter on tessellations.

Possibly (along with the Gardner reference), this is where I first saw the Cairo tiling, in 1986, in the Grimsby College library.

Note also that there was another version of the text for the IBM PC, The Art of Graphics for the IBM PC.

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

Abstract
This article presents an attempt to employ a programming approach to mathematical formalization. Six self-instructional units (120 instructional hours) have been developed for the Open University in Israel.

Unit No. 5: Ornamental group.

The unit starts with a presentation of transformation in the plane…The main part of this unit deals with tessellating the plane with convex polygon tiles. This part starts by discussing the drawing of Dutch artist M. C. Escher and the art of filling the plane with similar interlocking figures…Then the possible ways of tiling with pentagons are explored, especially the Cairo tiling. There are 17 ways of tiling the plane with regular polygons if the “edge-to-edge” condition is imposed…

A brief mention of the Cairo tiling in passing in the context of tiling in general. Significant in that this is the first known instance where “Cairo tiling” is used (and has become the de facto term), as against “Cairo tessellation” or “street tile in Cairo”, etc.

George E. Andrews. Percy Alexander MacMahon: Number theory, invariants, and applications. MIT Press, 1986, pp. 195–210. Cairo reference p. 196. Chapter 15, “Repeating Patterns”.
Of the recent work on repeating patterns, a paper by D. Schattschneider (1978a) beautifully bridges the gap between the recreational and serious aspects of repeating patterns. Her article is reprinted in the next section…

It is said to appear as street paving in Cairo...

Here, Andrews simply repeats the Schattschneider article in full, where the Cairo tiling is mentioned, as detailed above. Consequently, the Cairo reference is by default (and already discussed).

1989 (4)
W. K. Chorbachi. Computers and Mathematics with Applications. ‘In the Tower of Babel: Beyond Symmetry In Islamic Design’. Vol. 17, No. 4–6, 1989, 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’.

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. The first known French reference.

Istvan Hargittai. Symmetry 2: Unifying Human Understanding. Pergamon, 1989, p. 784.

Repeats the Chorbaci text.

Michael Serra. Discovering Geometry: An Inductive Approach. Key Curriculum Press, 1989, pp. 316, 323. Also see a later edition of 1997. 

Page 316 …The tile pattern on the right is made of identical pentagons (non-regular) that could completely cover a floor without gaps or overlaps …The pentagon tessellation can be found in the street tiling of portions of Cairo and many other ancient cities in the Islamic world…

Page 323
1. Another very beautiful pentagon tessellation uses equilateral pentagons (the sides are congruent but not the angles). An example is the Cairo street tiling shown at the beginning of Lesson 7.5. The construction of an equilateral pentagon is shown below. On poster-board or heavy cardboard, construct an equilateral pentagon and use it as a template to recreate the Cairo street tiling.

Chapter 7, pp. 294–341, is on transformations and tessellation. Gives an interesting variation of the construction of the Cairo-like equilateral pentagon.

Internet Archive. Not downloadable

1990 (1)

Francis S. Hill. Computer Graphics. Macmillan Publishing Company, New York, 1990, 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.

1991 (3)

Ann E. Fetter. et al. The Platonic Solids Activity Book. Key Curriculum Press/Visual Geometry Project. Backline Masters, 1991, 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.

David Wells.The Penguin Dictionary of Curious and Interesting Geometry. Penguin Books, 1991, 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.

Arthur F. Coxford, Linda Burks, Claudia Giamati, Joyce Jonik. Geometry from Multiple Perspectives. Reston, Va.: National Council of Teachers of Mathematics, 1991, p. 68.

The duals of 3-3-4-3-4 and 3-3-3-4-4 are each made up of pentagons. In the former case, the pentagon is the Cairo pentagon, and in the latter it is like home plate in baseball.

The first known use of the term "Cairo Pentagon".

Internet Archive. Not downloadable.

1992 (1)

Steven P. Meiring. A Core Curriculum: Making Mathematics Count for Everyone. Curriculum and Evaluation Standards for School Mathematics Addenda Series, Grades 9-12. National Council of Teachers of Mathematics, 1992. p. 68. 

The duals of 3-3-4-3-4 and 3-3-3-4-4 are each made up of pentagons. In the former case, the pentagon is the Cairo pentagon, and in the latter it is like home plate in baseball.

This repeats the text from the 1991 book by Coxford.

Internet Archive. Not downloadable

1993 (1)

Nenad Trinajstić. 'The Magic of the Number Five'. Croatia Chemica Acta 66 (1), 1993, 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....

1994 (2)

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.

Carter Bays. Complex Systems Publications, Volume 8, Issue 2, 1994, 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. The first of two articles by Bays on the Cairo tiling; also see 2005.

1996 (1)

Michael O’Keefe and Bruce G. Hyde. Crystal Structures No. 1. Patterns & Symmetry. Mineralogical Society of America, 1996, p. 207.

The pattern is known as Cairo tiling, or MacMahon’s net and In Cairo (Egypt) the tiling is common for paved sidewalks…

Also see their 1960 entry in non-attributed instances. O'Keefe and Hyde are the source of "MacMahon's net".

1998 (1)

David A. Singer. Geometry Plane and Fancy, Springer, 1998, pp. 34.
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…

although it is not regular, it is equilateral, that is… 

1999 (1)
John Gregory. Investigating with TesselTiles. Vernon Hills, Ill.: ETA, 1999, p. 2.

The red Cairo pentagon—a shape used to create several walkways in Cairo, Egypt—contains two right angles. Although it is equilateral, this pentagon is not equiangular.

Internet Archive. Not downloadable

2000 (1)

M. Deza,  P. W. Fowler, A. Rassat, and K. M. Rogers. 'Fullerenes as tilings of surfaces'. Journal of Chemical Information Computer and Modelling. ACS Publications, 2000, pp. 550–558. See p. 554.

Another such example but with congruent convex pentagons is the Cairo tiling [the dual of the Archimedean tiling (3.4.3.4)].
Advanced. A brief mention in passing.

2001 (1)

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

For children. No evidence of a text discussion.

2003 (3)

Teacher’s Guide: Tessellations and Tile Patterns, p.30 (Cabri) Geometric investigations on the VoyageTM 200 with Cabri. Texas Instruments Incorporated. 2003.
….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.

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…

Chris Pritchard. The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching, 2003, 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 in possession, Google Books reference.

2004 (1)

Robert Parviainan. ‘Connectivity Properties of Archimedean and Laves Lattices’. Uppsala Dissertations in Mathematics 34. 2004, p. 9.
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.

Two mentions of Cairo. 

Free download

2005 (4)

David Mitchell. Sticky Note Origami: 25 Designs to Make at Your Desk, Sterling Publication Company, 2005, 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. Cairo tiles are a special type of pentagon...

The first page is text followed by two pages of diagrams.

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

Problem 6.5.3. Construct a dual of the 3. 3. 4. 3. 4 tessellation by joining the centres 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.
Problem 6.5.4 

Two references

Google Books

Sue Johnston-Wilder and John Mason (eds.). Developing Thinking in Geometry. The Open University, 2005, pp. 182, 184.

P. 182. Some pentagons do tessellate… The one shown in Figure 10.1k is often referred to as the Cairo tessellation as it appears in a mosque there.

As you look at the tiling, you may become aware of switching your attention between pentagons and hexagons. The hexagons make it easier to see why the tiling can be extended indefinitely…

P. 184. Task 10.2.2 Generalising the Cairo Dual 

Using the Cairo tiling draw its dual using Fig. 10.2c. Convince yourself, and preferably someone else, that there are relationships between the numbers of edges adjacent to a vertex in a tiling and its dual, and the number of edges of the tiles.

Poor research, with the repeated falsehood of a mosque appearance and with no mention of the true street paving appearance, especially so after many previous references to street paving going back 34 years! 

Not in possession, Google Books limited preview; by chance, both pages are featured.

Carter Bays. Complex Systems Publications, Volume 15, Issue 3, 2005, pp. 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, which in its most regular form is composed of equilateral "isosceles" pentagons…’
On CA (Cellar Automata). The second of Bays' article on the Cairo tiling; also see 1994.

2006 (1)
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.

2007 (4)
B. G. Thomas and M. A. Hann. "Patterned Polyhedra: Tiling the Platonic Solids". In Sarhangi, Reza (Ed). Bridges. Mathematical Connections in Art, Music, and Science. (Ninth) Conference Proceedings 2007. Donostia, Spain. 

The regular pentagon, with five-fold rotational symmetry, cannot tile the plane without gap or overlap. There are however various equilateral pentagons that can tessellate the plane. Probably the best known is the Cairo tessellation, formed by convex equilateral pentagons (equal-length sides, but different associated angles). Using knowledge of the Cairo tessellation, the method used by Schattschneider and Walker [16] presents one solution to this problem, but this is an area requiring substantial further investigation. 

This is apparently the first reference of three where Thomas & Hann state the equilateral model. Two references to Cairo.

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 (erroneously) as equilateral.

Mike Ollerton. 100+ Ideas for Teaching Mathematics 2007, p. 66.

This tessellation not only begs interesting questions about angle sizes and side ... The Cairo tessellation... The challenge is to use this tile to fill 2D space.

Google snippet.

Martin Kindt. ‘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.

L. J. Frobisher, Anne Frobisher, A. Orton, J. E. H. Orton. Learning to Teach Shape and Space: a handbook for students and teachers in the primary school. Cheltenham: Nelson Thores, 2007, p. 153.

The Cairo tessellation shown in Figure 6.48 is very famous. Why do you think it is considered special? How would you describe it over the telephone to a friend so that they could sketch it? Work out how to draw it accurately.

For children

Internet Archive 

2008 (5)

Anon. Key Curriculum Press. Chapter 7, 'Transformations and Tessellations', 2008, 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.

Merrilyn Goos et al. Teaching Secondary School Mathematics: Research and Practice for the 21st Century. 2008
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 in possession, Google Books reference. Very curious; the ‘famous mosque’ has evaded detection! Likely quoting and extrapolating from Gardner.

B. G. Thomas and M. A. Hann. "Patterning by Projection: Tiling the Dodecahedron and other Solids". In Sarhangi, Reza (Ed). Bridges. Mathematical Connections in Art, Music, and Science, 2008, p. 101.

…There are, however, equilateral convex pentagons that do tessellate the plane, such as the well-known Cairo tessellation shown in Figure 1. Using knowledge of the Cairo tessellation, the method presented by Schattschneider and Walker [4] provides one solution to the problem of applying a regularly repeating pattern to the dodecahedron. Figure 2 illustrates the manipulation of a pattern based on the Cairo tiling in order to tile the dodecahedron, in which the pattern is projected outwards from the faces of an inscribed cube….

Seven references to Cairo; six as tessellation, and one on tiling. 

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

Robert Fathauer. Designing and Drawing Tessellations, Tessellations, 2008, 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.

In a general discussion on paving stones. An equilateral pentagon is shown, although not stated.

