One of the few applications tessellation has is in the use of actual street pavings. Pavings as such have a long history, seen world wide. Indeed, from as far back as * the streets of Cairo were paved. Of interest mathematically is to what tilings 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? Undoubtedly, there is mostly a utilitarianism concern here. Mostly, pavings are ‘plain’, with no interior design, with a rectangle favoured, more or less accompanied in popularity with a square. Another favoured instance, albeit decidedly lesser so than those above, is a hexagon. Together, these three appear almost invariably by default, likely on account of their sheer simplicity of familiarity, as no real thought is needed on the part of the designer or person who is laying these as to how these pave; it is obvious, even to the layman. As such, it can be said that the instigator of such instances lacks imagination, and so aesthetically in this context such instances are excluded from this page. Another frequently to be seen paving is of narrow rectangles (block pavers), typically arranged in a herring bone or basket weave patterns. (A reason the rectangle being favoured for this is not only utilitarian, but is that in a herring bone pattern the pavers are ‘self locking’, and so need less frequent replacing; other formats are ‘looser’). 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’, and so again such instances are thus excluded on those grounds.
As such, pavement designs are to all intents and purposes anonymous, with any one design 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, with the perhaps more well known ‘Cairo tiling’ in Egypt and lesser known ‘Baldaso de Bilbao’ in Spain. Indeed, although the Cairo is indeed notable in frequency of sightings, it has really only come to prominence by chance matters. I document the full story on my Cairo pages, but in brief, it was 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, 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 world wide fame. Furthermore, there is also another paving that has been given a name ‘Baldaso de Bilbao’ (Tile of Bilbao), although this lacks any tessellation element, and is rather of a geometric design of a circular composition, yet none the less, 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 below in more detail.
This page is thus dedicated to showing those pavings that are thus a little ‘out of the ordinary’, thereby showing more imagination from the instigator. These are thus likely (but not always necessarily) to be of more interest than otherwise (with squares and rectangles), and 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 of what I term as ‘interesting’. Also included are some pavings that are not strictly tessellations, but of optical illusions, and so as by default are ‘of interest’, showing imagination, despite being of rectangles, squares, or hexagons. 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. Curiously, some countries seem to favour certain types. Spain seems to have a loose ‘tradition’ of illusion types, with three instances. Morocco 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.
The ‘wavy rectangle’ is possibly favoured due to a variety of placement possibilities, seen in no less than eleven countries: Australia, China, Japan, Norway, Romania, Slovakia, South Korea, Syria, Tunisia, US, Thailand and Vietnam (indeed, so prevalent is this design is that I have stopped adding any further examples). Another favoured one is the ‘Dumbbell’, with instances in Egypt, India, and US. The fusion of three hexagons is seen in Australia, China, Mongolia, and Ukraine. Note that this also allows for different placements.
Of necessity, due to the obvious practical difficulties involved of visiting all the world’s towns and cities is that the selection here is necessity of an ad hoc nature, and cannot by any stretch of the imagination be described as ‘scientific’ as in ‘all encompassing’, as with the whole world to find pavings from, it is simply impractical to undertake systematic research in the field. Consequently, this can be described as a ‘just for fun’ page, a snapshot of the world’s most interesting pavings, but nonetheless, even in these early days of the investigation, one can see patterns emerge. As might be imagined, finding these on my part is restricted to web searches, mostly on Flickr, and a few personal contacts that have a like interest, with photos taken on whim as according to country of vacation visited. Unfortunately, it was not always possible to obtain copyright (no reply received), and where this is so, a link to the picture is given. Consequently, some countries here are (unfairly) heavily biased in their instances. For example, my two chief correspondents in this matter, Frank Housholder, in Germany, and Pam Garnett, in Egypt, have found many examples, and so instances from these two countries are likely over represented as against others. In addition, countries that are ‘off the beaten track’ are less likely to appear in picture databases. Furthermore, the pavings found are very much subject to the whims of the web; a paving could (surprisingly) be very popular indeed, and yet not show up in search engines. Indeed, before my researches, the frequently quoted and well-known Cairo tiling was to all intents and purposes invisible to Google (only with a specific place search would it show), despite its now established prevalence in many districts of Cairo! And so if that situation could occur in so widespread a paving, then other popular tilings may also have evaded detection. I also might just add that I am very interested in the ‘Cairo paving’ mentioned here, of which I have compiled a separate page, with many subsections. Any further details not covered, or indeed just general observations concerning this, would in particular be warmly welcomed.
