A miscellaneous collection of various Cairo aspects, generally individually of a less substantial nature that with my other researches, and so for the sake of a more succinct listing, of which the Cairo section is by far the most numerous, I thus in effect amalgamate disparate elements:
2. Bisymmetric Hendecahedron
7. Penta-Graphene Discovery
8. Escher-like Cairo tilings by Alain Nicolas
9. Tamara Restaurant Panels
10. IKEA Jall tabletop ironing board
11. Le Noeud Papillon, Sydney, Australia, bow ties
13. Picture links: Pan-American Games, 2015
14. 'Bretzel Love’ Café chair
15. Torus, by Kenneth Peel
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 such 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 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 in 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)
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,
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 a specially aesthetic and interesting tiling:
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 commonly 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.
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.
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 in 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.
2. Bisymmetric Hendecahedron
Another curiosity of the Cairo tiling is that it can be seen in an instance of polyhedra, specifically of the space-filling Bisymmetric Hendecahedron. Such a polyhedron is a relatively recent discovery, as reported in a 1996 paper by Guy Inchbald. For further details see:
Although the Cairo tiling as an entity was not directly noticed in Inchbald’s paper, an architect, Jak Drinnan, happened to noticed this, and being familiar with the Cairotiling made the association.
Although I am interested in polyhedra, this is very much secondary to my interest in tessellation. Indeed, I lack the mathematics here for a true mathematical treatment. Would any reader more familiar with this have anything to say?
For more on Jak Drinnan and the bisymmetric hendecahedron per se, see:http://www.jd-d.co.uk/2012/07/material-designing-complexity.html
© Jac Drinan
3a. Flooring, Cairo Apartment
3b Flooring, UK, 1950s Style? Natural Floors
interesting historical sighting, and in a ‘foreign’ context, is a sighting as a
flooring, in the UK, of c. 1950s. However, details here
are a little sketchy. A picture was posted on Natural Floors NW (North West) Limited website (and Facebook
page) with an intriguing reference to the Cairo tiling in a bathroom of the 1950s:
However, upon correspondence with the company, the picture as shown is not the original flooring, but is rather a modern day recreation as best as is possible. Furthermore, the attributed date (1950s), is a best guess, from the age of the house and the material used, with printed tile effect linoleum. I quote excerpts from the company mails:
(Mail 1, 19 October 2015)
We had a customer buy a house built in the 1950's & in the bathroom was a printed tile effect linoleum, printed lino was a very popular floor covering in the UK from as early as the 1930's but the early ones were quite often floral patterns. As this house was from the 50's we assume the lino was of the same period. No one makes a printed lino anymore, (Vinyl is not lino!) so we made the [modern day] pattern by hand cutting sheet linoleum in 4 colours using Forbo Walton Linoleum as this was the closest we could get to the original colours.
(Mail 2, 20 October 2015)
We assume it is a 1950's pattern as this type of patterned linoleum was not made much beyond the 50's. Once vinyl became popular Marmoleum was used mainly in commercial locations. It is only now becoming popular again as a domestic flooring….
However, much still remains unknown here. Despite extensive web searching, who manufactured the vinyl, and what date these were installed is not known for certain, although 1950s does indeed seem a distinct possibility. Can anyone shed any more light on these background matters? Or indeed, have anything else to say on this?
© Natural Floors NW
3c Bathroom Flooring
instance of a modern-day self design is of bathroom floor at the home of John
Mansfield, in North Potomac, Maryland, USA (about 25 miles north by northwest
from the center of Washington, D.C). This was built
during the summer of 2014. The material is porcelain. The Cairo tiling was specially
chosen for the project. John told me;
Some table designs as shown at design fairs, from Andreas Hopf, the designer.
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 Giza. However, 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 its 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 north east 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.