2009 (2)

Lisa Iwamoto. Digital Fabrications: Architectural and Material Techniques. New York: Princeton Architectural Press, 2009, p. 112.
— project uses a pentagonal Cairo tessellation pattern, flexibly aggregated to yield multiple overall arrangements. Each vertical layer of the cell was formed using CNC-milled molds, and the resulting shells fit together like a plastic container. The parts were rigid, thin, light, and easily transportable.

Caption— Sidewalk in Cairo featuring pentagonal Cairo tessellation. Photo: Craig Scott

Of note as the first known purposeful in situ picture. Iwamoto and Scott are partners. The book also has later editions of 2013 (and 2019?)

Internet Archive, not downloadable. The book was uploaded to IA on November 3, 2021.

Craig Scott is not a new name to me, with correspondence dating back to 2011, from a Flickr posting of his, when I first became aware of this.
https://archive.org/details/digitalfabricati0000iwam/page/112/mode/2up?q=Digital+Fabrications

Craig S. Kaplan. Introductory Tiling Theory for Computer Graphics. Morgan & Claypool Publishers, 2009, p. 33.

In Chapter 4.2, on Laves Tilings

12, The Laves tiling [3.3.4.3.4] is sometimes known as the “Cairo tiling” because it is widely used there. Place the prototile from the Cairo tiling…

Implying that the dual is the in situ model.

Not in possession, Google Books reference.

Faith H. Wallace. Teaching Mathematics through Reading. methods and materials for grades 6-8. Linworth Pub, 2009, p. 50.

The Cairo Tessellation is an attractive and intriguing pattern, frequently used in Islamic design, is created by tessellating irregular pentagon tiles. Tiles can be combined as a simple irregular hexagon or in more complex ways. Contrasting color sticky notes add effect.

For children

Internet Archive 

2010 (4)
Claudi Alsina and Roger B. Nelsen. Charming Proofs: A Journey Into Elegant Mathematics. Dolciani Mathematical Expositions, 2010, pp. 163–164.

In Chapter 10. Adventures in Tiling and Colouring, pp. 159–

P. 163. Another pentagonal tiling can be created by overlaying two non-regular hexagonal tilings illustrated in Fig. 10.6. This rather attractive monohedral 

P. 164. pentagonal tiling is sometimes called the Cairo tiling, for its reported use as a street paving design in that city.

Also see Alsina & Nelsen’s Panology of Polygons, 2023.

Not in possession, Google Books limited preview.

Richard Elwes. Maths 1001: Absolutely Everything You Need to Know about Mathematics in 1001 Bite-Sized Explanations. Quercus, 2010, p. 109.
... some irregular convex pentagons which do. There are 14 essentially different known ways for this to happen. One of these is the Cairo tessellation (illustrated) which adorns the pavements of that city.
One reference. Internet Archive.

Eric Ressouche, Virginie Simonet, Benjamin Canals, Marin Gospodinov, Vassil Skumryev. 'Magnetic frustration in an iron-based Cairo pentagonal lattice'. Physical Review Letters. 2009
The pentagon, a 5-edges polygon, is an old issue in mathematical recreation... It exists however several possibilities of tessellation of a plane with nonregular pentagons, a famous one being the Cairo tessellation whose name was given because it appears in the streets of Cairo and in many Islamic decorations (Fig. 1).

Seemingly the first of “high-level physics” papers. It repeats the misapprehension. 11 references to Cairo, but none of a “Cairo Pentagon”.
I asked Copilot AI for any earlier instances of such high-end physics-type papers, but it could not find any. It stated:
The 2009 paper is the first to show that Bi₂Fe₄O₉ contains a Cairo‑type Fe³⁺ lattice — that discovery created the field. 

Freely available. From this date, there are many others of the same broad type.

Subhash Chandra Saxena. College Geometry: A New Paradigm: (a rigorous exploratory approach using technology) Myrtle Beach. S.C.: Sherian Press, 2010, pp. 13, 15

Exploration 4.2.3 creates tessellation with a Cairo pentagon.

and

P. 13 There is an interesting equilateral pentagon, which is not equiangular and is used in tessellations in Chapter 4. It is constructed as follows: Example 1.2.2: (Cairo pentagon):

P. 13…This pentagon is occasionally called the Cairo pentagon, because several streets in that capital of Egypt have tessellations of this pentagon, as shown on the right of Figure 1.2.10.

P. 108 — It is interesting to note that while a regular pentagon cannot produce a pure tessellation, some equilateral pentagons can. One of them is generated by the so-called Cairo pentagon (Figure 4.2.6 (a)). This tessellation is quite well known in the Muslim world, especially in some streets of Cairo.

P. 109 — Another pentagon that can tessellate in the style of Cairo pentagon, and is easier to construct, is shown in Figure 4.2.7. It is not equilateral, but has four congruent sides. Those four enclose three angles, two of them right angles, and one measuring 120°. The remaining two angles also measure 120°. Call it De Figure 4.2.7 “Almost Cairo pentagon” as it tessellates in exactly the same way. The details are left for the reader to explore (Exercise 4.2.12).

12 mentions of a Cairo pentagon. Of note is that Saxena distinguishes what she believes to be the in situ model with a variant, which she terms “almost Cairo pentagon”!

Internet Archive. Not downloadable

2011 (5)

M. Rojas,  Onofre Rojas, and S. M. de Souza  'Frustrated Ising model on Cairo pentagonal lattice'. Physical Review E 86(5-1):051116 November 2012.
More recently Urumov [9] considered the Ising model on the Cairo pentagonal lattice using the decoration transformation [10]... 

11 references to "Cairo pentagonal lattice", never tiling or tessellation. No reference is made to the origin. Seemingly one of the first of 'high-level physics' of the Ising model, from this date there are many others. 

Mike Askew and Sheila Ebbutt. The Bedside Book of Geometry: From Pythagoras to the Space Race: The ABC of Geometry. 2011, Murdoch Books Pty Limited.

Mike Askew. The Bedside Book of Geometry: from Pythagoras to the Space Race: the ABC of Geometry. London: New Burlington Books, 2011, p. 109, two references

(4, 3, 3, 4, 3). Marking the centre of each of these and joining these points produces the dual tiling of Cairo pentagons.

Internet Archive. Not downloadable.

Also see a US edition of the book, of the same year, but under a different title.

Mike Askew. Geometry: The Size and Shape of Everyday Math. New York: Metro Books, 2011, p. 109, two references
(4, 3, 3, 4, 3). Marking the center of each of these and joining these points produces the dual tiling of Cairo pentagons.

Internet Archive. Not downloadable

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 ...
Not in possession, Google Books

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

Iron-Based Cairo Pentagonal Lattice

Academic physics.
Not in possession, Google Books

Abdul Karim Bangura. African Mathematics: From Bones to Computers. University Press of America, 2011. 

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

A solitary reference in passing.
Not in possession, Google Books

2012 (1)

Edited by Christoph Gengnagel, A. Kilian, Norbert Palz, Fabian Scheurer. Computational Design Modeling: Proceedings of the Design Modeling Symposium. Springer, 2012, p. 229.

…on the mathematical configuration of a Cairo tessellation

Not in possession, Google Books, Snippet view only, two references. 

Calvin T. Long, Duane W DeTemple, Richard S. Milman. Mathematical Reasoning for Elementary Teachers. Boston: Pearson Addison Wesley, 2012, p. 648.

(b) Trace the semiregular tiling of Figure 11.25 whose vertex figure consists of two non-adjacent squares and three equilateral triangles. Then construct the dual, which is known as the Cairo tiling, since the paving stones of the streets of Cairo make this pattern.

In Chapter 8, Transformation Symmetries and Tilings.

2013 (4)

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

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 in possession, Google Books reference.

Ruairi Glynn 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. 

A recurring pattern in our office is the Cairo pattern, where four pentagons combine to comprise a hexagon. The streets of Cairo are paved with stones of this geometry. … Cairo tower

When considering structure for a three-story pavilion, we explored the 3D potential of a 2D pattern: the Cairo hexagon. Comprised by an arrangement of four pentagons, we folded the Cairo along its pentagonal seams up into a version of a space frame. 

An architectural project. 10 references to Cairo with various appendages.

Glenn Ellison. Hard Math for Elementary School: Answer Key for Workbook. CreateSpace Independent Publishing Platform, 2013, p. 36. 

The pentagon below has three 120° angles and two 90° angles. It is sometimes called the Cairo pentagon because there are streets in Cairo, Egypt that are paved with stones in this shape. Make some Cairo pentagons by tracing this shape and see if you can figure out how they tile.

Internet Archive. Not downloadable
Three references.

Wassim Jabi. Parametric Design For Architecture. Laurence King Publishing, 2013, pp. 64–66. (2 June 2026)

p. 64…In this project, the studio focused on the implementation of a new surface iconography based on the digital manipulation of a traditional tiling pattern, called Cairo tessellation. The tiles designed by the studio were used to compose a relief panel for a residential entry foyer, in which the original pattern of the Cairo tiling is also maintained, but only as a geometric background.

For the design of the tiles, the firm began with a 30-degree tilted square grid that underwent four stages of tessellation. The result was a grid of irregular pentagons that maintained the topology of the original tiles. The studio identified five focal points on each pentagon, inspired by the pattern of the Cairo tiling, and applied a network of curvilinear streamlines — a visualization of lows — between those points. The pattern that was produced in this manner visually suggested a 3D interpretation of the original 2D floral pattern of the Cairo tiles…

p. 66. ...These final tiles are made from polyurethane and they possess a highly reflective, undulating surface that lacks any classical geometric clarity or symmetry, but which echoes the repetitive geometry of the traditional Cairo tile. In the final product, the geometric essence of the Cairo tile pattern has been morphed into a freer, more seductive form, which extends into the third dimension.

No rationale is given for the naming. Their model is the dual.

2014 (3)

Ralf Benölken, 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'.

Not in possession, Google Books reference.

David E. Laughlin and ‎Kazuhiro Hono. Physical Metallurgy, p. 76, 5th edition, 2014.

Dual to the Archimedean snub square tiling 32.4.3.4 is the Cairo pentagonal tiling ... nets in … the Catalan Cairo pentagonal tiling V32.4.3.4

Not in possession, Google Books reference.

David G. Wells. Motivating Mathematics: Engaging Teachers And Engaged Students. Imperial College Press, 2015, p. 218. 

Next is the well-known Cairo tessellation…

Not in possession, Google Books reference.

2016 (1)

Walter Steurer and Julia Dshemuchadse. Intermetallics: Structures, Properties, and Statistics. OUP Oxford, 2016, pp. 40, 42, 366, 387, 498.

3.2.2 2D Archimedean (Kepler) tilings

P. 40. …For example, the Cairo pentagon tiling V3.3.4.3.4, consisting of smashed pentagons, is the dual of the Archimedean snub square tiling  3.3.4.3.4…

(many minor references not shown) 

I am at a loss to understand “smashed pentagons”! The subject text is outside my field of interest, and is far too advanced fro my limited knowledge..

Not in possession, Google Books reference.

2017 (4)

Robert J. Lang. Twists, Tilings, and Tessellations: Mathematical Methods for Geometric Origami. CRC Press, 2017, pp. 385–387, 402. 707 pp. 

