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In Situ Oddments

Non-Standard Instance

Recently (22 December 2012), another instance of the tiling has been found, as a flooring (above), rather than as a pavement as previously. Such a development is most surprising and totally unexpected. My correspondent, Greg Niemeyer (of the US), tells me he saw this in a ‘downtown Cairo apartment block’ (in 2008), and on this flooring taking his interest, upon enquiring with local friends, they informed him that this was ‘old stuff from the 1950s’, but both with no more detail forthcoming. However I am extremely wary here with the latter detail in particular, but for now at least I am prepared to take this at face value, with due caution. (I am aware that in Cairo what is reported as fact is not necessarily so; see manufacturing page.) Such an ‘early’ date for the tiling, if indeed true, thus likely predates the earliest pavement instances, of 1956 (Heliopolis, Al-Galaa club). Therefore, this sighting is of the utmost significance on two counts (with due reservation):

1. Possibly the oldest sighting

2. A new form of the design (as flooring)

Of note is that the tiles are exactly of the same type of pentagon as of the pavings (with the same collinearity), but are of a much smaller scale, with tiles of about 3-4 inches sides – observe the photographer’s feet on the right-hand side of the picture for an indication as to scale. Five colours are used; black, white, yellow, reddish-brown, blue. Curiously, black, white, yellow, reddish-brown are commonly used as colours for the larger single pentagon. However, there has never been an instance of blue. Note that the colouring‘arrangement’ is arbitrary.

From this sighting, therefore, questions arise as to the order of developments – was the tiling first made as a floor tiling, and then later made as a paving? This is how it would appear at first glance. However, this is conjecture; it may well be that the pavements are 1950s as well! One cannot be too dogmatic here with so much uncertainty as to order and date. It would seem unlikely that it would be formed independently; the coincidence would be too great i.e. two pentagon pavers, and of the same type. However, what can explain the previous lack of sightings of this kind (flooring)? It would seem unlikely to be a one-off occurrence; presumably it must have been made in bulk, in relative terms, but if so why is it not seen elsewhere? Seemingly as ever with the investigation, there are inexplicable aspects. There are no previous accounts of this, and none of my investigators has seen it elsewhere.

As ever, a series of open-ended questions arise, of which I invite responses from interested readers:

Has anyone seen this flooring elsewhere?

What is its background?

What company made it?

Is a 1950s date feasible?

Where exactly is the ‘downtown Cairo apartment block’? (We would like to examine this flooring in situ for more possible clues as to background)

Or indeed, any other comments or suggestions on this are warmly welcomed.

Created 3 January 2013



Manufacturer

An open question is to where the pavings were, and indeed likely still are, being produced and distributed. Details on this aspect are sketchy, to say the least. However, following information provided by a correspondent, Mohammed Elmokadem, he told me that these are made in small workshops in Basateen, Al Darb al Ahmar, and in GizaHowever, these have not been found, some of the areas are either far from my two investigators, Pam Garnett and Gregg De Young to visit, or not suitable places to visit due to security concerns. Also, Mohammed told me that frequently details of the date are inscribed on the reverse side, and this had led to further detail, thanks to the ‘in the field’ investigations of Pam in examining the reverse side of a square format loose tile, as shown below. This shows not the date, but the place of manufacture, which translates as it being made in the district of Basateen, with the ‘Basateen United supply company’, New Maadi, but here the investigation has come to a stop, as Pam has not been able to find the company, despite having a street address upon Googling the above reference; the shop there present knows nothing of this; possibly they have moved. However, this at least provides a lead. Does anyone know of this company? Also, following correspondence with Laëtitia Huré, general secretary to the manager of the Old Cataract Hotel in Aswan, there is a report of it being made in Aswan, 450 miles away from Cairo, who told me that upon asking the staff, it is made in ‘small factories in Aswan’. However, she was not able to be more specific than this. This report is certainly interesting. Upon reflection, it would appear that rather than ordering these directly from Cairo, with all the costs arising from such a far distance, one can readily imagine that these would be made local. Therefore, they could indeed be being made in a variety of locations, but if so, they have so far evaded detection! Does anyone know of these places?

A problem in determining where the pavings are (or were) being made is one of culture. Egyptians, upon being asked a question, do not like to disappoint the interviewer and tend to give what the person the answer they would like. For example (to give two actual instances) ‘do you know if the paving is made in Heliopolis?’, the answer is a yes, despite not necessarily knowing the answer! Of course, this may very well be the correct answer. A similar situation ensued as to an enquiry about the paving in Aswan; I was told it is made there, but no further details were forthcoming. So, what at first appear to be a definitive source may not necessarily be so.

Another anecdotal account of the manufacturing place is from Margaret Shabka, who told me it is made in the ‘Gulf States’. Unfortunately, I have not been able to elucidate any further detail. My collaborators John Lockerbie (who knows the Gulf States very well indeed, and was based there), and Gregg De Young considers this unlikely, on economic grounds. The expense, not to mention the time and trouble of transporting these to Cairo seems more trouble than it's worth. More or less ‘on site’ manufacturing remains more likely. However, if anyone has any comment to add to this I’d be delighted to here. Certainly, if they were made in the Gulf States, one would expect them to appear in those countries, but if so they have not been found on picture searches.

I also have an account that the pavings are manufactured at the El Obour tile factory, 35km northeast of Cairo, but details are sketchy.

However, despite the above tantalising reports, contact with the manufacturer/distributor (possibly more than one) has not been made. Would any reader of this page know of this? This aspect remains one of the ‘great unknowns’; if the manufacturer can be contacted, much new light could be shed on the inquiry, such as the all-important history, and the exact angles of the pentagon then determined (there are two possibilities), and so hence my desire to find this detail if at all possible.


