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Martin Gardner Mystery Resolved

Yet another mystery concerning the Cairo tiling arises from a quote by Martin Gardner in his Scientific American July 1975 article, 'On tessellating the plane with convex polygon tiles', pp. 112-118, where he states that the Cairo tiling is seen '…and occasionally in the mosaics of Moorish buildings', but he is not specific as to the type of buildings or exactly where this is. The term ‘Moorish’ covers a wide expanse, from Spain to Egypt! As such, there is no documentary evidence for this ‘sighting’ whatsoever in the form of a picture, or indeed any other indirect references. This quote has long puzzled me. In the course of my investigations I have also consulted various Islamic authorities on tiling, such as Brian Wichmann and David Wade, both of whom are unaware of it in this form. My chief investigator on the Cairo tiling, Gregg De Young, in Cairo since 1990, is also unfamiliar with it, and he also asked another authority, Bernard O’Kane, also in Cairo, and he too has not seen this. Gardner’s quote, in the plural, would, to me, imply that it is seen more than once, but all still to no avail. And so the matter remained unresolved, until recent times. However, subsequently to these personal inquiries, further details to the background of this account have been unearthed which solve the mystery! Upon contacting Tim Noakes, the curator of the Stanford special collection archive at Stanford University (where Gardner’s files that he composed during the preparations of his columns in ‘Scientific American’ are stored), much light has been shed on this thanks to Noakes and a visiting researcher of the archive, Bjarne Toft. Upon my request for further details, Noakes was unable to immediately find anything in the archive. However, he kept me in mind, and later mentioned my request to Toft, who was researching as regards his own interest in Gardner (of board games and Piet Hein), and subsequently sent me what can only be described as a treasure trove of papers pertaining to the matter.  Pleasingly, the full story can now be told. In short, beyond all reasonable doubt, Gardner was amazingly referring to the Taj Mahal as the ‘Moorish building’. Although quite why he should describe the Taj Mahal as ‘Moorish’ is yet another minor mystery. The background to this assertion is that Gardner made a series of notes on his source material for his columns, taken from various mathematics journals, of which Toft kindly forwarded.  Numerous indirect references to the Taj Mahal by Gardner can be seen:

  • In a paper by L. Fejes Tóth [9], Richard K. Guy adds a note as regards a pentagon tiling that Tóth has mentioned ‘I believe I recall seeing (3, 3, 4, 3, 4) among the many unusual tilings at the Taj Mahal’. Specific reference is made to the Taj Mahal by means of underlining.
  • A letter [7] from H. S. M Coxeter to Gardner mentions this Taj Mahal reference (Coxeter used the tiling on the front cover of his book, [1] of 1963 (it does not appear in subsequent editions).
  • A letter [8] to R. B. Kershner is also informative, in that Gardner quotes Guy’s Taj Mahal reference, in which Gardner asks Kershner (among other pentagonal matters) if he has seen a picture of it anywhere. 

Further to this account, upon having been previously been aware of Guy’s added note, I asked him myself in an email [3] if he could shed any further light. However, he could not add anything more; time having dimmed his memory. However, in the light of more detail from the archive, I got back to him again, to see if he could add anything more. Upon being aware that the Fatehpur Sikir has a tiling that at first glance could be mistaken for a Cairo tiling (although it is not), I mentioned this to him, and he told me [4] that he had indeed visited the Fatepur Sikir, but could not say definitely that this is where he saw it, with no less that 50 years since his visit to India! However, this thus remains a possibility for Guy’s account, but nothing more. The matter looks like it is going to remain unresolved. Despite much searching for a picture, this has been to no avail. Given that the Taj Mahal is photographically well documented, it now seems almost inconceivable that it could have been seen there. I also asked Ivars Peterson, who visited, and posted numerous pictures on his blog about this, but he too could not find a picture. Therefore, Guy’s so-called sighting at the Taj Mahal should be treated with due circumspection. The only conclusion one can draw is that it does not exist in the Taj Mahal. Likely, if Guy did indeed see this, it was at some other Indian landmark, or possibly he misinterpreted the tilling at the Fatehpur Sikri. Therefore, Gardner’s account of the Cairo tiling being seen as a ‘Moorish building’ is almost certainly without any foundation whatsoever! Likely, due to the uncertainty of Guy’s account, he was purposefully a little vague, which thus explains the omission of specifics.

 

Aside from the ‘Moorish building’ mystery, many other Cairo tiling aspects of various degrees of uncertainty in the article can also be nicely cleared up with the archive papers. It can be seen that Gardner based his article (as I long suspected) on J. A. Dunn’s 1971 paper [2], as he singles out the word ‘Cairo’, and adding to this with a question mark. Implied here is the uncertainty of its existence, of not having a picture of this to hand.  Also, he notes the equilateral aspect of Dunn’s account [2], of which Gardner repeats in his article [5], albeit this equilateral assertion is not correct. Dunn was mistaken. He also underlines Dunn’s quote of ‘favourite street-tiling in Cairo’, which becomes Gardner’s ‘frequently seen as a street tiling’, whilst the ‘…in the mosaics Moorish buildings’ is almost certainly the Taj Mahal. Gardner also used Kershner’s hexagon and pentagon diagrams [6] in his paper, expanding slightly on the number of tiles used.