Large tome, Chapter 5, Tilings, pp. 345–403.

P. 385. If two non-consecutive angles of an equilateral pentagon sum to 180°, as in Figure 5.49, then the pentagon and its mirror image can be assembled into a tiling in two different ways, as shown in Figure 5.50. This tiling is called a Cairo tiling and has been used as a floor tiling in the Arab world.

P. 386. Figure 5.49. Schematic of a Cairo tile. All sides are equal length; two non-consecutive angles sum to 180°.

P. 387. Figure 5.50. Two different forms of a Cairo tiling from an equilateral pentagon with corner angles of (105°, 87.61°, 143.33°, 75°, 129.06°). Each tiling is composed of equal numbers of the pentagon and its mirror image.

The subdivided pentagon can then be placed onto the Cairo tiling, giving rise to the tessellation in Figure 5.52.

Figure 5.51. A mirror-symmetric Cairo tile. Left: the tile. Right: dissected into isosceles triangle and symmetric trapezoid.

P. 402. Terms. Cairo tiling A particular tiling of identical pentagons

P. 453 (shows the Gjerde diagram). 

[Caption] Figure 6.39.

Left: “Cairo Tessellation” (2011), a flagstone tessellation by Eric Gjerde, based on the Cairo tiling.

Right: the reverse, showing simple folds at the vertices

There are multiple ways to realize flagstone tessellations, and while the construction in this section is quite general, for specific patterns, there are often simpler treatments of the folds around the vertices. A particularly elegant one is the flagstone tessellation by Eric Gjerde shown in Figure 6.39. This is based on the Cairo tiling that we met back in Chapter 5; fitting, perhaps, because the Cairo tiling is itself often used for flagstone paving.
Interesting in many ways—perhaps inadvertently on Lang’s part. He appears uncertain about what constitutes the Cairo tiling. This is unusual, as most authorities at least agree on the basic models, even if they sometimes misidentify the dual or the equilateral pentagon as “Cairo”. Both of those have two adjacent right angles, yet none of the examples Lang presents possesses this feature. He does include an equilateral pentagon, but again without the defining right‑angle condition. Trying to unravel his understanding is difficult, and I confess I baulk at the task. The investigation has already taken most of the day, and I cannot justify spending still more time on it.
My best guess is that he encountered a description of the Cairo tiling as “equilateral” somewhere, and adopted a superficially similar pentagon under that assumption. But even this does not explain the Gjerde model, which is a traditional Cairo tiling. His bibliography contains very few tiling references, despite the otherwise advanced mathematical level of the book.
He also neglects the standard explanation of the pattern as a street paving, referring instead to a “floor tiling” (although he later describes Gjerde’s example as “flagstone paving”). His geographical description is similarly imprecise: the broad reference to the “Arab world” is vague in the extreme and gives the misleading impression that the pattern is ubiquitous there, which it is not.
All that aside, the models he presents are genuinely interesting, especially Figure 5.50 on p. 387. The hexagon outlines give the superficial impression that (a) they meet at right angles, as in the standard Cairo‑type constructions, and (b) the hexagons are identical. Neither is true: the angles differ, and the hexagons are not congruent. Nonetheless, offhand, I can’t recall having seen this before. It may be derived from Schattsneider’s “Tiling the Plane with Congruent Pentagons” listing on pp. 40–42 gives a listing of equilateral pentagons known to tile. This has all the appearance of a Type 8, although there are other possibilities. Interestingly, only one since has “simple” angles, with others being of a more advanced mathematical description. In contrast, Lang gives “simple” angles. To try and find another example, I put the angles into Google and Copilot. None could find such a pentagon. On second thought, perhaps it is best described in the advanced notation of Fig. 5.51, but I lack the will to investigate.
Various terms appear throughout—“Cairo tiling”, “Cairo tile”, “Cairo pentagon”, and “Cairo tessellation”.

Simply stated, more time is required to study this than I have time for.

Aakash Moncy. 'Mechanics of Cairo lattices'. Thesis, 2017, 84 pp.
Abstract
A tessellation made up of equal sided pentagons, called the Cairo lattice, is being investigated for its applicability as a structural 2D planar lattice….Under periodicity, the Cairo tessellation has a higher elastic buckling strength…
P. 4. The aim of this thesis is to explore the mechanics of a unique tessellation made up

of equal-sided pentagons, more commonly known as the Cairo tessellation….
P. 13. Cairo tessellation is a five point geometrical pattern based on pentagons, otherwise also called Cairo pentagonal pattern because of its origins in Egypt. It has been realized that such a tessellation has not been investigated as a potential geometry for a lattice material. But certainly among architects, the Cairo tessellation is of great interest with respect to modern design. There are many ways of generating the Cairo pentagon [37]... 

“Materials science”. Thre are 231 references to “Cairo”! Numerous references throughout, too many to list. The title varies throughout, with the default description being “Cairo lattice” (141). “Cairo tiling” is not used at all, “Cairo tessellation” (24), and “Cairo pentagon” (2).

A single sentence describes the origin.

Quotes Lockerbie in the references [37]

https://aaltodoc.aalto.fi/items/524b1ada-147e-4a18-bdfe-fd8e3c13d448

Ponnadurai Ramasami (ed). Computational Sciences. De Gruyter, 2017, p. 57.

Another proposed allotrope is penta-graphene [181] which is composed of pentagons with C at its vertices, which closely resembles MacMohon’s [sic]  net, a semiregular tiling of the Euclidean plane similar to Cairo pentagonal tiling. 

A single reference in a 248-page academic book on “high-end” chemistry, for want of a more technical description,

Not in possession, Google Books. A PDF is on ResearchGate.

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

(Pentagraphene)… resembling the Cairo pentagonal tiling

Not in possession, Google Books

Toshikazu Sunada. “Topics on Mathematical Crystallography”. In Tullio Ceccherini-Silberstein, Maura Salvatori, Ecaterina Sava-Huss. Graphs And Random Walks. London Mathematical Society Lecture Note Series 436, Cambridge University Press, 2017, pp. 475–519. See p. 511.

Figure 16.14 exhibits a few more examples of standard realizations (the picture on the right side is a tiling of pentagons with picturesque properties that has become known as the Cairo pentagon).

Advanced crystallography. Nothing more than in passing.

Laurent Najm, ‎Pascal Romon (eds.). Modern Approaches to Discrete Curvature. Springer, 2017, p. 180.

Next, we discuss examples of non definite curvature, see Fig. 6.2 as well. Example 6.2 The so called Cairo tiling consists of pentagons... 

2019 (3)

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

In April 2019, I set off on a pilgrimage to see for myself a sidewalk incarnation of the best tiling by unit-area pentagons (Figure 1). It is called a Cairo tiling, because, I had been told, similar tilings can be found in Egypt on the streets of Cairo…

References throughout. On his recent visit of the same year, following up on the report on my webpage as to sightings. I get an in-person mention. His picture, of the Gezira Sporting Club, was used as the front cover, possibly a first!

Mircea Pitici (ed). The Best Writing on Mathematics. Princeton University Press, 2019. 114–116. 

A portion of the Cairo tiling. 

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

Not in possession, Google Books

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.

The ‘Cairo tiling’ (Fig. 1) has many names, as we will see in Section 2. In particular, it is the dual of the snub version of the familiar square tiling.

[Section 2] The Cairo tiling is shown in Fig. 1. This beautiful tiling has many names. It has also been called the Cairo pentagonal tiling, the MacMahon net (O’Keeffe & Hyde, 1980), the mcm net (O’Keeffe et al., 2008), the dual of the 32:4:3:4 tiling [Grunbaum & Shephard, 1987, pp. 63, 96, 480 (Fig. P5-24)], the dual-snub-quadrille tiling, or the dual-snub-square tiling (Conway et al., 2008, pp. 263, 288). We will refer to it simply as the Cairo tiling. There is only one shape of tile, an irregular pentagon, which may be varied somewhat.

29 references to “Cairo”, which is 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 with The tiling is named from its use in Cairo, where this pentagonal tile has been mass-produced since at least the 1950s and is prominent around the city, but not by name.

In their discussion within the main study, Goodman‑Strauss & Sloane give a survey of alternative names applied to the Cairo tiling (and to other pentagonal tilings), pp. 123 and 130. My primary concern is with O’Keeffe & Hyde’s terms “MacMahon net” and “mcm net”. Why introduce additional terminology, and so complicate an already established name. It does not make sense. At least Goodman‑Strauss and Sloane remain consistent with “Cairo tiling”, although they misattribute it to the dual.

Furthermore, on p. 130, O’Keeffe (and Conway) give alternative names to the snub‑632 tiling, more commonly known as the “floret”, namely “fsz‑d net”. Why? O’Keeffe in particular does this twice.

2022 (2)

Eli Maor (author), Eugen Jost (illustrator). Pentagons and Pentagrams. Princeton University Press, 2022, pp. 95–96.

Another example of pentagonal-hexagonal tiling is the Cairo tessellation, so called because it appears frequently in mosques across the Middle East and North Africa.

Two falsehoods in one sentence!

Cameron Browne, Akihiro Kishimoto, and Jonathan Schaeffer (eds.). Advances in Computer Games: 17th International Conference, ACG 2021, Virtual Event, November 23–25, 2021. Springer, 2022, p. 243.

5 Graph Operators

…based on tiling 3.3.4.3.4 to produce the well known Cairo tiling:

A mention essentially in passing. Gives the dual.

Not in possession, Google Books reference.

2023 (2)

Claudi Alsina and Roger B. Nelsen. Panoply of Polygons. American Mathematical Society, 2023, pp. 25, 44.

1.8. Voronoi Diagrams and Dual Tilings

P. 25. …The one on the right is called a Cairo tiling, whose pentagons have two 90° angles and three 120° angles.

Shows the dual.

Also see Alsina & Nelsen’s Charming Proofs, 2010.

Not in possession, Google Books. A limited preview curtails a fuller discussion.

Colin Adams. The Tiling Book: An Introduction to the Mathematical Theory. American Mathematical Society, 2023, p. 156.

P. 156. Frank Morgan and his students proved that both the Cairo tiling and the prismatic tiling, two of our Laves tilings as in Fig. 2.8, are minimal perimeter pentagonal tilings, yielding…

? Cairo Pentagonal Tiling Prismatic Pentagonal Tiling Figure 2.88: The Cairo and prismatic tilings pentagons are both least perimeter

Quoting Morgan et al.

Not in possession, Google Books reference. A limited preview curtails a fuller discussion.

2025 (1)

Benedek Nagy and Tibor Lukić. “Binary Tomography on the Cairo Pattern”. Combinatorial Image Analysis: 23rd International Workshop, IWCIA 2025, Szeged, Hungary, September 24–26, 2025, Revised Selected Papers.

In this paper, we investigate a tomography reconstruction approach for binary images on the Cairo pattern. In discrete tomography the underlying grid plays an essential role, as the projection data and the structure of the grid together give the information that we can use for the reconstruction purpose. The Cairo pattern is one of the most known pentagonal tessellations of the plane and it has a nice symmetric structure

Many references, all advanced.

