Pentagon Confusion

 or,

 How Did the Cairo Pentagon Come to Be Misidentified?

Introduction

As has been clearly established, the in situ pentagon possesses a condition of collinearity and is not the commonly cited equilateral pentagon or the dual of the 3.3.4.3.4 tiling (referred to simply as the ‘dual’ below), since neither possesses this property. The obvious question, therefore, is how this misattribution arose. To address this, I examine the four earliest accounts, two of which are first-hand observations by the mathematicians James Dunn and Robert H. Macmillan, and analyse how these erroneous descriptions came to be adopted.

1. James Dunn, 1971. "Tessellations with Pentagons", in The Mathematical Gazette

It would therefore appear that the error originated in the first published reference linking the tiling to Cairo, namely Dunn's article, from which it has since been repeated either directly or indirectly. Dunn states quite categorically that the pentagon 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.

More specifically, Dunn refers to Cundy and Rollett’s Mathematical Models, p. 63, which indeed shows an equilateral pentagon tiling. Dunn’s hand-drawn diagram is not the best in terms of accuracy, with considerable departure in places from an equilateral pentagon, but the intent is clear.

Given the contradictory in situ photographs, which do not support this statement, I later sought out Dunn in order to establish whether he could substantiate the claim that the pentagons were equilateral. In correspondence and phone calls during 2010, he confirmed that the report was based on a chance first-hand sighting shortly after arriving at Cairo Airport, in Heliopolis, probably in 1969.

More significantly, when I asked how he had determined that the pentagons were equilateral, and 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 by Cundy and Rollett above. Giving the matter little further thought, which was understandable in the circumstances of a chance observation, he subsequently mentioned the tiling in his article on pentagon tessellations. In doing so, he appears inadvertently to have initiated the misconception that the Cairo tiles are equilateral. As an aside to the above, in a later email, he was far less certain that the sighting was at Cairo airport. This is all quite understandable, as at the time a chance sighting of no particular importance at the time would not stick in the memory to be recalled totally accurately 40 years later!  

To assess whether this explanation is plausible, Figure 1 compares the two relevant pentagon types side by side: the equilateral form and the in situ form rendered as a line drawing. At first glance, the two exhibit a marked superficial resemblance, particularly when viewed as a complete tiling, where the intersecting parhexagons reinforce the same overall impression. A casual observer lacking the means to make direct measurements with a ruler or protractor could readily mistake one for the other, as Dunn apparently did at the time, especially under the influence of the Cundy and Rollett diagram. Nevertheless, the differences become much more apparent when the two are presented side by side.

Fig. 1: (a) Equilateral pentagon (b) In situ pentagon

 2. Martin Gardner, 1975. "On Tessellating the Plane with Convex Polygon Tiles", in Scientific American

The second apparent instance is Martin Gardner's account, which also, somewhat surprisingly given the contradictory nature of the in situ photographs, depicts an equilateral pentagon:

… This beautiful tessellation [referring to a diagram of equilateral pentagons] is frequently seen as a street tiling in Cairo and occasionally in the mosaics of Moorish buildings.

Gardner does not directly identify the source of this statement. Nor is there any indication that he had personally observed the tiling in Cairo. Given what follows in the article, it is therefore highly unlikely that this was a first-hand account. The bibliography accompanying the article includes a reference to Dunn's paper, making it plausible that Gardner derived the Cairo attribution, at least in part, from that source. Indeed, the wording is strikingly similar, with Dunn's “favourite street-tiling in Cairo” becoming Gardner's “frequently seen as a street tiling in Cairo”. At first sight, this appears convincing.

The matter is not entirely straightforward, however. Gardner adds a detail not found in Dunn's account, namely that the tessellation is “occasionally” seen in the mosaics of Moorish buildings.

“Moorish buildings” is a frustratingly vague description. Why not be more specific? Moorish architecture is a style within Islamic architecture that developed in the western Islamic world, including al-Andalus (the Iberian Peninsula) and parts of North Africa. However, research suggests that Gardner was using the term more loosely to denote Islamic decorative architecture in general.

Research has revealed that Gardner was in fact referring to the Taj Mahal! Notes for the column preserved in the Gardner archives at Stanford University show that the source was an incorrect report supplied by Richard K. Guy:

Richard K. Guy adds a note as regards a pentagon tiling that Tóth has mentioned, “I believe I recall seeing (3, 3, 4, 3, 4) among the many unusual tilings at the Taj Mahal.

