Defining a Cairo Tiling

Defining a Cairo Tiling

Or

Is Your Cairo Tiling the Same as My Cairo Tiling?

Is the definition a simple matter, as one might otherwise think? Anything but! Much depends on which “Cairo tiling” one has in mind—very much in the spirit of Branko Grünbaum’s celebrated question, “Are Your Polyhedra the Same as My Polyhedra?” In that paper, Grünbaum highlights what he calls the “Original Sin” of polyhedral theory: the long‑standing failure, from Euclid through Kepler and Cauchy, to define a polyhedron rigorously before attempting to classify the regular ones. I adapt his title advisedly.

The situation with the Cairo tiling echoes this problem. There are not merely slight variants but three different “contenders”, each sharing the same broad form, yet differing in mathematically significant ways, namely having a collinearity feature, equilateral, and the dual of the 32.4.3.4 tiling (Fig. 1a-c). 

Strictly, one could also say there are others: two with alignments on adjoining vertices, the Cordovan and Bailey pentagon. 

There is also a wider range, where the Cairo tiling degenerates to a “near rectangle” and “near square”. I discuss these under the page “Minima to Maxima”. Be that as it may, for the sake of a more condensed study, I restrict the discussion to the three main types as given. That said, the generic definition also applies to these others, although I don't refer to them.

To my mind, the definition ought to be straightforward: use the in‑situ paving as the canonical model. What could be more natural? Yet authors repeatedly adopt other forms. From the outset, the Cairo tiling has been dogged by inaccurate descriptions. James Dunn (1971) states that the tile is equilateral; Martin Gardner (1975) repeats Dunn’s claim; Doris Schattschneider (1978) presents it as the dual of the 3.3.4.3.4 tiling; and Robert Macmillan (1979) introduces a collinearity condition. These cannot all be correct! In fact, Macmillan’s account is the one that matches the in‑situ model, whereas the others do not. The result is a literature permeated with misattributions, as the examples below make clear.

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

Despite the correct geometry being identified by Macmillan as early as 1978—following his visit to Cairo—subsequent authors have almost universally adopted either the equilateral model first proposed by Dunn and repeated by Gardner, or the dual of the 3.3.4.3.4 tiling introduced by Schattschneider. What is striking is that, aside from Subhash Chandra Saxena in 2010, no one seems to have noticed the discrepancy or checked the original sources. Even after I presented clear, in‑situ evidence in 2010, with the webpage ranking prominently on Google, little has changed. To this day, mathematicians who should know better continue to describe the equilateral model or the dual of 3.3.4.3.4 as if they were the in‑situ paving.

So, to paraphrase Grünbaum: which Cairo tiling are we going to define? The in‑situ model must obviously take precedence, yet it is also useful to provide a “catch‑all” description that accommodates all three established versions. After all, they share certain core features — notably the two adjacent right angles and the intersecting hexagons formed at right angles — and incidentally, it is this hexagonal structure that largely explains the tiling’s enduring appeal. 

My first step is to examine the literature for formal definitions. Surprisingly, these are rare. Most authors simply present one of the three models without further comment and leave the matter there. Indeed, I find myself struggling to locate anything in print that attempts a rigorous definition at all. 

Online, the Wikipedia article offers a detailed account, but it inevitably strays into a discussion of various pentagon types, which makes the reading less straightforward than one might hope. Still, it does identify the essential features: “two non‑adjacent right angles, with a pair of sides of equal length meeting at each of these right angles.” It then proceeds to list special cases. The main shortcoming, however, is that nowhere does it show a photograph of the in‑situ paving itself. There are plenty of line drawings, and examples from Croatia and Denmark, but not the most obvious instance — the paving in Cairo! The page also includes, rather prominently in its opening paragraph, peripheral remarks about Percy MacMahon and John Horton Conway, whose connections to the tiling are extremely loose and quite out of place in a definition‑focused introduction. And there are other issues besides. The article could be considerably improved.

Doris Schattschneider, in a tiling‑group discussion responding to this very question I had posed, offered the following definition: 

“…the pentagon in this tiling is Type 4 in the list of convex pentagons that tile the plane, characterized by the conditions that it has two non‑adjacent angles that are 90 degrees, each enclosed by two equal sides. A stronger condition would be that all four of these sides are equal.”

As definitions go, this is entirely sound: it captures the “standard model” of the Cairo pentagon accurately and succinctly. However, it does not cover every legitimate instance. In particular, one can construct a Cairo tiling in which all five sides of the pentagon are equal — an equilateral pentagon — and this special case is excluded. For completeness, therefore, any definition should acknowledge this equilateral instance alongside the standard form.

Building on the core description by Schattschneider, for a generic Cairo tiling to accommodate the equilateral and dual possibilities, I propose the following:

Generic Cairo pentagon (covering all three models)
A symmetric pentagon with two non‑adjacent right angles, each right angle enclosed by a pair of equal‑length sides. These four sides may all be equal, with the fifth side either shorter or longer; and in one exceptional case, all five sides may be equal.

In‑situ Cairo pentagon (canonical model)
A symmetric pentagon with two non‑adjacent right angles, each enclosed by a pair of equal‑length sides. All four of these sides are equal, while the fifth side is longer, in accordance with the collinearity property of adjacent hexagonal units in the paving.

A definition framed in this way may fall short of full mathematical rigour — and I gladly leave room for any mathematician to refine or strengthen it — but it more than suffices as a clear, accessible description for a general audience. 

References (the main “early” discussions)

James Dunn, 1971. ‘Tessellations with Pentagons’, in The Mathematical Gazette

… 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. 

Martin Gardner, 1975. ‘On tessellating the plane with convex polygon tiles’, in Scientific American

… 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. 

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

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

Robert H. Macmillan, 1979. ‘Pyramids and Pavements: some thoughts from Cairo’, in The Mathematical Gazette

(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 often adopted in Cairo… 

Branko Grünbaum. “Are Your Polyhedra the Same as My Polyhedra?” Discrete and Computational Geometry: The Goodman–Pollack Festschrift, edited by Boris Aronov, Saugata Basu, János Pach, and Micha Sharir, Springer, New York, 2003, pp. 461–488.

https://faculty.washington.edu/moishe/branko/Your%20polyhedra,%20my%20polyhedra/grun.pdf 

David Bailey. “Pentagon Confusion”. Retreived 21 October 2026

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

Wikipedia. Cairo Tiling. Retreived 21 October 2026

https://en.wikipedia.org/wiki/Cairo_pentagonal_tiling

Page History
20–21 October 2026. Although I originally set out to revise and update a page on the same broad premise, Defining a Cairo‑like Tiling, the old page now survives only in the smallest of remnants (some diagrams and some table detail in list form). With new and expanded text and a much improved presentation, the revision has effectively left the old version far behind, and I consider it justifiable to treat this as a new page and retire the former one. The earlier page can still be viewed on the Wayback Machine if desired.