1. Attraction

An open question is just what is so ‘special’ about the Cairo pentagon tessellation that it attracts so much attention, both mathematical and non-mathematical (the latter in the form of actual street pavings), in contrast to other tilings that do not get much attention. Therefore, I thus address this issue of its attractiveness and try and answer this query. Note that here I use the term ‘Cairo’generically, to refer to both types of pentagon, namely the ‘equilateral’ and ‘dual of 32. 4. 3. 4’, as generally distinctions are not made between the types. For the sake of brevity, these are referred to as ‘equilateral’ and ‘dual’ below. Contributions are sought from 'other people'.

Mathematicians' Interest

Mathematicians have long been interested in this as a tiling per se, even before the Cairo attribute, such as with P. A. MacMahon, in New Mathematical Pastimes, dating back to at least 1921 (the earliest discussion I can find), and H. M. Cundy and A. P. Rollett, Mathematical Models, 1951, to give but just two of the most prominent examples. Likely, the reason for this is that it has many interesting properties. Perhaps of most note is the interpretation, in that a secondary’ grid of par hexagons is formed, overlapping at right angles to each other, of which I detail below. Further interest is in the two types, in that one can transpose between them, as detailed in Macmillan. Additionally, this has been used decoratively, as a book cover (Coxeter’s Regular Complex Polytopes). M. C. Escher also used this in his work, with periodic drawings 131, 132, 133, 134. Non-mathematicians, such as paving companies have also used it, with examples as far apart as in England, Japan (see Hargittai, page 174), and Cairo (Dunn and Macmillan). Therefore, with such a diverse interest, it must consist of a little something ‘extra’, aside from an ordinary, ‘run-of-the-mill’ tiling. Indeed, it has been much commented upon, and described in flowery terms by mathematicians, with:

The tessellation…is one of the most remarkable (MacMahon, 1921)
… a very elegant arrangement of pentagons… (Wells, 1956)

This beautiful tessellation… (Gardner, 1975)

… special aesthetic appeal (Schattschneider, 1978)

… the tessellation is particularly pleasing to the eye (Macmillan, 1979)

The beautiful Cairo tessellation (Martin, 1982)
One particularly elegant tiling of the plane by pentagons (Singer, 1998)
... is an attractive and intriguing pattern of tiles (Mitchell, 2005)


Very few tilings are so described as above, and so just what is it about this tiling that attracts such descriptions? As such, I believe that there is no one single factor, but of combinations that conflate to give an especially aesthetic and interesting tiling:

  • Certainly, one curious aspect to this is that it has different interpretations of composition, aside from the basic pentagon tessellation. One possibility is that of outlining par hexagons, consisting of a block of four pentagons that tile at right angles to each other, giving the impression of overlapping, which is quite striking. That said, other tilings can indeed also possess this feature, and so it’s not that rare, or even unique, although it is certainly unusual.
  • Another attraction is in the composition, in that the 32. 4. 3. 4 type is derived from a basic set of tiles, namely the 11 semi-regular tilings. One can say that this as a tiling is thus of a ‘fundamental’, rather than of an arbitrary nature, and so appeal more. However, as this is but one of the 11 duals of the semi-regular tilings, this cannot be the complete reason for such interest.
  • Another attraction is the tile itself, based upon an equilateral pentagon, of which, being of a basic, fundamental nature is of interest, this being in contrast to an arbitrary pentagon tiling. However, as this is but one of five different equilateral pentagon tiling types, and so again this cannot be the complete reason for such interest. Furthermore, of the five equilateral pentagon types, this is arguably the closest in ‘proportion’ to a regular pentagon, and so it appeals.

Paving Interest

Also of note is that this has been used for actual pavings, of which presumably this must have caught the manufacturer’s eye, and so presumably thought the tiling was of more interest than others. But again, other tilings aside from the common to be found squares and rectangles have also been used, and so the Cairo tiling in not unique in this, although it is certainly unusual, in that relatively few tilings have been produced as actual street pavings.

Other People:

By Bruce Bilney:

The Cairo Pentagon Tiling is to my eye one of the most beautiful regular designs of all. Its proportions and layout are simple yet remarkable. Sets of four identical, attractively-proportioned, but slightly irregular pentagons form elongate hexagonal “lozenges”, which then tessellate in a robust arrangement, reminiscent of an art-deco design. The overall effect is nearly mesmerizing. It seems to me that it would make an ideal pattern for a faux-brick paving tiling.

Summary

From the above, the Cairo pentagon has no unique attributes. Therefore, I consider that it is more likely in that it appeals as it possesses all the features discussed above, something which other tilings lack. That said, I think the primary reason is the ‘overlapping’ of the par hexagons, which in combination consist of the four pentagons. Somehow, it seems an ‘unlikely’ situation, and so this catches the eye, with further aesthetic appeal provided by the apparent ‘regularity’ of the pentagons themselves. Furthermore, the pattern is visually ‘simple’, and so has immediate appeal, consisting of just one tile, with a ‘basic’ nature, either equilateral of dual, with one line of mirror symmetry. Contrast this simplicity with the typically more ‘involved’ Islamic patterns, where one could say that there is simply too much detail for the eye to take in, and so such tilings arguably lack elegance. Also, tilings based on pentagons per se somehow seem more ‘interesting’ than those of, say, triangles or quadrilaterals.

Created 7 December 2010. Added A. F. Wells reference, and rearranged the listing chronologically 9 February 2016.

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