I would be pleased to receive any pictures from readers of this page of any ‘interesting pavements’ from around the world, and for obvious reasons, the more ‘off the beaten track’ countries so much the better, to which these would not necessarily show up in web searches.
Of interest is to who exactly has designed any particular paving; generally this is not known, and the circumstances behind its installation. If any pavement designer has anything to say on this I would be pleased to receive details.
The history of such pavement designs is apparently a surprisingly relatively recent phenomena, of which although the history is somewhat uncertain, probably dates back to the 1950s at the earliest, with the 1960s more likely. This can be determined by the introduction of such concrete pavers, as against clay pavers, of a rectangle form of much earlier dates (with information kindly supplied by John Knapman, a paving expert), which thus dates from this period. But exactly when these started appearing is unclear. Indeed, as yet I do not have an ‘early’ pavement picture of this type. As a rule, such matters are not mentioned in books on tiling with a strictly mathematical perspective. However, ‘historic pavements’ are discussed, briefly, in the bible of tilings, Tilings and Patterns, by Grünbaum and Shephard, page 9, accompanied by the caption ‘Some unusual street tilings seen by the authors in Europe and North America’. Their Figure 12 gives four instances, two of which are seen on this page. Indeed, the second and fourth are very popular indeed; ‘unusual’ here is (I believe) applied to the tile itself, rather than that of intrinsic rarity. Unfortunately, no indication is given as to when these pavings were seen. The book is of 1987. Has anyone any pictures of interesting street pavings before this date?
Finally, a few web links showing manufacturers and machines. Such imaginative designs are now becoming more prevalent, with many more companies making these than previously. Prominent of these is the Srikrishnaplasto company in India with a line of many different types, mostly of a simple nature (albeit this is not to imply denigration in this; indeed, they remain most attractive, a welcome change from the hackneyed rectangle):
© 'dmsperkins', Buenos Aires
© 'Terrell' Chacarita cemetery in Buenos
© Ross F. Housholder, Melbourne
© Ross F. Housholder, Tasmania
© Maria Cristina Travaglio, Vienna
© Huseyneli Huseyneliev
most geometrical pavings are ‘abstract’, occasionally an Escher-like tiling is
used, as in this example of a ram from Azerbaijan, by Mammad Huseynaliyev a physicist
from Azerbaijan National Science Academy. The ram was chosen, not by chance,
but for a specific reason, in that it is highly symbolic for the city of Nakhchivan, the capital of the eponymous
Nakhchivan Autonomous Republic of Azerbaijan, located 450 km west of Baku. Many sculptures from medieval
times featuring a ram can be seen across the region, hence the choice here. The
curved figure with the ram paving inside is called a ‘Buta’ and is a typical
detail of Azerbaijani national ornament and is considered the symbol of Azerbaijan.
© Ju Row Farr San Paulo Note that the tiles are a stylised picture of the state of San Paulo
© Carlos Strauch Santo Antonio de Jesus
© Tom Magliery Downtown Vancouver, West End.Observe the curious clustering, of four circles combined
© Tom Magliery Vancouver, Bentall Tower Two
© Ross F. Housholder Puerto Rico
© Ross F. Housholder St Maarten
© David Barrington Playa del Ingle, Gran Canaria
© Melisande Also see an instance in Norway
© Lance Belville, Beijing, close to the temple of heaven
© J. Eric Lynxwiler. Havana, Cuba in May 2015.