Not in possession, Google Books reference.

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 ...

2. A simple listing of non-attributed references of the Cairo tiling, in chronological order, with quotes and comments where considered appropriate

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. A reference to a jali, albeit, matters of provenance are a little understated. See p. 178 in the Indian and Islamic Art catalogue, 2016.

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’.

1900 (1)

Max Brückner. Vielecke und Vielfläche: Theorie und Geschichte. (Translated: Polygons and Polyhedra) Leipzig: B. G. Teubner, 1900, p. 158

Within a book on polyhedra. Highly technical, with much abstruse text, albeit liberally illustrated with line drawings, and latterly plates and polyhedral models. Of interest as regards tiling p. 109 dual tiling (Cairo) p. 158.

1909 (1)
Herbert C. Moore. ‘Tile’. United States Patents 928,320 and 928,321 of 20 July 1909.
...Referring to Figs. 1-4, inclusive, A is a unit formed as a five-sided or pentagonal tile having its sides A, A, A, and A' equal each to each...
The first recorded instance of a patent for a flooring. The tiling also appears in the second patent. Of note here is that Moore in principle shows a minimum and maximum deformation of the pentagon, to an implied rectangle and square, as outlined in more detail by Macmillan in his 1979 paper.

1921 (1)
Percy A. MacMahon. New Mathematical Pastimes. Cambridge University Press, 1921 and 1930. (Reprinted by Tarquin Books 2004)

The reader will observe that the tessellation (fig. 124 b) is composed of two sets of hexagons at right angles to one another.

Each hexagon contains four equilateral pentagons, one of each aspect…

81. The tessellation of pentagons that has been depicted is one of the most remarkable that can be met with in the subject.

If the patterns that have been given above, for the contact system under examination 1 to 4, 2 to 3, be inspected they will be found to include three convex pentagons, out of an infinite number that it is possible to construct. In particular the first given of the three is equilateral.

A lengthy discussion is given on the Cairo tiling amid other tilings. Cairo diagram p. 101.

MacMahon is another who comments on its properties.

1922 (1)
Percy A. MacMahon. ‘The design of repeating patterns for decorative work’. Journal of the Royal Society of 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.

1925 (1)
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.

1926 (1)
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.

1931 (1)
Fritz Laves. ‘Ebenenteilung in Wirkungsbereiche’. Zeitschrift für Kristallographie 76, 1931, pp. 277–284. 

Fig. 4 shows a Cairo-like tiling within the context of 'Wirkungsbereiche' (Influence). The paper has been translated into a Google Doc for a potential better understanding.  

1933 (1)
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.

1951 (2)
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.

'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?

1954 (2)
'Croton'. Cairo tiling used again as a crossword puzzle, in The Listener, 22 April 1954, puzzle 1251 Hexa-Pentagonal II, by 'Croton'.

Cyril Stanley Smith. “The Shape of Things.” Scientific American, vol. 190, no. 1, 1954, pp. 58–65.

In a general discussion on tiling.

1955 (1)
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.

1956 (2)
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.

A. F. Wells. The Third Dimension in Chemistry. University Press, Oxford, 1956, p. 24.

Polygons and Plane Nets
…The inverse of (b), resulting from the midpoints of the polygons, is a very elegant arrangement of pentagons which has the further point of interest that it can be obtained by superposing two hexagonal nets in the way shown in Fig. 20 (c).

Wildely quoted in tiling mathematics despite being a chemistry book! The premise here is the duality of the Archimedean tilings. The tiling diagrams use the term “nets” of the day for tilings, although he also uses “division of a plane”. Cairo tilings pp. 24–25, in the context of Laves tilings, although not named as such.

This is pleasing as Wells singles out the aesthetics, as well as making the premise of two hexagonal nets by emphasising, something he does not do for other tilings.

1958 (1)
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.

1963 (1)
H. S. M. Coxeter. Regular Complex Polytopes. Second edition. Dover Publications Inc., New York, 1961.
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.

1967 (1)
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 regard 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.

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.

1970 (2)
Ernest R. Ranucci. Tessellation and Dissection. J. Weston Walch, 1970.
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.

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

1972 (1)

Robert Williams. Natural Structure: Toward a Form Language. Eudaemon Press, 1972. pp. 38, 43. 263 pp.

A “significant expansion” of Handbook of Structure, 1968. This does not seem to be available.

The next edition was in 1979. There is no material difference.

P. 38. Fig. 2-8. (b) 32.4.3.4 and its dual tessellation. 

A Cairo tiling is shown, derived from the dual, side by side.

P 43.  Wood, D. G. 1967. "'Space Enclosure Systems," Bulletin 203. Columbus, Ohio: Engineering Experiment Station, The Ohio State University, describes a tessellation of the plane with equal edge pentagons and vertex angles of 131°24'; 90°; 114°18'; 114°18’ and 90°. (i.e. a Cairo tiling)

1974 (2)
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.

1975 (1)
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.

1976 (2)

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.

Phares O’Daffer. G; Clemens, Stanley R. Geometry. An Investigative Approach 1st edition 1976 (check), 2nd edition 1992, Addison-Wesley Publishing Company. (Note that I have the second edition, not the first.)
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).

1977 (1)
Lorraine Mottershead. Sources of Mathematical Discovery. Basil Blackwell, 1977, pp 106–107.
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 nearly.

1977 (1)
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!

1978 (2)
Peter Pearce and Susan Pearce. Polyhedra Primer. Dale Seymour Publications, 1978, pp. 35 and 39.
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.

Ernest H. Lockwood and Robert H. Macmillan. Geometric symmetry. Cambridge University Press, 1978 (and reprint 2008), p. 88.
‘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 something of a moot point, as its description of the Cairo tiles is extremely loose. Even so, Macmillan’s authorship makes this fragmentary account worth noting, not least because it does not quite align with the more formal treatment he later published in the Mathematical Gazette. 

1979 (1)

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.

1980 (1)

Michael O’Keefe and Bruce G. Hyde. “Plane Nets in Crystal Chemistry”, Philos Trans A Math Phys Eng Sci, 1980, 295 (1417): pp. 553–618. 

P. 557. 3. Tessellations Involving Irregular Polygons

It is known (see, for example, Cundy & Rollet 1961; MacMahon 1921) that the plane can be covered by equal-sided (but not regular) pentagons with two angles of 2X (MacMahon's net no. 18 below).

[Equations]

P. 567...Pentagon-only nets can be derived as the duals of the five-connected nets described above (i.e., derived by joining the centres of the polygons of those nets). One such, the dual of 32.4.3.4, is topologically equivalent to MacMahon's net mentioned above; the others appear to be of less interest.…as already discussed, there is some arbitrariness in assigning coordinates and cell dimensions and for some less common nets we give the coordinates found in crystal structures.

(18) [(53)2, 54, 53, 54], MacMahon's net. A = 3. Figure 20. We have already discussed this net above. 

P. 585 Net 18 ('MacMahon's net') is also a member of this family. 

Four references to “MacMahon’s Net” and likely the earliest by the authors, although the term is unique to them. A Cairo tiling is shown, p. 567, of the equilateral pentagon model as given by MacMahon (1921), hence the choice of term. Probably, O’Keefe & Hyde were unfamiliar with the now-established term (Cairo tiling) and, for the sake of a better description, named it after the source after seeing New Mathematical Pastimes. In other publications, they retain the name. They also note that the dual is topologically identical to the equilateral. No other tiling they show is similarly named. 

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 to 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 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 have credited MacMahon as the first known instance (at the time), but without naming it after him. Also see a later paper of 1996.

1982 (1)

Patrick Murphy. Modern Mathematics Made Simple. Heinemann London. Tessellations, Chapter 10, 1982, pp. 194–205, 262.

Cairo diagram, of equilateral pentagons (but not attributed), p. 200.

1983 (2)
John Willson. Mosaic and Tessellated Patterns. How to Create Them. Dover Publications, Inc. 1983. Plate 3

Cairo tiling plate 3.

Cyril Stanley Smith. A Search for Structure. The MIT Press, 1983.

Has a non-attributed Cairo tiling.

1986 (5) Jim McGregor and Alan Watt. The Art of Graphics for the IBM PC. Addison-Wesley Publishers, 1986, pp. 196–197? 207-208.

The plane cannot be tesselated (sic) by regular pentagons, but there are 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.
I seem to have two versions of the book. A minor part of a chapter on tessellations. Diagram p. 197.

A. L. Loeb. 'Symmetry and Modularity'. Computers and Mathematics with Applications, Elsevier, 1986

Jay Kappraff. ‘A Course in the Mathematics of Design’. Computers and Mathematics with Applications, Vol. 12B, Nos. 3/4, 1986, pp 913–948.

Cairo tiling in the context of the set of 11 Laves tilings, p. 923, but as such, is inconsequential.

Lothar Collatz. Geometrische Oranamente (in German), 1986

Cairo tiling diagram in the context of 43433? classification.

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.

1987 (2)
Bob Burn. The Design of Tessellations. Cambridge University Press, 1987. 

Sheet 30. Equilateral pentagon.

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, but it appears to be so.

1988 (1)
Loeb, Arthur. “Polyherda: Surfaces or Solids?”, pp. 106-117. In Marjorie Senechal and George Fleck, editors. Shaping Space: A Polyhedral Approach. Boston: Birkhäuser, 1988, p. 113.

Figure 6-12 shows two of the pentagonal tessellations about which Doris Schattschneider is an expert.

The Cairo tiling (but not named as such) discussed (and illustrated) amid the three pentagons as the dual of the.3. 3. 4. 3. 4 tiling.

1989 (4)
Dale Seymour and Jill Britton. Introduction to Tessellations. Dale Seymour Publications Cairo tiling (but not attributed), 1989, p. 39.

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

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 in the same ‘unusual’ configuration.

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, and as such is a "secondary" reference.

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

1991 (1)
Jay Kappraff. Connections: The Geometric Bridge Between Art and Science. McGraw-Hill. 1991, 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 (1)

Arthur L. Loeb. Concepts and Images: Visual Mathematics. Birkhäuser, 1993, pp. 94, 101.

Page 94. Another tessellating pentagon is shown in Figure 9-8 (a). Two of its angles are right angles; they are separated by an angle whose magnitude is still to be determined, and which we shall call a. Four such pentagons meet around a right-angled vertex, which will be a four-fold rotocenter. A mirror line bisects angle a, and the other right-angled vertex will also be a four-fold rotocenter, enantiomorphic to the first one. Accordingly, the symmetry of the tessellations, shown in Figure 9-8 (b), is 244.

Page 94. Fig. 9-8 (b). Another special pentagonal tessellation.

Page 101. Figure 10-1 shows how a pentagonal tessellation may be considered as the superposition of two mutually perpendicular and congruent hexagonal tessellations.
In a chapter on pentagonal tilings, various presentations of the Cairo tiling are shown, albeit unattributed. Curiously, Loeb never references the Cairo aspect in any of his writings. 

1999 (2)
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.

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

2005 (1)
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.

John D. Barrow. The Infinite Book. Vintage, 2005

Has brief tiling matters, with of significance the Cairo tiling p. 16, although without attribution, 

2006 (2)
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’.

