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Miscellaneous

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:

1. Attraction
2. Bisymmetric Hendecahedron
3. Flooring
4. Furniture
5. Manufacturer
6. Quilting/Patchwork
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
12. Cushion
13. Picture links: Pan-American Games, 2015
14. 'Bretzel Love’ Café chair
15. Torus, by Kenneth Peel
16. Kimono by Moriguchi Kunihiko
17. Canopy, Maple Square, Toronto, Canada
18. Chair, by Monsieur Meuble, a French furniture company
19. Wine Racks, by Grassi Pietre, for Zýmē Winery, San Pietro, Italy


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.



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:

http://www.steelpillow.com/polyhedra/five_sf/five.htm

Although the Cairo tiling as an entity was not directly noticed in Inchbald’s paper, an architect, Jak Drinnan, happened to notice this, and being familiar with the Cairo tiling 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


© Greg Niemeyer

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



3b Flooring, UK, 1950s Style? Natural Floors

A most 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:

https://www.facebook.com/Natural.Floors?fref=ts
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 the best guess, from the age of the house and the material used, with printed tile effect linoleum. I quote excerpts from the company emails:
(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.
and
(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

An 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 north-west from the centre 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;
I wanted to do something mathematically interesting. I considered Penrose tilings, but in the small space of a bathroom it would be hard to appreciate that that pattern, though non-periodic, is not merely random. Plus I didn't like the look of that so much. Looking about tesselations, I came upon periodic arrangements of pentagons. I liked the interlacing of the two systems of hexagons. Building the floor with equilateral pentagons would allow an interesting geometric construction. 

© John Mansfield

4. Furniture

Some table designs as shown at design fairs, from Andreas Hopf, the designer.

http://www.hopfnordin.se


   

   
© Andreas Hopf



5. 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 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 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.


Reverse side of a square matrix paving

6. Quilting/Patchwork

Josée Carrier

http://www.thecharmingneedle.com/2014/06/cosmic-voyage.html

© 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 excellent Escher-like instances by the French artist, Alain Nicolas, based on the Cairo tiling. Other contributions are sought.

   

© Alain Nicolas


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

As drawn to my attention by Josée Carrier, and independently Robert Ferréol


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 a limited edition of twenty, with a price of $165.

   
© Le Noeud Papillon Company


12. Cushion
A cushion, available from Macy's, of New York, US, and other stores



13. Picture Links


14. 'Bretzel Love’ Café Chair

A ‘Cairo Café Chair’, as seen in Paris, France, from the ‘Bretzel Love’ Café chain. 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.
    With my thanks to Robert Ferréol, of France, for drawing this sighting to my attention.


© Robert Ferréol


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.
The file for making this is at:
http://www.thingiverse.com/thing:2001922/#files
And other instances, of further tori and polyhedra:
http://www.thingiverse.com/kennyp1369/designs

© Kenneth Peel


16. Kimono, by Moriguchi Kunihiko 

An interesting sighting, the first of its type, comes from the world of fabric/clothing, with a Kimono by Moriguchi Kunihiko (1941-), of Japan. This was a part of an exhibition on Kimonos at Paris in 2016 from November 16 to December 17, 2016, at the House of Culture of Japan to Paris. Kunihiko is a household name in Japan and has acquired a degree of fame, described as a ‘living treasure’ of Japan.
    With my thanks to Robert Ferréol, of France, for drawing this sighting to my attention.

© Jin Angdoo


17. Canopy, Maple Leaf Square, Toronto, Canada

A pentagonal tiled canopy (inspired by the experience of walking through a forest’s dappled light) by United Visual Artists (UVA) a London-based art practice, of a 90-metre long light sculpture spanning the front facade of the Maple Leaf Square building in Toronto, Canada, of 2010. Their form, abstracted from the geometry of leaves, reflect nature. Whether the designer was familiar with the Cairo tiling is unclear; it is not mentioned on their web page, and a mail asking for further details to UVA went unanswered. Of note in the panoramic view is the famous landmark the CN Tower in the background.

   
© James Medcraft


18. Chair, at Monsieur Meuble, a French furniture chain store

A unique sighting is of a chair, from ‘Monsieur Meuble’ a well-known French furniture store and chain of over 40 years old, with branches all over France. The chair appears to have been part of a liquidation sale. Does anyone know more on this? I have no other details.

© Vincent Pantaloni


19. Wine racks, by Grassi Pietre, for Zýmē Winery, San PietroItaly

An unusual, and indeed unique instance, is by Grassi Pietre architects, for Zýmē Winery, of San Pietro, Italy, with the pentagons used as series of wine racks, in stone, with the wine bottles stored in a vault in a cave, as temperatures must remain constant. The use of pentagons is predicated on the logo of the company, of a wine leaf in the broad shape of a pentagon. Each pentagon measures 78 x 56 x 60 cm and weighs 85 kg.

   
© Daniele Domenicali

Page Created: 18 June 2012. Revised and enlarged subsequently
Overall page created 7 August 2014, for a more streamlined section, assembling previous single pages as a 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
IKEA 

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 department store added 28 January 2016
'Bretzel Love’ Café chair added 8 September 2016
Torus by Kenneth Peel added 23 February 2017
Kimono by Moriguchi Kunihiko added 9 June 2017
Pentagonal tiled canopy, Toronto, Canada text and pictures added 14 July 2017
Monsieur Meuble chair text and picture added 27 July 2017
Grassi Pietre for Zýmē Winery, San PietroItalytext and picture added 28 July 2017
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