Miscellaneous

Some Thoughts on Designing the Cairo Tile in Ancient History

Although a redundant statement, the Cairo tiling is but one of many tilings, both as artefacts and mathematical abstractions, simple and complex, and everything in between. It is interesting to speculate how it originally (of course without the attribution) it may have been found in ancient times, from the very beginning of tiling, about 5,000 years ago, although there is no evidence that this is so. Let us consider the matter of designing tiles/tiling in general, without a thought as to the Cairo tile, and we see how easily and quickly it can arise by a simple experiment, or play.

(i) Of course, we begin in the most simple and basic way, here with a grid of 3 x 3 squares (Fig.1), with an upright cross of equal length. 

(ii) Experiment, with upright crosses of equal length about the midpoint on a square. Translate (or reflect) successively (Fig. 2). This simply gives another square tiling; not very interesting. 

(iii) Let's try another idea. Now, retaining the square, instead of having an upright cross, rotate to a natural division (Fig. 3). 

(iv) Again, translate (Fig. 4). This simply gives an open grid. 

(v) An obvious thought is to connect the lines, forming a rotational order 4 tile (Fig. 5). On account of its symmetry, this is thus more interesting, but it is not outstanding or possessing unusual properties in an aesthetic sense. It is but one of many such tilings with order 4 rotational symmetry. 

(vi) On the same premise, let’s then try reflecting the rotated cross. This then (naturally) joins up the lines. Reflect successively in two directions, horizontally and vertically. This then gives the 'bowtie' tiling (Fig. 6). This is quite interesting; a symmetrical octagonal tile along two axes, appearing in two orientations, with each line the same length. 

(vii) An obvious thought is then to simply join the two indentations (Fig. 7). Voilà! A Cairo tile is thus formed. It is so basic and easy; even a ten-year-old (if not earlier) child could do this! No mathematical calculation is required. And further, this can be seen to be of the in situ model, with aesthetic features of collinearity and the overlapping of horizontal and vertical parhexagons which is surely an obvious attraction to the mooted designer. Further, it is the first such (Cairo) tiling to appear (as I show elsewhere, it is possible to have different proportions of the tile, in which it degenerates from minima to maxima to a rectangle and a square), in that it is not some arbitrarily high number, such as number 5, or 50, in a long series that is conceivable ‘difficult’ to find; it appears immediately. Indeed, it is entirely feasible to design this even quicker, at the reflecting stick cross stage (vi), without the given mooted beginning sequence. 

(viii) Further, from the tiling, it is easy to outline a suitable region in a square to repeat as an actual square tile (Fig. 8). Although it may be stating the obvious, by far the most common shape for a tile is a square, and so this thus is eminently suitable as an artefact. 

Therefore, given its ease of composing, and of more interest of most given the pentagon polygon, and obvious aesthetics, and its easy transposition to an actual square tile if so desired, why do we not see it from ancient times? I do not know. It is not even a question of perishable materials. Tiles, when fired, as an artefact are very long-lasting, much more so than wood, for instance. Quite frankly, I am baffled! Upon consulting dedicated tile books such as van Lemmen [1] and Kneale [2], there is not even the slightest impression of an appearance, whether as above or one of its many variations as I show on my ‘stick cross’ pages. There are far more complex and involved tiles than the Cairo tiling in the book (and in others of a similar nature), and so its omission is thus all the more surprising. A like situation also arises in the field of mathematics. Indeed, its first appearance, as a minor variation to the above, is as late as the 17th century as a jali (an Indian screen), albeit the date has not been verified for certain [3], and then again in the ‘Early to Mid 20th century’, of a flooring at Heidelberg Castle, Germany. This sighting is unpublished and unverified. It was first described in the mathematical literature as late as 1921, of Percy A. McMahon’s New Mathematical Pastimes. [4]

References
[1] van Lemmen, Hans. 5000 Years of Tiles. The British Museum, 2013
[2] Kneale, Nicholas. The Tile Book (Fired Earth). The Artisan Press, Leicester, 1991
[3] Ray, Simon. Indian & Islamic Works of Art. Self Published Catalogue, 2016, pp. 178-179.
[4] MacMahon, Percy A. New Mathematical Pastimes. Cambridge University Press 1921 and 1930. 

Web
http://www.tess-elation.co.uk/cairo-tiling/study-1
http://www.tess-elation.co.uk/cairo-tiling/study-2

29 July 2025. Reappraisal in New Sites. The page after the conversion was in a poor state as could possibly be imagined, with so many shortcomings. In particular, it was disjointed. This is a typical finding with long pages after the conversion process. As such, so bad it was, the page was irredeemable. Oddly, it had two distinct unrelated studies, as above (reinserted from Classic Sites, for reasons as above. This then intoduced a new shortcoming, in that the GeoGebra drawings were pixelated! To be corrected), as well as, oddly, 'The Cairo Tiling As...', of a single very long page, seemingly created from a page copy, although with the above study at the head, perhaps not. It may even have been added one aspect at a time. Whatever, quite what I was aiming for is unclear. Was it an attempt at a changed presentation, having become exasperated with the old presentation? Was I going to delete the above study and have 'The Cairo Tiling As...' as a long piece? Or both, despite the obvious incongruence in length? And possibly with the addition of other 'Miscellaneous' studies, presumably of short length, deemed undeserving of a distinct page? I simply don't recall. Given that I have now properly ordered the 'The Cairo Tiling As...' material, the text and imagery are redundant, which I thus delete. As such, the page is poorly titled; I may make a copy and rename.

Page created 14 July 2020