The terrazzo is in front of the 1912 train
station but that station was modernized in the 1950s or so. The terrazzo likely
dates to that renovation.
© Olga Schlyter, Copenhagen
Egypt has a multiplicity of street pavings, of which one of these, consisting of pentagons, has become so well-know as to have the attribute ‘Cairo tiling’ given (by mathematicians, of which the term is widely used) to it, this dating from the 1971. Of which incidentally this is the only tiling having a name. Recently, I have been researching this particular tiling, of which various aspects remain unresolved. Its actual age is uncertain; the first decided reference being from 1971. Has any one seen this in any earlier publication? And who, or what body, implemented this? Where are they being manufactured? As I state above, the streets of Cairo have been paved for centuries, however, I am unaware of what exactly these pavings took the forms of. Have any of these actually survived? I would be most surprised, but given the history of Cairo I would say that there is still a possibility of some still surviving. Has anyone seen a drawing, or picture of these? Even relatively modern photos would be appreciated.
© Helen Donnelly, Dokki, Cairo
© Pam Garnett Maadi, a suburb of Cairo
© Pam Garnett, Cairo
© Alia Nassar, Port Saad
© Carole Hurel, Sisteron
© Ross F. Housholder, Garmisch-Partenkirchen and
© Ross F. Housholder
© Ross F. Housholder, Bingen and Bingen
© Ross F. Housholder, Bingen, Innsbruck railway station
© Ross F. Housholder Landshut
© Anders Sandberg, Berlin Airport
An unusual tiling - a special type of octagon, a 'par octagon' (four sides are parallel). This is interesting mathematically regarding the parallel sides - is there some general rule as to why this would tile?
© Henry Segerman, Heidelberg,
on Berliner Strasse, north of the university
© Liane McCarthy, Polichronos
© Maria Cristina Travaglio, Budapest
© Ross F. Housholder, Akureyri
© Ramesh Thadani, Agra, Taj Mahal
The first of three pavings outside the Taj Mahal, Agra, which has some interesting pavings. This consists of two tiles. Different interpretation of this tiling are possible which this feature could explain its popularity: (i) This could be described as of ‘interlocked octagons’. (ii) Of two tiles, of star octagons and rhombs, or alternatively of either star octagons or rhombs as entities in their own right, with negative spaces.
© Tom Magliery, Agra, Taj Mahal
© Diana Biggs, Delhi. This particular paving is quite frequent in India, also being seen in Mumbai and Calcutta. Curiously, it is also found in Egypt and Romania, likely independently.
© Yazed Lord, Mumbai
© Yoshiaki Araki, In front of the Musashi-Koganei station of Japan Railway Chuo line, Tokyo. An instance of the 'Cairo tiling' in Japan!
© Jun Nishiseko, Hamamatsu, Shizuoka Prefecture. Another instance of the 'Cairo tiling' in Japan!
© 'Gunnsteinlye', Osaka
© Rob, Ritsumeikan University, Kusatsu
© 'seikinsou', Osaka, near Shin-Osaka Station.http://www.flickr.com/photos/seikinsou/
A feature of the ‘wavy rectangle’ here is that it can be placed in different ways to form a tiling (and so is of added interest); two instances can be seen in the same picture here.
© 'Furdis', Tokyo
© Isabelle Prondzynski, Mbotela, Nairobi
© Katharine M, Ulaan Baatar
Morocco is interesting in that it is one of the few Islamic countries (indeed the only on
e so far) that uses traditional Islamic-type designs for its pavements.