Gardner seems to have conflated the equilateral tiling of Dunn with the 3.3.4.3.4 tiling mentioned by Guy. Furthermore, Guy himself was mistaken, as the 3.3.4.3.4 tiling does not occur at the Taj Mahal. Rather, the appearance arises from a visual resemblance observed at a distance. What Guy was actually referring to was a clamshell tiling in the “Cairo configuration”, high up. Viewed from a considerable distance high above eye level, the resemblance can be deceptive, particularly to someone already familiar with the Cairo tiling. This matter is discussed in more detail in “Martin Gardner Mystery Resolved”.

No other supporting examples of such mosaics have come to light, and Dunn makes no mention of them.

On the balance of evidence, it is now near certain that Gardner's statement ultimately derives from Dunn's account, with the additional reference to Moorish mosaics originating from Guy's erroneous report. The complete absence of equilateral pentagons in the known in situ photographs lends support to this interpretation. Indeed, it would be remarkable if two independent sources had separately arrived at the same incorrect conclusion.

Fig. 2: (a) In situ pentagon (b) Dual of 3.3.4.3.4 tiling

3. Doris Schattschneider, 1978. "Tiling the Plane with Congruent Pentagons", in Mathematics Magazine

The third apparent instance is due to Doris Schattschneider, who writes:

Discovering tilings by pentagons

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; …

Tiling (3) is the dual of the 3.3.4.3.4 tiling. Of the 35 references cited in the article, only Dunn and Gardner mention the Cairo tiling, and so the attribution must ultimately derive from one or both of these sources. It should be noted, however, that the Cairo tiling is mentioned only in passing within what is otherwise a broad discussion of pentagonal tilings.

Particularly noteworthy is Schattschneider's wording: "it is said to appear as street paving in Cairo". This is clearly not a first-hand account, as otherwise such phrasing would not have been used. As so often with the history of the Cairo tiling, however, the precise source remains uncertain. Both Dunn and Gardner are cited in the references.

The wording leans towards Dunn, with similarities between Dunn's "is a favourite street-tiling in Cairo" and Schattschneider's "it is said to appear as street paving in Cairo". The resemblance is suggestive, although not sufficiently close to establish this with certainty. However, a further complication arises. Neither Dunn nor Gardner illustrates or discusses the dual of the 3.3.4.3.4 tiling. Schattschneider therefore appears to have combined the inherited Cairo attribution with a model that was already familiar within her own discussion of pentagonal tilings.

In this respect, her reasoning seems analogous to Dunn's. Just as Dunn appears to have interpreted the in situ tiling through the lens of an equilateral pentagon familiar from Cundy and Rollett, Schattschneider appears to have associated the Cairo attribution with the dual tiling that formed part of her own mathematical framework.

Numerous later authors appear to have adopted the dual as the Cairo pentagon from this account, thereby creating a second line of misattribution within the literature.

4. E. H. Lockwood and Robert H. Macmillan , 1978. Geometric symmetry. Cambridge University Press
The fourth apparent instance is a more tantalising case. Owing to various uncertainties, it is not possible to state with complete confidence that Lockwood and Macmillan are referring specifically to Cairo and its paving. Nevertheless, it seems a reasonable possibility. The text itself does not mention Cairo, but in view of Macmillan's subsequent paper of 1979, in which he explicitly refers to a visit there, the passage merits consideration:

…the reciprocals of the tessellations 3².4.3.4 and 3⁴.6 are patterns of congruent pentagons as are often used for street paving in Moslem countries.

The phrase "Moslem countries" is obviously vague and could refer to any one of a number of countries. However, what lends weight to the conjecture that Egypt, and by extension Cairo, is intended is Macmillan's subsequent statement in The Mathematical Gazette:

On a recent visit to Cairo… Many of the streets of Cairo are paved with a traditional Islamic tessellation of pentagonal tiles…

It would therefore seem likely that Egypt was among the countries Macmillan had in mind. Proceeding on that assumption, the notable feature is the type of pentagon illustrated, namely the dual of the 3.3.4.3.4 tiling. As evidenced by the known in situ photographs, there is no corroboration for such a paving. An obvious question therefore arises: why does Macmillan apparently associate this form with street paving in Moslem countries?