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.

2008 (1)  B. G. Thomas, B. G. and M. A. Hann. In Bridges. Mathematical Connections in Art, Music, and Science, 2008.

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.

2014 (1)
Hans-Joachim Gorski and Susanne Müller-Philipp. Leitfaden Geometrie: Für Studierende der Lehrämter, Springer 2014, p. 186.

2016 (2)
Mark Neyrinck. ‘The Origami Cosmic Web’, The Paper, No. 122, 2016, pp. 26–27.

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

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.

2017 (1)
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 für 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.

3. References of all instances of the Cairo tiling, based on the in situ model or otherwise, attributed or not, in chronological order, with quotes and comments where appropriate

17th Century?
Simon Ray. Indian & Islamic Works of Art. Self-Published, 2016, pp. 178-179.

Mughal jali. 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. 

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’.

1909 (1)
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.

1921 (1)
Percy A. MacMahon. A. New Mathematical Pastimes. Cambridge University Press, 1921 and 1930. (Reprinted by Tarquin Books 2004)

Cairo diagram p. 101.

1922 (1)
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 who exactly determined the tile.

1925 (1)
Friedrich Haag. 'Die pentagonale Anordnung von sich berührenden Kreisen in der Ebene’. Zeitschrift für Kristallographie 61, 1925, pp. 339–340.
Has Cairo tiling in the form of circle packing.

1926 (1)
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.

1931 (1)
Fritz Laves. ‘Ebenenteilung in Wirkungsbereiche’. Zeitschrift für Kristallographie 76 (1931): 277–284. 

1933 (1)
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, 1933, 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 whose articles he quotes.

1951 (2)
H. Martyn Cundy and A. P. Rollett. Mathematical Models. Oxford University Press, 1951 (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.

 '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 does anyone know who he is?

1954 (2)
'Croton'. Cairo tiling used again as a crossword puzzle, in The Listener, 22 April 1954, puzzle 1251 Hexa-Pentagonal II, by 'Croton'.

Cyril Stanley Smith. “The Shape of Things.” Scientific American, vol. 190, no. 1, 1954, pp. 58–65.

In a general discussion on tiling.

1955 (1)
A. P. Rollett. ‘A Pentagonal Tessellation’. The Mathematical Gazette, Vol 39, No. 329, September 1955, note 2530, 1955, 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.

1956 (1)
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.

1958 (1)
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.

1963 (1)
H. S. M. Coxeter. Regular Complex Polytopes. Second edition. Dover Publications Inc., New York, 1963, cover.
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.

1967 (1)
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 regard 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, but the observation of his appears to be his own.

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.

1970 (2)
Ernest R. Ranucci. Tessellation and Dissection. J. Weston Walch, p. 36, 1970
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.

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

1971 (1)
James A. Dunn. ‘Tessellations with Pentagons’. The Mathematical Gazette, Vol. 55, No. 394, December, pp. 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.

1972 (1)
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.

1974 (1)
Stanley R. Clemens. ‘Tessellations of Pentagons’. Mathematics Teaching, No. 67, June, 1974, pp.18–1.
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.

1975 (2)
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.

Martin Gardner. Scientific American. Mathematical Games, July. ‘On tessellating the plane with convex polygon tiles’, 1975, 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 is based upon Dunn's account. 

1976 (2)
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.

Phares O’Daffer. G. Clemens, Stanley R. Geometry. An Investigative Approach 1st edition 1976 check, 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).

1977 (2)
Lorraine Mottershead. Sources of Mathematical Discovery. Basil Blackwell, 1977, pp.106–107.
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 myself thought until 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.

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!

1978 (3)
Doris Schattschneider. Tiling the Plane with Congruent Pentagons’ Mathematics Magazine. Vol. 1, 51, No.1 January 1978, pp. 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.

Peter Pearce and Susan Pearce. Polyhedra Primer. Dale Seymour Publications, 1978, p. 35.
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.

Ernest H. Lockwood and Robert H. Macmillan. Geometric symmetry. Cambridge University Press 1978 (and reprint 2008), p. 88.
‘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 (1979) Mathematical Gazette article.

Robert H. Macmillan. Mathematical Gazette, ‘Pyramids and Pavements: some thoughts from Cairo’, 1979, 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 is the depth of detail Macmillan gives. Notably, he describes an in situ pentagon possessing 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!

1980 (1)
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.

1982 (2)
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.
Gives the ‘45°’ construction.

Patrick Murphy. Modern Mathematics Made Simple. Heinemann London Tessellations, Chapter 10, 1982, pp. 194–205, 262.

Cairo diagram, of equilateral pentagons (but not attributed), p. 200.

1983 (2)
John Willson. Mosaic and Tessellated Patterns. How to Create Them. Dover Publications, Inc. 1983. Plate 3

Cairo tiling plate 3.

Cyril Stanley Smith. A Search for Structure. The MIT Press, 1983

Has non-attributed Cairo tiling.

1984 (1)

William Blackwell. Geometry in Architecture. John Wiley & Sons, New York, 1984, pp. 54–55.

The tile pattern of Figure 5.12 has the appearance of interlocking hexagons but consists of identical equal sided (but not equal angular) pentagons. The hexagonal patterns cross at right angles and the whole 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 mosaics of Moorish buildings, combines elements of four-, five-, and six-sided polygons and is another of the shapes in geometry in which, mathematically, the square root of seven plays a part. (The diagonal dimension of the rectangle enclosing the equilateral triangle includes the square root of seven.) 

Like all other tile patterns, this pattern can provide the basis for a three-dimensional assembly of prisms. The photograph of a model (Figure 5.13) shows such an assembly with the pentagonal prisms proportioned to minimize the total surface area of each piece. With this proportion, the height as well as the dimensions of the base embodies the square root of seven.

[Caption] A pentagonal tile pattern with pieces equilateral but not equiangular. The pattern retains a right-angular relationship to the walls of a rectangular room and can be subdivided in several ways.

Likely taking his lead from the Gardner quote, as the latter part is almost word-for-word. In so many words, leaning on Gardner, states that the in situ paving is equilateral. Gives a nice diagram showing different unit cells for a repeating square matrix.

Shows the Donald Wood pentagonal prisms.

Snippet view in Google Books, showing what appears to be an equilateral pentagon

https://archive.org/details/geometryinarchit0000blac

Search only, but there is a time-consuming workaround for single pages of interest.

1986 (5)
Jim McGregor and Alan 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.

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

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

A. L. Loeb. 'Symmetry and Modularity'. Computers and Mathematics with Applications, Elsevier, 1986

Jay Kappraff. ‘A Course in the Mathematics of Design’. Computers and Mathematics with Applications Vol. 12B, Nos. 3/4, 1986, pp 913–948.

Cairo tiling in the context of the set of  11 Laves tiling; p. 923 but as such, inconsequential.

Lothar Collatz. Geometrische Oranamente (in German), 1986.

Cairo tiling diagram in context of 43433 classification.

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.

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))

1987 (3)
Branko Grünbaum and Geoffrey C. Shephard. Tilings and Patterns. W. H. Freeman and Company, 1987, p. 5.
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.

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

Equilateral pentagon.

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.

1988 (1)
Loeb, Arthur. “Polyherda: Surfaces or Solids?”, pp. 106-117. In Marjorie Senechal and George Fleck, editors. Shaping Space: A Polyhedral Approach. Boston: Birkhäuser, 1988, p. 113.

Figure 6-12 shows two of the pentagonal tessellations about which Doris Schattschneider is an expert.

The Cairo tiling (but not named as such) discussed (and illustrated) amid the three pentagons as the dual of the.3. 3. 4. 3. 4 tiling.

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

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.

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.

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.

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

Michael Serra. Discovering Geometry: An Inductive Approach. Key Curriculum Press. 1989, pp. 316, 323. Also see a later edition of 1997. 

Page 316 …The tile pattern on the right is made of identical pentagons (non-regular) that could completely cover a floor without gaps or overlaps …The pentagon tessellation can be found in the street tiling of portions of Cairo and many other ancient cities in the Islamic world…

Page 323
1. Another very beautiful pentagon tessellation uses equilateral pentagons (the sides are congruent but not the angles). An example is the Cairo street tiling shown at the beginning of Lesson 7.5. The construction of an equilateral pentagon is shown below. On poster-board or heavy cardboard, construct an equilateral pentagon and use it as a template to recreate the Cairo street tiling.

Chapter 7, pp. 294–341, is on transformations and tessellation. Gives an interesting variation of the construction of an equilateral pentagon.

Internet Archive. Not downloadable

1990 (1)
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.

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.

David Wells. The Penguin Dictionary of Curious and Interesting Geometry. Penguin Books, 1991, p. 23.

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.

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.

Arthur F. Coxford. Geometry from Multiple Perspectives. Reston, Va.: National Council of Teachers of Mathematics, 1991, p. 68.

The duals of 3-3-4-3-4 and 3-3-3-4-4 are each made up of pentagons. In the former case, the pentagon is the Cairo pentagon, and in the latter it is like home plate in baseball.
Internet Archive. Not downloadable 

1992 (1)
Steven P. Meiring. A Core Curriculum: Making Mathematics Count for Everyone. Curriculum and Evaluation Standards for School Mathematics Addenda Series, Grades 9-12. National Council of Teachers of Mathematics, 1992. p, 68, 

The duals of 3-3-4-3-4 and 3-3-3-4-4 are each made up of pentagons. In the former case, the pentagon is the Cairo pentagon, and in the latter it is like home plate in baseball.

This repeats the text from the 1991 book by Coxford.

Internet Archive. Not downloadable

1993 (1)
Nenad Trinajstic. 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.

1994 (2)
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.

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.

1996 (1)
Michael O’Keefe and Bruce G. Hyde. Crystal Structures. 1. Patterns & Symmetry. Mineralogical Society of America, 1996, 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? Also Dover 2020, corrected.

1998 (1)
David A. Singer. Geometry Plane and Fancy, Springer-Verlag, 1998, p. 34.
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.

1999 (3)
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.

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

John Gregory, Investigating with TesselTiles. Vernon Hills, Ill.: ETA, 1999, p. 2.

The red Cairo pentagon—a shape used to create several walkways in Cairo, Egypt—contains two right angles. Although it is equilateral, this pentagon is not equiangular.

Internet Archive. Not downloadable 

2001 (1)

Edward Duffy, Greg Murty, Lorraine Mottershead. Connections Maths 7. Pascal Press, 2001, p. 83.
Cairo streets have this Islamic pattern
Not in possession, Google Books reference.

2003 (5)

Teacher’s Guide: Tessellations and Tile Patterns, p. 30 (Cabri) Geometric investigations on the VoyageTM 200 with Cabri. Texas, 2003 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…
A brief discussion in the wider context of pentagonal tilings. Begins by quoting David Wells’ book The Penguin Book of Curious and Interesting Geometry and so likely the text is based on his reference. However, the ‘Teacher’s Guide’ gives a different tiling to that in the Wells book, interestingly, a ‘collinear’ pentagon based on the in situ paving.

