The first of three pages (so far) on tessellations as actual pavements, and related aspects, of which the first has as its premise images from around the world, with a short introduction, the second, patents, and third, a miscellany, covering more topics but rather of a less extensive extent, that quite simply do not justify distinct pages.

Street Pavements

One of the relatively few applications tessellation has is in the use of actual street pavings. Pavings as such have a long history, seen worldwide. Indeed, street paving has been found from the first human settlements around 4000 BC in cities of the Indus Valley Civilization on the Indian subcontinent in modern day Pakistan, such as Harappa and Mohenjo-Daro. Roads in the towns were straight and long, intersecting one another at right angles. However, quite when the first side pavings, or sidewalks were instigated is under investigation.

Of interest mathematically is to what tessellations have been chosen for this purpose, in that what is it exactly that makes a certain tiling attractive enough to be manufactured as a paving? Or is there no real understanding of the issue, with any tiling too hand largely arbitrarily selected? There are seemingly instances of both aspects to be seen. Undoubtedly, there is mostly a utilitarianism concern here, with aesthetics largely secondary. Mostly, pavings are ‘plain’, as squares or rectangles, likely for the sake of convenience of the worker placement, and indeed cost; after all what can be easier than to lay paving slabs with such obvious shapes. A little more imagination is shown with regular hexagons, but these too are of an everyday nature. As such, it can be said that the instigator, or installing body of such instances lacks imagination, and so aesthetically in this context such instances are largely excluded from this particular page. However, when for some reason there is indeed an aspect of interest, I do occasionally make an exception to this rule. However, rectangles can indeed be made to tile in a variety of interesting ways, and so these I show on the ‘Miscellaneous’ page. Although these can be described as slightly more ‘interesting’ than those described above, to me at least it still does not display enough imagination to qualify as ‘of interest’ here, and so again such instances are thus excluded on those grounds.

As such, pavement designs are to all intents and purposes largely anonymous as to a named designer, with any one design, where indeed accompanied by a description, is simply described as according to a trade name or catalogue number. However, on occasions, some do indeed acquire more attention than others, and not necessarily due to sheer frequency as might otherwise be thought. Simply stated, there are two pavements with named associations, not after a named designer, but rather of cities. Undoubtedly, the more well known instance, at least to mathematicians (if not the general public), is the ‘Cairo tiling’, or has at least acquired this name, in Egypt, and the decidedly lesser known ‘Baldaso de Bilbao’ in Spain. Indeed, although the Cairo tiling is indeed notable in terms of frequency of sightings (although now sadly diminishing, with constant renovations; it not being replaced and not currently being manufactured), it has really only come to prominence by chance matters. I document the full story on my Cairo pages, but in brief, it was instigated in 1957 by Ramzi Omar, but not named as such at the time, and only much later was it so named from a single sighting, by James Dunn in a 1971 mathematics article who stated it was a ‘favourite’ (due to the sheer extent of a sighting, at Cairo airport, but not frequency, although likely it was indeed by then). Martin Gardner in a mathematics 1975 article picked up on this, in a sense misreporting from Dunn’s article, stating it was ‘frequently seen’, from which it then gained worldwide fame. As the paving is more fully discussed elsewhere, I thus defer to those writings. Furthermore, there is also another paving that has been given a name after its city of origin, namely the ‘Baldaso de Bilbao’ (Tile of Bilbao), although this lacks any tessellation element, and is rather of a geometric design of a circular composition, yet nonetheless, it is interesting on account of its popularity. Pleasingly, it has a documented history, as well as a named designer although both a little hazy as to specifics; different dates and designers are quoted, all of which I discuss on the miscellaneous page in more detail. This page is thus dedicated to showing those pavings that are thus a little ‘out of the ordinary’, having some degree of aesthetic interest, thereby showing more imagination from the instigator than otherwise with the more commonly seen squares and rectangles. Furthermore, of interest in that likely a non-mathematician has selected a certain tiling to manufacture, and so it is interesting in observing what a likely outsider to mathematics considers to be attractive. Below I show the world’s offerings, or at least what I can find, of what I term as ‘interesting’. However, by its very nature, this is a daunting task! It may very well be some have been excluded. Even after many years of searching, it is not uncommon for me to find numerous instances from any one country. Romania was a case in point. Nothing at all for many years, and then someone with an appreciation for such designs posts a page replete of examples. Also included are some pavings that are not strictly tessellations, but of optical illusions, and so as by default are ‘of interest’. And likewise where, showing some degree of imagination, despite being of rectangles, squares, or hexagons I nonetheless make an exception to my rule for these polygons. Also, on occasions, I show some consisting of interesting geometrical designs.

The countries shown are: Argentina, Australia, Azerbaijan, Brazil, Bulgaria, Canada, Canary Islands, China, Egypt, Ecuador, Germany, India, Japan, Morocco, Netherlands, Nigeria, Norway, Poland, Portugal, Romania, South Korea, Spain, Syria, Thailand, Ukraine, United Arabic Emirates, US and UK. Likely there are many more countries, but they have simply not been documented by travellers. Curiously, some countries seem to favour certain types. Spain seems to have a loose ‘tradition’ of illusion types, with three instances. Morocco, not unsurprisingly, favours those based on traditional Islamic patterns. This would thus not seem to appear by chance; does anyone know of the circumstances behind this? The distribution of interesting placements is seemingly at whim; some countries seem to have a rich variety, whilst others are totally lacking! Of the pavings shown, it can be seen that some instances are more popular than others, despite these being separated by continents, and so likely of some connection, although I do not see how. Is there an ‘international association of paving manufacturers’ to explain this? Of note is what I descriptively term as the ‘Wavy Rectangle’, ‘Dumbbell’, and ‘fused hexagons’ below. All of these feature extensively.