One possibility, although it may do Macmillan an injustice, is that, as with Dunn, he paid little attention to what was perhaps initially a chance observation and interpreted the pattern through a familiar mathematical model. However, there is an important difference. Whereas the equilateral pentagon proposed by Dunn bears a genuine superficial resemblance to the in situ tiles, the dual is noticeably less convincing as a match. Put simply, the in situ pentagon and the equilateral pentagon both possess a long base, whereas the dual possesses a short base. The distinction should be immediately apparent to an experienced geometer.

This raises a difficulty. Would a mathematician of Macmillan's stature really make such a mistake? It seems improbable, yet no other explanation readily presents itself. Certainly, not a single instance of the dual has been found in any known photograph, nor for that matter any example of the 3⁴.6 reciprocal.

Another peculiarity is that Macmillan's 1979 article largely abandons the dual interpretation and instead correctly identifies the in situ tiling as possessing a condition of collinearity. Here again, however, a degree of uncertainty remains. In both accounts Macmillan uses the word "often" rather than "always" or "invariably" when referring to the appearance of such tilings. Strictly speaking, "often" does not exclude the possibility of alternative forms.

Perhaps too much should not be read into a single word, and such scrutiny risks becoming pedantic. Equally, Macmillan was not a writer known for carelessness in matters of geometry. The ambiguity therefore remains.

Be that as it may, because Geometric symmetry does not explicitly mention Cairo, I have been unable to find any later reference that appears to derive from this account. Indeed, Schattschneider's discussion of the dual seems to be independent of it. Consequently, this source appears unlikely to have contributed significantly to the subsequent identification of the dual as the Cairo pentagon.

To assess the plausibility of this conjecture, Figure 2 compares the in situ pentagon with the dual. The differences are considerable, and the resemblance is only superficial, certainly much less so than in the comparison shown in Figure 1. Would Macmillan really have mistaken one for the other? Given the visual differences, this seems unlikely, yet it remains the only explanation currently available.

5. Robert H. Macmillan, 1979. “Pyramids and pavements: some thoughts from Cairo”, in The Mathematical Gazette

The fifth apparent instance is a first-hand sighting by Macmillan. As detailed in the article, he clearly, albeit with a proviso discussed below, refers to the condition of collinearity (which both the equilateral and dual types lack):

(iv) If 𝝓 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 often adopted in Cairo…

At first sight, this would appear finally and correctly to establish the nature of the Cairo pentagon, since the stated condition of collinearity accords with the in situ photographs. However, a note of caution remains. Macmillan describes this proportion as being "often adopted" in Cairo, rather than "always" or "invariably" adopted. This wording potentially implies the existence of more than one form of Cairo pentagon.

The question therefore arises as to whether Macmillan intended to distinguish between different types, or whether "often" was simply an imprecise choice of wording. The latter interpretation appears more likely. All known photographic evidence points towards collinearity, and Macmillan's own observations are consistent with this conclusion (albeit tempted by his account in Geometric symmetry, above). Furthermore, I have now assembled perhaps more than 100 distinct sightings from a variety of locations throughout Cairo. These occur in two principal forms, either as square matrices or as individual pentagons. While the square-matrix arrangements can sometimes make the intended geometry more difficult to assess, all are either demonstrably collinear or strongly suggest collinearity.

Nevertheless, the wording remains intriguing. Macmillan was a careful and precise observer, and elsewhere in the same article (p. 253) he again uses the word "often", this time in the context of colouring. In that instance, the term is entirely appropriate, as multiple colourings are indeed encountered. Whether the same nuance was intended here cannot now be known with certainty.

Despite this slight ambiguity of expression, the cumulative evidence strongly supports the interpretation that Macmillan regarded the Cairo pentagon as possessing collinearity. Taken together with the photographic record, there can be little doubt that collinearity, rather than the equilateral or dual forms, was the characteristic he intended to highlight.