Catherine A Gorini. The Facts on File Geometry Handbook. 2003, 2009 revised edition. Facts on File Inc, and imprint of Infobase publishing

Cairo tiling illustrated p. 22, equilateral. Gives the following definition: Cairo tessellation: A tessellation of the plane by congruent convex equilateral pentagons that have two nonadjacent right angles; so called because it can be found on streets in Cairo.

Oddly, Gorini shows an accompanying picture of a pentagon that is not equilateral, a 4, 1 type…

Chris Pritchard. The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching, 2003
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 in possession, Google Books reference.

Eric W. Weisstein. CRC concise encyclopedia of mathematics, 2003, 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.

2004 (1)
Robert Parviainan. ‘Connectivity Properties of Archimedean and Laves Lattices’. Uppsala Dissertations in Mathematics 34, 2004, p. 9.
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.

2005 (2)
David Mitchell. Sticky Note Origami: 25 Designs to make at your desk, Sterling Publication Company, 2005
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 in possession, Google Books reference.

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

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 in possession, Google Books reference.

Sue Johnston Wilder and John Mason. Developing Thinking in Geometry, 2005, 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 in possession, Google Books reference.

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…’

On CA (Cellular Automata) A curiosity, with the Cairo tiling acting as a backdrop on the Game of Life.

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

2006 (2)
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’.

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. 2006

2007 (3)
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…

Mike Ollerton. 100+ Ideas for Teaching Mathematics, p. 66
This tessellation not only begs interesting questions about angle sizes and side ... The Cairo tessellation... The challenge is to use this tile to fill 2D space.

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, pp. 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.

L. J. Frobisher, Anne Frobisher, A. Orton, J. E. H. Orton. Learning to Teach Shape and Space: a handbook for students and teachers in the primary school. Cheltenham: Nelson Thores, 2007, p. 153.

The Cairo tessellation shown in Figure 6.48 is very famous. Why do you think it is considered special? How would you describe it over the telephone to a friend so that they could sketch it? Work out how to draw it accurately.

For children

Internet Archive 

2008 (5)
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’.

Merrilyn Goos et al. Teaching Secondary School Mathematics: Research and Practice for the 21st Century, 2008.
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 in possession, Google Books reference.

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
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.

Robert Fathauer. Designing and Drawing Tessellations, Tessellations, 2008, 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.

B. G. Thomas, B. G. and M. A. Hann. In Bridges. Mathematical Connections in Art, Music, and Science, 2008.
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).

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

Translated: A very well-known pattern with pentagons is the Cairo Tiling. The name comes from a street paving pattern that is said to exist in Cairo. 

2009 (3)

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 in possession, Google Books reference.

Eric Ressouche, Virginie Simonet, Benjamin Canals, Marin Gospodinov, Vassil Skumryev. 'Magnetic frustration in an iron-based Cairo pentagonal lattice'. Physical Review Letters. 2009

The pentagon, a 5-edges polygon, is an old issue in mathematical recreation... It exists however several possibilities of tessellation of a plane with nonregular pentagons, a famous one being the Cairo tessellation whose name was given because it appears in the streets of Cairo and in many Islamic decorations (Fig. 1). 

Mike Ollerton. The mathematics teacher's handbook, p. 148.

… use four different colours to make the 'Cairo' tiling design.

Not in possession, Google Books reference.

Faith H. Wallace. Teaching Mathematics through Reading. methods and materials for grades 6-8. Linworth Pub, 2009, p. 50.

The Cairo Tessellation is an attractive and intriguing pattern, frequently used in Islamic design, is created by tessellating irregular pentagon tiles. Tiles can be combined as a simple irregular hexagon or in more complex ways. Contrasting color sticky notes add effect.

For children..

Internet Archive 

2010 (2)
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 in possession, Google Books reference.

Subhash Chandra Saxena. College Geometry: A New Paradigm: (a rigorous exploratory approach using technology) Myrtle Beach. S.C.: Sherian Press, 2010, pp. 13, 15

Exploration 4.2.3 creates tessellation with a Cairo pentagon.

and

P. 13 There is an interesting equilateral pentagon, which is not equiangular and is used in tessellations in Chapter 4. It is constructed as follows: Example 1.2.2: (Cairo pentagon):

P. 13…This pentagon is occasionally called the Cairo pentagon, because several streets in that capital of Egypt have tessellations of this pentagon, as shown on the right of Figure 1.2.10.

P. 108 — It is interesting to note that while a regular pentagon cannot produce a pure tessellation, some equilateral pentagons can. One of them is generated by the so-called Cairo pentagon (Figure 4.2.6 (a)). This tessellation is quite well known in the Muslim world, especially in some streets of Cairo.

P. 109 — Another pentagon that can tessellate in the style of Cairo pentagon, and is easier to construct, is shown in Figure 4.2.7. It is not equilateral, but has four congruent sides. Those four enclose three angles, two of them right angles, and one measuring 120°. The remaining two angles also measure 120°. Call it De Figure 4.2.7 “Almost Cairo pentagon” as it tessellates in exactly the same way. The details are left for the reader to explore (Exercise 4.2.12).

12 mentions of a Cairo pentagon. Of note is that Saxena distinguishes what she believes to be the in situ model with a variant, which she terms “almost Cairo pentagon”!

Internet Archive. Not downloadable

2011 (4)
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.

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 in possession, Google Books reference. Cursory mention in passing.

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 ...

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

Iron-Based Cairo Pentagonal Lattice

Google Books

2012 (2)
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

Calvin T. Long, Duane W. DeTemple, Richard S. Milman. Mathematical Reasoning for Elementary Teachers. Boston: Pearson Addison Wesley, 2012, p. 648.

(b) Trace the semiregular tiling of Figure 11.25 whose vertex figure consists of two non-adjacent squares and three equilateral triangles. Then construct the dual, which is known as the Cairo tiling, since the paving stones of the streets of Cairo make this pattern.

In Chapter 8, Transformation Symmetries and Tilings.

2013 (5)

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 ...

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

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 in possession, Google Books reference.

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.

Glenn Ellison. Hard Math for Elementary School: Answer Key for Workbook. CreateSpace Independent Publishing Platform, 2013, p. 36.
Three references

The pentagon below has three 120° angles and two 90° angles. It is sometimes called the Cairo pentagon because there are streets in Cairo, Egypt that are paved with stones in this shape. Make some Cairo pentagons by tracing this shape and see if you can figure out how they tile.

Internet Archive. Not downloadable

Wassim Jabi. Parametric Design For Architecture. Laurence King Publishing, 2013, pp. 64–66.

p. 6…In this project, the studio focused on the implementation of a new surface iconography based on the digital manipulation of a traditional tiling pattern, called Cairo tessellation. The tiles designed by the studio were used to compose a relief panel for a residential entry foyer, in which the original pattern of the Cairo tiling is also maintained, but only as a geometric background.

For the design of the tiles, the firm began with a 30-degree tilted square grid that underwent four stages of tessellation. The result was a grid of irregular pentagons that maintained the topology of the original tiles. The studio identified five focal points on each pentagon, inspired by the pattern of the Cairo tiling, and applied a network of curvilinear streamlines — a visualization of lows — between those points. The pattern that was produced in this manner visually suggested a 3D interpretation of the original 2D floral pattern of the Cairo tiles…

p. 66. These final tiles are made from polyurethane and they possess a highly reflective, undulating surface that lacks any classical geometric clarity or symmetry, but which echoes the repetitive geometry of the traditional Cairo tile. In the final product, the geometric essence of the Cairo tile pattern has been morphed into a freer, more seductive form, which extends into the third dimension.

No rationale is given for the naming. Their model is the dual.

2014 (2)

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. 2014, p. 203, ‘Cairo tiling’

In abbildung 133 ist die parkettierung ‘Cairo tiling’ dargestellt.
Translated: Figure 133 shows the 'Cairo tiling' and What other symmetries does this charming tiling possess? 

The first of three German references. Also see Helmerich (2015), Steurer (2016).

David E. Laughlin and ‎Kazuhiro Hono. Physical Metallurgy. 2014, p. 76, 5th edition.

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

Not in possession, Google Books reference.

2015 (1)
Markus Helmerich, ‎Katja Lengnink. Einführung Mathematik. Springer, 2015, p. 108.

Translated: (Introduction to Primary School Mathematics).

Fig. 4.5 “Cairo Tiling”.

The second German reference.

2016 (3)

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

... 34 Cairo pentagon tiling 496 

Not in possession, Google Books reference.The third and final German reference

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

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

Robert J. Lang. Twists, Tilings, and Tessellations: Mathematical Methods for Geometric Origami. 2016? CRC Press.

Left: “Cairo Tessellation” (2011), a flagstone tessellation by Eric Gjerde, based on the Cairo tiling.

Not in possession, Google Books reference.

2017 (1)
Ed Southall and Vincent Pantaloni. Geometry Snacks. Tarquin, 2017, 90 pp. 

Cairo tiling as an equilateral pentagon. Area problem p. 74, #47. Solution p. 82.
Not in possession

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

Numerous references throughout, too many to list.

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

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

Not in possession, Google Books.

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

Toshikazu Sunada. “Topics on Mathematical Crystallography”. In Tullio Ceccherini-Silberstein, Maura Salvatori, Ecaterina Sava-Huss. Graphs And Random Walks. London Mathematical Society Lecture Note Series 436, Cambridge University Press, 2017, pp. 475–519. See p. 511.

Figure 16.14 exhibits a few more examples of standard realizations (the picture on the right side is a tiling of pentagons with picturesque properties that has become known as the Cairo pentagon).

Advanced crystallography. Nothing more than in passing.

Internet Archive

Laurent Najm, ‎Pascal Romon (eds.). Modern Approaches to Discrete Curvature. Springer, 2017, p. 180.

Next, we discuss examples of non definite curvature, see Fig. 6.2 as well. Example 6.2 The so called Cairo tiling consists of pentagons...

2019 (3)

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.

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

A portion of the Cairo tiling. 

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

Google Books

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.
29 references to “Cairo”, which is 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.

The ‘Cairo tiling’ (Fig. 1) has many names, as we will see in Section 2. In particular, it is the dual of the snub version of the familiar square tiling.

[Section 2] The Cairo tiling is shown in Fig. 1. This beautiful tiling has many names. It has also been called the Cairo pentagonal tiling, the MacMahon net (O’Keeffe & Hyde, 1980), the mcm net (O’Keeffe et al., 2008), the dual of the 32:4:3:4 tiling [Grunbaum & Shephard, 1987, pp. 63, 96, 480 (Fig. P5-24)], the dual-snub-quadrille tiling, or the dual-snub-square tiling (Conway et al., 2008, pp. 263, 288). We will refer to it simply as the Cairo tiling. There is only one shape of tile, an irregular pentagon, which may be varied somewhat.

I see further on in the paper (p. 130) O’Keefe (and Conway) give alternative names to other pentagonal tilings (fsz-d net), namely the snub-632 tiling (more commonly known as the “floret”). Why O’Keefe in particular does this (twice!), I am unclear, as introduced in a 1980 paper with B. G. Hyde. Why introduce more terms (and so confuse matters) to an established name? At least Goodman-Strauss & Sloane are consistent with “Cairo tiling”, albeit they misattribute it to the dual.