© Richard Chester, Marrakech
© 'Kitticot', Marrakech
© Robert Ferreol, Marrakech Detail
© Christopher Rose
© Roi Shiratski, Funepark, outskirts of Amsterdam. As designed by Martien Osch. An interesting design, of two pentagons
© Jennifer Ehidiamen, Lagos
© AquaZeiss, Oslo
© Tom Magliery, Oslo
© 'Shimmertje', Muscat
I now have more detail as to the ‘Zakopane Heesch paving’ sighting, due to the
sterling efforts of Tom Jacobs, who has been investigating this recently (September
2016), and who, like myself, sought further detail. Previously, first shown on
Wikipedia  in 2005, all that was known was of a Zakopane reference, of a
single, close-up picture of the paving, of a brief, one-line comment, with
photograph of a street pavement in Zakopane, Poland.
In short, although basics are stated, this can plainly to be seen lacking. All
references, on the web and in articles and books  are seemingly taken from
this single report.
extensive searching, on the web, reveals two more pictures, but this time with
the paving shown in context with its surroundings:
(Third image down and third over)
From this, with
the captions given, the pavers are in an outdoor market outside of a funicular
stop at the northern edge of Zakopane. However, many open questions remain,
discussed below, with occasional clarifications below the question:
1. Why is
it in Zakopane, a smallish town with no known mathematical background?
the web, Zakopane is a smallish town about 28,000 people, on the southern
border of Poland at the foot of the Tatra Mountains, known for mountaineering, skiing
and tourism rather than a centre of mathematics, and so is not a town with any
obvious mathematical connection that may explain its appearance. Heesch’s
university was at Göttingen, Germany, of which although Poland shares a border
with Germany, Göttingen is in the centre of Germany, many hundreds of miles
away, and so it seems most unlikely he has a connection with Zakopane
2. Why is
it outside the market place?
For such a
specialised tiling, it seems a somewhat ‘everyday’ place to install. Given the
nature of the design I would have expected a more notable building or public
3. Why only
tiling No. 9 (in particular) of the set of 28?
background to Heesch tilings. Heesch, with Otto Kienzle in 1963 , compiled a
set of 28 tilings that can tile the plane in an isohedral manner without using
reflections. Although the exact Heesch tiling number is not identified here,
upon research this is based on Heesch tiling No. 9.
4. Is it
confined to the two places as detailed?
instigated the project and to what reason?
6. When was
this is it a 2001 installation, as at this time the market place area was
add anything to this? No detail is too small to mention.
 Conway, John H., Heidi Burgiel and Chaim
Goodman-Strauss. The Symmetries of Things. A. K. Peters Ltd, 2008.
 Heesch, Heinrich and Otto Kienzle. Flächenschluss. System der Formen
lückenlos aneinanderschliessender Flachteile. Berlin, 1963.
© Claire McCann, Almeria. An optical illusion, or effect, of which Portugal and Brazil are famous for
Romania, or in particular Bucharest, has an abundance of interesting tessellating pavements.
© Matthew Reames
Note a different arrangement of the tiles, top left
Yet another instance of the favoured ‘dumbell’ tile, seen worldwide. Note that the ‘dumbell’ has variations, and can be shortened or lengthened and yet still tile, as in the next photo.
Another of the popular instance, which also has a variation as to placement; see following picture
This can be interpreted as a variation of the well-known Cairo tiling, with circle arcs replacing the four ‘variable’ straight line sides. The simplest interpretation of this can be described as that of the ‘axe head’ tessellation, with a dividing line. Note that other, circle arc variations can be determined; New Mathematical Pastimes, by P.A. MacMahon, p. 105, gives some others.
Probably the most favoured instance of the eight semi-regular tilings, of 4. 8. 8.
© 'Jak 388', Incheon, near Seoul
© Keith Sharps
This (flat) paving is based on a well-known optical illusion, in which the impression of three-dimensionality is the premise. As can be seen be seen, the effect is highly effective. Of interest is that this paving was pictured in a British national newspaper The Sun (a first on the subject matter?)