Conclusion
With numerous photographs now available, drawn from a variety of locations, the confusion surrounding the Cairo pentagon appears largely explicable, and the various strands of evidence fit together reasonably well. I conjecture that most references describing the pentagon as either equilateral or the dual of the 3.3.4.3.4 tiling ultimately derive from one of three sources: Dunn (the equilateral pentagon), Gardner (also the equilateral pentagon and probably the most influential source owing to the wide circulation and accessibility of Scientific American), or Schattschneider (the dual of the 3.3.4.3.4 tiling). These descriptions then appear to have been repeated without direct reference to the in situ tiles and gradually acquired the status of accepted fact.

That situation is problematic enough in itself. However, the confusion appears to have been compounded by inadequate cross-checking within the literature. The two incorrect pentagon types are cited in roughly equal measure, yet surprisingly little attention seems to have been paid to the discrepancy between them. One might reasonably have expected such a contradiction to have prompted closer scrutiny.

More surprising still is the apparent neglect of Macmillan's account. Despite extensive searching, I have been unable to find a single reference to his 1979 article (see addendum), even though it correctly identifies the essential condition of collinearity. This is particularly noteworthy because, although Gardner's article in Scientific American was undoubtedly the most widely read and accessible source, one might nevertheless have expected academics working in the field to have been familiar with The Mathematical Gazette, in which Macmillan's article appeared.

Consequently, the subsequent literature appears to have been influenced largely by the earlier misconceptions, while the first published account to identify the defining characteristic of the in situ pentagon attracted remarkably little attention.

Addendum

For completeness, I list below those references in which the author either explicitly states, or implicitly indicates through an accompanying diagram, the type of pentagon associated with the Cairo tiling: equilateral, dual, or collinear. Incorrect attributions are remarkably common. Of the references located, eleven give the equilateral pentagon, ten the dual, and only one the collinear pentagon. The sole exception is Macmillan, whose account alone accords with the in situ evidence. Well played, Macmillan!

James Dunn, Equilateral, 1971

Martin Gardner, Equilateral, 1975

Doris Schattschneider, Dual of 32.4.3.4, 1978

Robert H. Macmillan, Collinearity, 1979

George E. Martin, Equilateral, 1982

Jim McGregor and Alan Watt, Equilateral, 1986

Francis S. Hill, Equilateral, 1990

Arthur F. Coxford, Dual of 32.4.3.4, 1991

Anne Fetter et al., Equilateral, 1991

David Wells, Dual of 32.4.3.4, 1991

Steven P. Meiring, Dual of 32.4.3.4, 1992

Audrey Leathard, Dual of 32.4.3.4, 1994

David A. Singer, Equilateral, 1998

John Gregory, Equilateral, 1999

Catherine A. Gorini, Equilateral, 2003

Eric W. Weisstein, Dual of 32.4.3.4, 2003

George McArtney Phillips, Dual of 32.4.3.4, 2005

Briony G. Thomas and Michael A. Hann, Equilateral, 2007

Robert Fathauer, Equilateral, 2008

Craig S. Kaplan, Dual of 32.4.3.4, 2009

Mike Askew, Dual of 32.4.3.4, 2011

Glenn Ellison, Dual of 32.4.3.4, 2013

David E. Laughlin and ‎Kazuhiro Hono, Dual of 32.4.3.4, 2014

Page History

28–30 October 2026. In general, this revision involved a substantial reworking of each section, with expanded text aimed at better untangling the various misattributions and the often convoluted chain of reasoning that led to them. To document every change in detail would be unnecessarily tedious, but the full extent of the revisions can be seen in the working Google Doc.
The previous printed presentation was very poor. For reasons that were initially unclear, a number of diagrams were reproduced at an excessively large scale, sometimes occupying approximately half a page and occasionally appearing alone on an otherwise largely empty page. Furthermore, the diagrams were heavily pixelated, and there was an obvious gap between the table and the page history section. Collectively, these issues gave the document a noticeably less professional appearance than intended. The 2026 return has not improved matters! To be addressed. 

30 July 2025. Reappraisal in New Sites. A general 'tidying up', without a major upheaval being required, as with other pages. Corrected minor typos in the text with Grammarly before a later, more extensive analysis.  As is common with tables, the conversion has left the table not ideal, but it is not wholly unacceptable either. In short, it serves, pending a new look.

Seemingly, no material change of note has occurred since its creation in 2011. I may update with the Stanford archive finding, which greatly clarifies Gardner's reasoning. Also, add further latter-day references to the table; the situation remains (shamefully) the same!

Created 16 November 2011