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.

2020 (1)

Robert Fathauer. Tessellations: Mathematics, Art, and Recreation 2020, p. 291. 

A link design created by decorating each pentagon in a Cairo pentagon tiling with knot graphics (Caption)

Eli Maor (author), Eugen Jost (illustrator). Pentagons and Pentagrams. Princeton University Press, 2022, pp. 95–96.
Another example of pentagonal-hexagonal tiling is the Cairo tessellation, so called because it appears frequently in mosques across the Middle East and North Africa.

Two falsehoods in one sentence!

2023 (3)

Hisashi Naito. Trivalent Discrete Surfaces and Carbon Structures, Springer Nature, Singapore, 2023, p. 43.

Example 3.20 (Cairo pentagonal tiling, Sunada, [54]). …A Cairo pentagonal tiling. 

Quoting Sunada and MacMahon’s nets. Also uses “Cairo tiling”

Nguyen Thanh Tien, ‎Thi Dieu Hien Nguyen, ‎Vo Khuong Dien. Chemical Modifications of Graphene-like Materials, World Scientific, 2023, p. 484. 

… resembling Cairo pentagonal tilling with a central sp3 carbon sublattice…

Also uses “pentagonal Cairo tessellations”

Colin Adams. The Tiling Book: An Introduction to the Mathematical Theory. American Mathematical Society, 2023, p. 156.

Cairo Pentagonal Tiling Prismatic Pentagonal Tiling Figure 2.88: The Cairo and prismatic tilings by pentagons are both least perimeter

Quoting Morgan et al.

Internet Archive

2024 (1)

Yong Liu, Liangchao Yuan, Wenwen Chi, Wang-Kang Han, Jinfang Zhang, Huan Pang, Zhongchang Wang & Zhi-Guo Gu. ‘Cairo pentagon tessellated covalent organic frameworks with mcm topology for near-infrared phototherapy’. 2024, Nature. 

We herein present the Cairo pentagonal tessellated COFs,...aligning precisely with the criteria of Cairo Pentagon.

Typical academic chemistry.

2026 (1)
Péter Balázs, ‎Reneta P. Barneva, ‎Valentin E. Brimkov. “Binary Tomography in the Cairo Pattern”. In Combinatorial Image Analysis: 23rd International Workshop, Springer, 2026, pp. 230–244.

In this paper, we investigate a tomography reconstruction approach for binary images on the Cairo pattern...

Their preferred term is “Cairo Pattern” with many references. Their mode is the dual.

Google Books, not in possession

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.

4. “Cairo Tiling”. The exact term, no variations, no matter how minor.

Much to my amazement, there are only six references to “Cairo tiling” at the Internet Archive, and only two were new!

1986 (1)
Ehud Bar-On. ‘A programming approach to mathematics’. Computers & Education 10(4): pp. 393–401. December 1986. Elsevier.

… especially the Cairo tiling.

1990 (1)
Francis S. Hill. Computer Graphics. Macmillan Publishing Company, New York, 1990, 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. 

1994 (1)

Mathematics Teaching in the Middle School, [Reston, VA]: [National Council of Teachers of Mathematics], 1994, p. 168.

In Infinity issue 2, David Mitchell showed how to create an origami Cairo tiling. This suggested some new types of Su Doku puzzles… What constraints can you invent? For example, all these puzzles are based around the idea that each overlapping hexagon (which is made up of four Cairo tiles) must contain the numbers 1 to 4.

Internet Archive. Not downloadable (15 May 2026)

1996 (1)

Michael O’Keefe and Bruce G. Hyde. Crystal Structures No. 1. Patterns & Symmetry. Mineralogical Society of America, 1996, p. 207.

The pattern is known as Cairo tiling, or MacMahon’s net and In Cairo (Egypt) the tiling is common for paved sidewalks…

2000 (1)

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

2005 (1)
Carter Bays. Complex Systems Publications, Volume 15, Issue 3, 2005. 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…’

2009 (1)

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

2010 (1)

Claudi Alsina and Roger B. Nelsen. Charming Proofs: A Journey Into Elegant Mathematics. Dolciani Mathematical Expositions, 2010, 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. 

2012 (2)
Calvin T. Long. Mathematical Reasoning for Elementary Teachers. Boston: Pearson Addison Wesley, 2012, p. 648.
(a) What are the duals of the three regular tilings shown in equilateral triangles. Then construct the dual, which is known Figure 11.22 as the Cairo tiling, since the paving stones of the streets of Cairo make this pattern.

Internet Archive. Not downloadable (15 May 2026)

Robert A. Meyers. Computational Complexity Theory, Techniques, and Applications. New York, NY: Springer Science+Business Media, LLC, 2012, p. 1344

GL rules are supported in pentagonal and hexagonal grids. The pentagonal grid (left) is called the Cairo Tiling, supposedly named after some paving tiles in that city. There are many different topologically distinct pentagonal grids; the Cairo Tiling is but one.
On gliders.
Internet Archive. Not downloadable (15 May 2026)

2014 (1)
Ralf 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'.

2017 (1)

58. Robert J. Lang. Twists, Tilings, and Tessellations: Mathematical Methods for Geometric Origami. CRC Press.

Left: “Cairo Tessellation” (2011), a flagstone tessellation by Eric Gjerde, based on the Cairo tiling.

Not in possession, Google Books reference.

2019 (2)
Mircea Pitici (ed). The Best Writing on Mathematics. Princeton University Press, 2019. 114–116.

A portion of the Cairo tiling. 

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

Not in possession, Google Books reference.

5. “Cairo Pentagon/s”. The exact term, no variations, no matter how minor.

1991 (1)
Arthur F. Coxford. Geometry from Multiple Perspectives. Reston, Va.: National Council of Teachers of Mathematics, 1991, p. 68.

The duals of 3-3-4-3-4 and 3-3-3-4-4 are each made up of pentagons. In the former case, the pentagon is the Cairo pentagon, and in the latter it is like home plate in baseball.

Internet Archive. Not downloadable

1992 (1)
Steven P. Meiring. A Core Curriculum: Making Mathematics Count for Everyone. Curriculum and Evaluation Standards for School Mathematics Addenda Series, Grades 9-12. National Council of Teachers of Mathematics, 1992. p. 68 

The duals of 3-3-4-3-4 and 3-3-3-4-4 are each made up of pentagons. In the former case, the pentagon is the Cairo pentagon, and in the latter it is like home plate in baseball.

This repeats the text from the 1991 book by Coxford.

Internet Archive. Not downloadable

1999 (1)
John Gregory, Investigating with TesselTiles. Vernon Hills, Ill.: ETA, 1999, p. 2.

The red Cairo pentagon—a shape used to create several walkways in Cairo, Egypt—contains two right angles. Although it is equilateral, this pentagon is not equiangular.

Internet Archive. Not downloadable

2010 (1)
Subhash Chandra Saxena. College Geometry: A New Paradigm: (a rigorous exploratory approach using technology) Myrtle Beach. S.C.: Sherian Press, pp. 13, 15

Exploration 4.2.3 creates tessellation with a Cairo pentagon.

and

P. 13 There is an interesting equilateral pentagon, which is not equiangular and is used in tessellations in Chapter 4. It is constructed as follows: Example 1.2.2: (Cairo pentagon):

P. 13…This pentagon is occasionally called the Cairo pentagon, because several streets in that capital of Egypt have tessellations of this pentagon, as shown on the right of Figure 1.2.10.

P. 108 — It is interesting to note that while a regular pentagon cannot produce a pure tessellation, some equilateral pentagons can. One of them is generated by the so-called Cairo pentagon (Figure 4.2.6 (a)). This tessellation is quite well known in the Muslim world, especially in some streets of Cairo.

P. 109 — Another pentagon that can tessellate in the style of Cairo pentagon, and is easier to construct, is shown in Figure 4.2.7. It is not equilateral, but has four congruent sides. Those four enclose three angles, two of them right angles, and one measuring 120°. The remaining two angles also measure 120°. Call it “Almost Cairo pentagon” as it tessellates in exactly the same way. The details are left for the reader to explore (Exercise 4.2.12).

12 mentions of a Cairo pentagon. Of note is that Saxena is one of the few who distinguishes between the various models, with what she believes to be the in situ model with a variant, which she terms “almost Cairo pentagon”!

Internet Archive. Not downloadable

2011 (2)

Mike Askew. The Bedside Book of Geometry: from Pythagoras to the Space Race: the ABC of Geometry. London: New Burlington Books, 2011, p. 109, two references
(4, 3, 3, 4, 3). Marking the centre of each of these and joining these points produces the dual tiling of Cairo pentagons.

Internet Archive. Not downloadable

Also see a US edition of the book, of the same year, but under a different title.

Mike Askew. Geometry: The Size and Shape of Everyday Math. New York: Metro Books, 2011, p. 109, two references
(4, 3, 3, 4, 3). Marking the center of each of these and joining these points produces the dual tiling of Cairo pentagons.

Internet Archive. Not downloadable

2013 (2)
Toshikazu Sunada. Topological Crystallography: With a View Towards Discrete Geometric Analysis. Springer, 2013.

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 in possession, Google Books reference.

The first clear reference to the exact term “Cairo Pentagon”.

Glenn Ellison. Hard Math for Elementary School: Answer Key for Workbook. CreateSpace Independent Publishing Platform, 2013, p. 36. 

The pentagon below has three 120° angles and two 90° angles. It is sometimes called the Cairo pentagon because there are streets in Cairo, Egypt that are paved with stones in this shape. Make some Cairo pentagons by tracing this shape and see if you can figure out how they tile.

Internet Archive. Not downloadable

2016 (1)
Walter Steurer and Julia Dshemuchadse. Intermetallics: Structures, Properties, and Statistics. OUP Oxford, p. 565.

Cairo pentagon tiling (many references) 

Not in possession, Google Books reference.

2020 (1)
Robert Fathauer. Tessellations: Mathematics, Art, and Recreation p. 291 NEW

A link design created by decorating each pentagon in a Cairo pentagon tiling with knot graphics (Caption)

2024 (1)
Yong Liu, Liangchao Yuan, Wenwen Chi, Wang-Kang Han, Jinfang Zhang, Huan Pang, Zhongchang Wang & Zhi-Guo Gu. ‘Cairo pentagon tessellated covalent organic frameworks with mcm topology for near-infrared phototherapy’. Nature. 

We herein present the Cairo pentagonal tessellated COFs,...aligning precisely with the criteria of Cairo Pentagon.

Typical academic chemistry.

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Liu, Yong, Liangchao Yuan, Wenwen Chi, Wang-Kang Han, Jinfang Zhang, Huan Pang, Zhongchang Wang & Zhi-Guo Gu. ‘Cairo pentagon tessellated covalent organic frameworks with mcm topology for near-infrared phototherapy’. Nature, 2024.