 

References

[1] Coxeter, H. S. M. Regular Polytopes. Dover Publications Inc., New York Third edition 1973. The first edition is 1947, the second edition is 1963. An earlier edition, of 1963? has the Cairo tiling featured on the front cover (but is not referenced as such; the term ‘Cairo’ was not appended to the tiling until 1971)

[2] Dunn, J. A. ‘Tessellations with Pentagons’. The Mathematical Gazette, Vol. 55, No. 394 (Dec. 1971) pp. 366-369

[3] Email from Guy to myself 12 March 2013

[4] Email from Guy to myself 13 December 2013

[5] Gardner, M. ‘On tessellating the plane with convex polygon tiles’. Scientific American June 1975 112-117. Note that this is repeated and updated in Gardner’s Time Travel and Other Mathematical Bewilderments. W. H. Freeman and Co pp. 174-175.

Contains the second recorded reference to the Cairo pentagon.

[6] Kershner, R. B. ‘On Paving the Plane’. The American Mathematical Monthly. Vol. 75, No. 8 (October 1968) 839-844.

[7] Letter by H. S. M. Coxeter to Martin Gardner, not dated

[8] Letter by Martin Gardner to R. B. Kershner, 13 April 1975

[9] Tóth, Fejes, L. ‘Tessellation of the Plane with Convex Polygons Having a Constant Number of Neighbours’. American Mathematical Monthly, 82, 1975, 273-276

Has a Cairo tiling diagram on p. 274, with a possible later reference by Richard K. Guy to a sighting at the Taj Mahal: ‘I believe I recall seeing 3, 3, 4, 3, 4 among the many unusual tessellations at the Taj Mahal’

 

 

Of note is that subsequent to Gardner’s account, various other authors have also reported this ‘Moorish building’ sighting in one form or the other, likely taking this line from Gardner, or copying after one another, and occasionally embellishing it (with vague descriptions ‘Islamic decorations’ or such like, and seen in any mosque or like centre), sometimes outrageously so (with descriptions of ‘centuries old’, which is a complete nonsense; the Cairo tiling is at the earliest of the late 1950s, and more likely the early 1960s), with:

Books

 

1984 Blackwell, William, Geometry in Architecture, Wiley 1984, p. 54
… This unusual pattern, which is seen in street tiling in Cairo and occasionally in the mosaic of Moorish buildings.
The latter part of the quote is taken from
Gardner, word for word.

 

1991 Wells, David. The Penguin Dictionary of Curious and Interesting Geometry. Penguin Books, p. 23
So called because it often appears in the streets of
Cairo, and in Islamic decoration.

 

1991 Fetter, Ann E. et al. The Platonic Solids Activity Book. Key Curriculum Press/Visual Geometry Project. Backline Masters p. 21.
This pattern is seen in street tiling in
Cairo and in the mosaics of Moorish buildings.
Almost certainly this quote above is taken from
Gardner, word for word.

 

2003 Weisstein, Eric W. CRC concise encyclopedia of mathematics, p. 313
A tessellation appearing in the streets of
Cairo and in many Islamic decorations.

 

2005 Mitchell, David. Sticky note origami: 25 designs to make at your desk, Sterling Publication Company
…of its frequent occurrence on the streets of
Cairo and in other Islamic centers and sites.

 

2005 Johnston-Wilder, Sue and ‎John Mason. Developing Thinking in Geometry, p. 182

…often referred to as the Cairo tessellation because it appears in a mosque there.

 

Web

 

Wolfram MathWorld
A tessellation appearing in the streets of
Cairo and in many Islamic decorations.

 

John Szinger

http://www.zingman.com/origami/subj_tessn.php

….The Cairo tessellation is an ancient pattern used in Moorish and Mideastern art and architecture for centuries.

 

Paul Scott
http://paulscottinfo.ipage.com/polyhedra/tessellations/other_tessellations.html

This is a tessellation which appears in the streets of Cairo, and in Islamic art.

 

Jennifer Green

http://mathematics.gulfcoast.edu/mathprojects/Tessellations.htm

One example of an ancient tessellation is the Cairo tessellation named so because the design can be found as a tiling pattern in the streets of Cairo.

 

Pearson Publishing

This tessellation is often found in the streets of Cairo and in Islamic decorations.

 

Although without going to the trouble of questioning each author, these accounts are without a shadow of doubt all copying from Gardner’s quote in one form or the other. No references are given for these assertions. Such instances can be summarily dismissed.
 
Acknowledgements
Tim Noakes, for taking the trouble to look in the archive, and keeping my request in mind, along with answering numerous questions arising from this, all above and beyond the call of duty
Bjarne Toft, for noticing the vital details in the archive
Richard Guy, for twice clarifying his account in Tóth's paper

 

Created 27 December 2013

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