© Ross F. Housholder
© Ross F. Housholder
Puerto de la Cruz, Tenerife Interlacing bands
Wikipedia Commons Barcelona, Rambla
An optical illusion; believe it or not, the tiles are rectangular!
© 'Brit and Armand', Malaga. Although not strictly a tessellating pattern; interesting geometrically nonetheless
As broadly outlined in the introduction, Bilbao has become well known for a design on pavements, with ‘Baldosa de Bilbao’ (Tile of Bilbao), and cannot possibly be overlooked, of which it is used to pave the streets literally all over the town. Indeed, estimates are that 600,000 tiles are replaced each year, of a total through the years of 66 million.
The design consists of a rosette motif with or without carved channels, the latter mostly serving a utilitarian purpose. The design was aimed at making pedestrians a little safer in Bilbao’s rainy climate (which is captured in almost all the photos), by allowing the water to drain into the carved channels and away from the surface of the tile, thus making it less slippery. Curiously, there are many other tile designs similar to the rosette style, and in different sizes, with some even pre-dating the Bilbao tile.
The design is linked so strongly to Bilbao’s identity that museum and souvenir shops are stocked with ceramic mugs, posters, soaps, t-shirts, shoes, ties, towels, key chains, pastries and chocolates all featuring the rosette tile design. They even have a Facebook page (in Spanish) dedicated to them!
Date of Introduction and Designer
Both the date of introduction and the designer of the tiles is a little uncertain. Some reports date their first production to the 1920s and 1930s, attributed to Eduardo Saenz Venturi, brother of the sculptor Federico Saenz Venturi from Bilbao, while others say they came about in the 1940s and 1950s in a municipal workshop.
Initially, the tiles were made using concrete and coarse sand, with a covering of iron shavings in a 15 x 15 cm format, though this kind of material didn’t adhere well to the ground and left many pedestrians who stepped on loose tiles soaked in rain water. Nowadays, almost all of the old tiles have been removed and replaced with new ones that are manufactured with cement at a standard size of 30 x 30 cm, although other, both smaller and larger formats can be seen occasionally, such as a ‘giant’ 60 x 60 cm tile.
Initially endemic to Bilbao, the tile design was eventually exported to many cities throughout Spain and Argentina and other South American countries, and Africa (in the former colony of Equatorial Guinea).
All pictures with permission, from: http://groundbeneathmyfeet.wordpress.com/tag/pavement/
© Ekaterina Petrova, Bilboa, 1-3, Barcelona 4
© Maria Cristina Travaglio
© Maria Cristina Travaglio
© Miroslaw Majewski Bangkok
© Miroslaw Majewski Bangkok
© Miroslaw Majewski Bangkok Temple
© J. Brew Ankara
© Joshua Vorbis Abu Dhabi
© Benjamin Esham New York
© Virginia Knowles Lake Lily
© Joe Schumacher Pittsburg
Although I wasn't going to include the commonly to be seen hexagon, I here make an exception. Of note is the sheer size of the hexagons - contrast with the doorway!
© Cathy Myers Newburgh, New York
© Susan Sermoneta New York City
© 'b.ca5ual' Washington DC near L'Enfant Plaza
© Keith Sharps, Newbiggin-By-The-Sea, Nothumberland
© Donna Smillie London, Southall Railway Station
© 'IQ89', Green Park, London
© Roger Marks Birmingham
© Jay Zhang, Saigon
© Eric Broug
Masala market, in Ndola
occasions, pavements are designed with optical illusions in mind, and in
particular, some countries almost make this a feature, namely with Portugal and Brazil, with wavy designs.
Created: 2 March 2012. Updated: 14 September, 13, 21 November, 12, 17 December 2012; 9 March 2013. 11 January 2016: Zambia added. 15 January 2016 Austria and Switzerland added. 24 February 2016 Budapest. 1 August 2016 Heidelberg. 19 Janaury 2017 Azerbaijan, Huseyneli Huseyneliev