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Data Analysis

“Street-tiling in Cairo” or similar (29)

Dunn (1971), Gardner (1975), Schattschneider (1978), Macmillan (1979), Martin (1982), Blackwell (1984), McGregor & Watt (1986), Andrews (1986), Grünbaum & Shephard (1987), Chorbachi (1989), De La Harpe (1989), Hill (1990), Fetter (1991), Wells (1991), Trinajstić (1993), Serra (1997), Singer (1998), Teacher’s Guide (2003), Gorini (2003), Pritchard (2003), Parviainan (2004), Mitchell (2005), Key Curriculum Press (2008), Iwamoto (2009). Ressouche (2009), Fathauer (2010), Alsina (2010), Saxena (2010), Glynn & Sheil (2013), Ellison (2013), Morgan (2019).

Lockwood & Macmillan (1978)*

Only four are first-hand: Dunn, Macmillan, Iwamoto, and Morgan, with those by Dunn and Macmillan being the most influential, with succeeding authors taking the text from their accounts and likely often copying each other. The impression generally given is that this is original research on the author’s part, rather than what it actually is, i.e. relying on preceding accounts.

“Equilateral pentagon” (12)

Dunn (1971), Gardner (1975), Martin (1982), Balacwell (1984), Serra (1989), Hill (1990), Bays (1994, 2005), Singer (1988), Gregory (1999), Gorini (2003), Thomas & Hann (2007 (2), 2008), Saxena (2010). 

As has been clearly established, the in situ pentagon is one with a condition of collinearity, and not the commonly quoted equilateral (or dual of the 32. 4. 3. 4 tiling), as both these lack this aspect. Therefore, an obvious question to ask is how this incorrect situation arose, which I now discuss with the first reference of any claim:
James Dunn, ‘Tessellations with Pentagons’, Mathematical Gazette, 1971.

As such, it would appear that the error arose in the very first reference with the Cairo attribute in Dunn’s article (of which it has largely then been either directly or indirectly repeated from this), in which he quite categorically states that it is equilateral:

… 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 illustration shows—or intends to show, for the drawing is somewhat carefree in places—an equilateral pentagon. In light of the contradictory in‑situ photographs, which do not support this claim, I eventually sought out Dunn to ask for further details about his relatively lightweight report and to see whether he could substantiate an equilateral pentagon determination. In a phone call and subsequent correspondence (2010), he told me that the sighting was indeed first‑hand, occurring purely by chance after arriving at Cairo airport, in Heliopolis, likely in 1969. More germane to this enquiry, when asked how he had determined that the pentagons were equilateral—specifically, whether he had measured them—he explained that he had not. Rather, he had assumed them to be equilateral, influenced by a diagram of equilateral pentagons in a Cairo‑like arrangement that he had seen in Mathematical Models by Cundy and Rollett (p. 63). Thinking little more of it (a not unreasonable response, given the circumstances of a chance sighting), he later wrote up the finding in an article on pentagonal tessellations and thus, apparently inadvertently, initiated the myth that the tiles are equilateral. Martin Gardner then repeated the claim, taking his cue from Dunn (whom he cites), and subsequent accounts likely drew on Gardner’s more widely read description. This, however, does not explain the separate claim that the in‑situ model is the dual, which I address next. 

“Dual of 3.3.4.3.4 tiling” (11)

Schattschneider (1978), Coxford (1991), Wells (1991), Meiring (1992), Leathard (1994), Weisstein (2003), Johnston-Wilder & Mason (2005), Phillips (2005), Askew (2011), Laughlin & Hono (2014), Goodman-Strauss & Sloane (2019).

The other incorrect attribution is by Doris Schattschneider, who then apparently begins the references to the dual type, apparently referring to either or both Dunn's or Gardner’s reference with:

Tiling (3) of Figure 1 [illustrated with the dual] has special aesthetic appeal. It is said to appear as a street tiling in Cairo;…

Doris Schattschneider. ‘Tiling the Plane with Congruent Pentagons’, Mathematics Magazine, 1978.

How this account originated is less clear than Dunn’s. Dunn and Gardner (but not Macmillan) are mentioned in the references, Dunn and Gardner both give the equilateral model, and so it would thus naturally follow from this that Schattsneider would repeat their claim. But no. Instead, she asserts that the in situ model is the dual. Whatever the reason, the claim is then widely used in the literature, with eleven other such references.

Descriptions of the in situ paving - Cairo tiling, Cairo pentagon, etc.
Of note is that the in situ pentagon paving is described in a variety of ways, with “Cairo Tiling”, “Cairo Pentagon”, “Cairo Tessellation", “Cairo tile”, and others, of which I now examine the various terms used. Some are more popular than others, and some pale into insignificance. The standard term used is “Cairo tiling”, and it has become the de facto description. Curiously, it was not immediately titled as such, with descriptions such as "street paving in Cairo” initially used instead, before the first use of “Cairo tiling” by Ehud Bar-On in 1986, and it then became the standard term. A curious instance is “MacMahon’s net”, which is (inexplicably) gaining traction, much to my dismay. There is no need for an additional term. 

“Cairo Tiling” (although implied in earlier paper, it was not described in such a term) (16)

Ehud Bar-On (1986), Hill (1990), O’Keefe & Hyde (1996), Deza (2000), Bays (2005), 

Johnston-Wilder & Mason (2005), Thomas & Hann (2007, 2008), Kaltenmorgen (2008), Kaplan (2009), Alsina & Nelsen (2010), Benölken, Gorski & Müller-Philipp (2014), Lang (2014), Najm (2017), Morgan (2019).

“Cairo Tessellation” (19)

Martin (1982), Leathard (1994), Bays (1994), Singer (1998), Teacher’s Guide (2003), Gorini (2003), Mitchell (2005), Phillips (2005), Johnston-Wilder & Mason (2005), Thomas & Hann (2005, 2008), Frobisher (2007), Ollerton (2007), Goos (2008), Iwamoto (2009), Elwes (2010), Ressouche  (2010), Goldemberg (2011), Gengnagel et al (2012), Lang (2017), Moncy (2017), (2019), Pitici (2019), Goodman-Strauss & Sloane (2019).

“Cairo Pentagon” (9)

Coxford (1991), Meiring (1992), Gregory (1999), Saxena (2010), Sunada (2013), Ellison (2013), Steurer & Dshemuchadse (2016), Lang (2017), Moncy (2017).

“Cairo Pentagonal tiling” (9)
Askew (2011), Bangura (2011), Laughlin & Hono (2014), Steurer & Dshemuchadse (2016), Ramasami (2017), Wu (2017), Goodman-Strauss & Sloane (2019), Adams (2023), Naito (2023).

“Cairo (Pentagonal) Lattice” (4)

Parviainen (2004),  Rojas, Rojas & de Souza (2011), Acton (2011), Moncy (2017).

“Cairo Tile” (2)

McGregor & Watt (1986), Sharp (2006).

“MacMahon’s Net” (5)

O’Keefe & Hyde (1980, 1996, 2020), Sunada (2012), Naito (2023).

“Islamic”, “Mosque”, “Ancient” etc. (11)

Gardner (1975), Macmillan* (1979), Serra (1989), Wells (1991), Duffy, Murty & Mottershead (2001), Teacher’s Guide (2003), Weisstein (2003), Mitchell (2005), Ressouche et al (2009), Wallace (2009), Maor (2022). 

It can be seen in many books and articles that a claim is made to the tiling as an entity appearing in Islamic decoration, in mosques, or is ancient or some other embellishment. I can assure you that it is not! The earliest such reference is Martin Gardner, "On tessellating the plane with convex polygon tiles", Scientific American, 1975, from which his account has been embellished and extended beyond all reason:

… is frequently seen as a street tiling in Cairo and occasionally on in the mosaics of Moorish buildings. 

However, he is not specific as to the type of buildings or exactly where this is, but is instead deliciously vague. The term ‘Moorish’ covers a wide expanse, from Spain to Egypt! As such, there is no documentary evidence for this ‘sighting’ whatsoever in the form of a picture, or indeed any other indirect references. This quote has long puzzled me. In the course of my investigations, I have also consulted various Islamic authorities on tiling, described in detail in the link below. Gardner’s quote, in the plural, would, to me, imply that it is seen more than once, but all still to no avail. Upon contacting Tim Noakes, the curator of the Stanford Special Collection Archive at Stanford University (where Gardner’s files that he composed during the preparations of his columns in Scientific American are stored), much light has been shed on this thanks to Noakes and a visiting researcher of the archive, Bjarne Toft. Pleasingly, the full story can now be told. In short, beyond all reasonable doubt, Gardner was amazingly referring to the Taj Mahal as the ‘Moorish building’. Although quite why he should describe the Taj Mahal as ‘Moorish’ is yet another minor mystery. The background to this assertion is that Gardner made a series of notes on his source material for his columns, taken from various mathematics journals, which Toft kindly forwarded. Numerous indirect references to the Taj Mahal by Gardner can be seen.

https://www.magnificent-tessellation.com/cairo-tiling/history-origins/martin-gardner-mystery-resolved

Page History
18 May 2026. Return with additions found at the Internet Archive and Google Books after the last major update (2019).
19 May 2026. Added two new sections with entries that, if not literally extracted from the above listing, are in essence so:
[Section] 4. “Cairo Tiling”. The exact term, no variations, no matter how minor.
[Section] 5. “Cairo Pentagon/s”. The exact term, no variations, no matter how minor.
20 May 2026. Materially expanded entries of: Lang, Moncy, Morgan, Goodman-Strauss, Sloane and a few others.
21 May 2026. Materially and in depth expanded the entries of: Dunn, Gardner, Schattschneider and Macmillan (these being the most significant historically). Also added to the introduction with annotation explanations, bias and colour coding.
22 May 2026. Expanded entry on Wells. Single "Analysis" by term.
25 May 2026. Changed the presentation of the entries, removing the sequential numbering and year at the start of each entry, with the year now moved above the first entry.
26–27 May 2026. Added an alphabetical, "bare bones" listing of basic bibliographical details for easier finding of names. Expanded on Serra.

28–29 May 2026. Data analysis.
29 May 2026. Expanded the entries of Andrews and Bar-On. New entry on Wells.

1 June 2026. “High-end” references discussion - Chemistry and physics-type papers. Added Maor.
2 June 2026. Added/expanded Loeb, Long, Sunada, Wassim, and Williams.

3 June 2026. Added Najim, Frobisher, and Wallace. Added "Outlets".

4 June 2026. Added rationale for references in print only, in which I exclude web references.
5 June 2026. Added Adams, Browne et al, Kaplan, Lang, Nagy et al, Steurer et al, Alsina et al (2), Johnston-Wilder et al, of new and expanded entries. 

6 August 2025. Reappraisal in New Sites. The conversion has left wide open spaces between the text for any one entry, which made for a very bloated page, which I now correct by taking the 'old' text at the Wayback Machine and reinserting, this being quicker than manually making the changes with the converted text. I also took the opportunity of updating the text, replacing hyphens with dashes for the page numbering. Also corrected typos.

I have also removed the update history, save for the date of creation and introduction of a new section, as it is unnecessary to detail when an entry deemed of note was added. This can be seen at the Wayback Machine if required.  

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.

Created on: 9 September 2011.