© Josée Carrier
7. Penta-graphene Discovery
A recent development (February 2015) concerning the Cairo tiling is in the form of a scientific discovery, of ‘Penta-graphene’. Researchers at Virginia Commonwealth University (VCU) and universities in China and Japan have discovered a new structural variant of carbon called ‘Penta-graphene’ – a very thin sheet of pure carbon that has a unique structure inspired by the Cairo tiling. The newly discovered material is a single layer of carbon pentagons that resembles the Cairo tiling, and that appears to be dynamically, thermally and mechanically stable and can withstand temperatures up to 1,000 K (730 °C; 1,340 °F).
The background to discovery is relayed by one of the researchers, Qian Wang, Ph.D., a professor at Peking University and an adjunct professor at VCU, who was dining in a restaurant in Beijing with her husband when she noticed artwork on the wall depicting pentagon tiles from the streets of Cairo. 'I told my husband, "Come, see! This is a pattern composed only of pentagons,'" she said. "I took a picture and sent it to one of my students, and said, 'I think we can make this. It might be stable. But you must check it carefully.' He did, and it turned out that this structure is so beautiful yet also very simple."
The researchers' paper, ‘Penta-Graphene: A New Carbon Allotrope’, will appear in the journal Proceedings of the National Academy of Sciences, and is based on research that was launched at Peking University and VCU.
Restaurant image © Qian Wang
8. Escher-like Cairo Tilings
Some Escher-like instances, by Alain Nicolas, based on the Cairo tiling. Other contributions are sought.
9. Tamara Lebanese Bistro Restaurant, Cairo
Some interesting, and indeed novel use is to be seen at the Tamara Lebanese bistro, in Cairo, with numerous geometric patterned panels suspended from the ceiling, one of which includes the Cairo tiling. Such an instance gives credence as to its being a traditional Islamic tiling, although there is scant evidence of this. Upon enquiring with the company, of a Lebanese premise, on the possibility of these being from the Lebanon, they told me that these were simply generic Islamic designs.
© Tamara Lebanese Bistro
10. IKEA Jäll Tabletop Ironing Board
11. Le Noeud Papillon, Sydney, Australia
Le Noeud Papillon, a bow tie company, of Sydney, Australia, has among their range two silk bow ties, of a pentagon theme, one of the recent (2015) Type 15 pentagon discovery, and another of a Cairo tiling. This is of a limited edition of twenty, with a price of $165.
A cushion, available from Macy's, of New York, US, and other stores
13. Picture Links
Pan-American games, Toronto, July 2015: http://www.gettyimages.co.uk/detail/news-photo/woman-walking-on-the-colorful-pentagonal-pattern-marked-on-news-photo/491613412
14. 'Bretzel Love’ Café Chair
A ‘Cairo Café Chair’, as seen in Paris, France, from the ‘Bretzel Love’ Café chain, drewn to my attention by Robert Ferréol. The background of this as to the designer and manufacturer are unknown. However, the chronology is roughly known; it appears to be ‘fairly recent’, the chain having opened in 2009. Of note is the interlinking structure, reminiscent of Rinus Roelofs’ work.
15. Torus, by Kenneth Peel
The Cairo tiling realised as a 3D model onto
the surface of a torus by Kenneth Peel, a student of mathematics at Bellingham, Washington state, US.
© Kenneth Peel
Created: 18 June 2012. Revised and enlarged subsequently
Overall page created 7 August 2014, for a more streamlined section, assembling previous single pages as an miscellaneous entity
Quilting/Patchwork section added 2 October 2014
Penta-graphene discovery added 17 June 2015
Natural Floors NW added 20 October 2015
Escher-like Cairo tilings by Alain Nicolas added 9 November 2015
Tamara Bistro Restaurant added 11 December 2015
Jäll tabletop ironing board added 5 January 2016
Picture Links: Pan-American games added 6 January 2016
Le Noeud Papillon, Sydney, Australia added 25 January 2016
Cushion from 'Macy's' added, 28 January 2016
'Bretzel Love’ Café chair by Robert Ferréol added 8 September 2016.
Torus by Kenneth Peel added 23 February 2017
Cairo Tiling >