Variations of the Cairo Tiling

Introduction

Within the standard model of the Cairo tiling, it is possible to vary, or relax, the conditions somewhat and have what I term a “variation”, whilst still retaining a broad, albeit sometimes very broad, Cairo premise. For example, one can:

Relax the core number of pentagons, with increases of two, three, or four pentagons

Relax the side conditions

Relax the 90° angle conditions

Relax the symmetry condition with asymmetric pentagons.

Relax the convexity of the parhexagon condition with concave parhexagons.

Relax the parhexagon condition altogether, with paroctagons* and pardecagons*
* The naming of these polygons is by myself, of an apparently unofficial naming without precedent, an extension of the parhexagon principle. Although unofficial, the intent is clear.
With the revised conditions above, the possibilities here are so vast as to be impractical to fully undertake “all possible” tilings. To this end, I thus show a few arbitrary examples of the possibilities. It should be pointed out that none here are an “improvement” on the original model, but nonetheless have a modicum of interest. To the best of my knowledge, such a study has not been attempted before.

An open question is how best to present the material. One could adopt different formats, reflecting the varied conditions under which these tilings arise. However, the most natural—and the approach taken here—is to order the studies by the most basic parameter: the number of pentagons involved. The following examples, grouped under 2, 3, or 4 pentagons, are broadly instances that arguably qualify as ‘Cairo‑variation’ tilings.

A consistent feature throughout is the effort to show the overlapping hexagons clearly and unambiguously, using a variety of colours to distinguish them. 

For ease of comparison, I begin by showing the standard model of the Cairo tiling, with the subsequent variations understood as relaxations of its defining conditions.

1 Pentagon

One pentagon, symmetrical, 4, 1 sides (four short, one long), two 90° opposite angles. As a tiling, two subsidiary convex parhexagons at right angles. As an aside, this example has lines of collinearity. A whole host of ‘special cases’ of the 1-pentagon can be formed, as discussed on the ‘aesthetics’ page.

2 Pentagons

3 Pentagons

4 Pentagons

As shown, several new variations on the Cairo‑tiling premise emerge, though their value is open to debate. In relation to the true Cairo tiling, they are best regarded as novelties—situated somewhere between the trivial and the genuinely worthwhile. 

Page History

19 October 2026. This return was in the midst of a general overhaul of the Cairo‑tiling pages. The text was revised and updated, while the diagrams remained unchanged. The change of title from Defining a Cairo‑type tiling to Variations of the Cairo Tiling required a new introduction. Although the former title was not incorrect, it was less descriptive than it might have been and risked confusion with the similarly named page Defining a Cairo‑like tiling, whose distinctions are now made clearer.
In the text and captions, I had repeatedly used “par hexagon”, along with “par octagon” and “par decagon”, rather than the correct “parhexagon”. The standard usage—limited though it is—is “parhexagon”, which I now adopt. A short summary has also been added, though it was arguably optional. The captions introduced on 22 July 2025, attached directly to the images, have been retained rather than replaced with separate text‑box captions. Overall, the changes are modest, amounting chiefly to improved wording under the new title, whilst still retaining core aspects of the old text. 

22 July 2025. New Sites 'adjustments'. The conversion had left the captions disjoint, which I now attach to each image. Minor corrections were made to the text in Grammarly pending a more extensive reappraisal. Unlike other pages, no changes of note have had to be made.

Created? : Revised 25 January 2013