Cairo Tiling

In various mathematical books and articles on tilings, a reference to a pentagonal tiling, with certain aesthetic properties, known as the 'Cairo Tiling' (or similar, such as 'Cairo pentagon/tessellation') is commonly to be found (with descriptions such as 'frequently seen' and 'many of the streets are paved…'), said to be found in Cairo, Egypt, hence the title. However, amazingly, although often discussed in the literature, and illustrated with a line drawing, this is not shown in its most obvious incarnation, as portrayed on the streets of Cairo as a purposeful photo. A distinction should be made between the photographs. Although the paving is indeed previously pictured, this was by chance, appearing as background matter to whatever is being photographed, for instance appearing in travel guides.

Until recent times, despite extensive searching, I have not been able to find an in situ picture of this. Note that here I emphasise 'picture', contrasting this against a line drawing, which is invariably found in all quoted sources. Indeed, I believed that this might at one time just be a 'mathematical urban myth', given that I had found no pictures, or had no referenced first-hand sources until recent times.

In the general story, much fundamental detail is omitted. Specifically, this includes the date of instigation, the designer, and the geometry of the paving. Further, the paving comes in two forms, in a square matrix, with the pentagons inscribed, and as single pentagons, with a curious 'many polygon' texture, along with different colouring. This account thus reveals these aspects for the first time.

My own interest in the Cairo tiling began around 1986-1987, at the beginning of my mathematical interest in tilings, although exactly in which publication I first saw it I do not recall. From the beginning, I was impressed by the aesthetics of it. A surprising, although explainable feature, is the grouping of four pentagons into a parhexagon, overlapping at right angles. Although other tilings can be seen to possess overlapping features, typically the tile will be of a polygon lacking the simplicity of the Cairo pentagon. Undoubtedly, this feature helps to explain its popularity among mathematicians, with much fanfare, unlike other 'everyday' tilings.

The study is in (?) distinct aspects, namely that of the originator, Ramzy Omar, in situ pavings, the mathematics and variations and examples, undoubtedly independently, seen elsewhere, in other countries, in a variety of forms.

Created 21 June 2024. Last Updated 21 June 2024