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Six Fallacies

The Cairo tiling seems dogged, from the very beginning of its first report (in 1971), by a series of misattributions and downright errors in matters of fact, not to mention some authors simply making up ‘facts’ as they go along! To clarify, I here list and correct six common fallacies I have found. Note that some of these aspects cross reference some other essays, where for their intrinsic worth I have examined these in more depth, in conjunction with other matters.

The format of the discussion is of a generally generic description of an author’s quote, accompanied by an actual example of the first such mistaken occurrence, with the author named, and followed by me correcting matters.


1. ‘The tile consists of an equilateral pentagon’

Finally, if the sides are all equal and x = x’ = 90°, the tessellation in Figure 5 becomes Figure 6 which is shown in Cundy and Rollett and is a favourite street-tiling in Cairo. J. A. Dunn [3] and others

No. From the in situ pictures, the tile is clearly not of an equilateral pentagon. In the light of the contradictory in situ pictures which do not corroborate this statement, I then made efforts to find Dunn, who first gave this description (and reported the Cairo connection), and ask him for more details as to his relatively lightweight report, to see if he could substantiate an equilateral pentagon determination. Upon a phone call and correspondence (2010) asking for specific details as to how he determined that the pentagons were equilateral, i.e. did he actually measure them, he told me that he did not, but had assumed the pentagons to be equilateral, this being influenced by a diagram (of equilateral pentagons in a Cairo-like diagram) that he had seen in Mathematical Models by Cundy and Rollett (page 63). Thinking little more of it at the time (in retrospect, fairly justifiable, given the circumstances of a chance sighting), he then wrote up the finding as equilateral in an article on pentagon tessellations per se at a later date, and thus inadvertently started the myth that the tiles are equilateral.


2. ‘The tile consists of an the dual of the 3, 3, 4, 3, 4 tiling’

Tiling (3) [3, 3. 4. 3, 4] of FIGURE 1 has special aesthetic appeal. It is said to appear as a street paving in Cairo. Doris Schattschneider [8] and others

No. From the in situ pictures, the tile is clearly not of a 3, 3, 4, 3, 4 tiling. Schattschneider seems to be simply mixing up the specific type of pentagon, basing her account on Dunn [3] and Gardner’s [4] papers; both are quoted.


3. ‘It appears as a mosaic in Moorish buildings’

‘…and occasionally in the mosaics of Moorish buildings'. Martin Gardner [4] and others

No. Upon recent investigations (December 2013) of which I detail more fully elsewhere (Martin Gardner Mystery Resolved), Gardner is beyond all reasonable doubt referring to a report (by Richard Guy) of the tiling possibly appearing at the Taj Mahal (and of which the possibility in itself now seems most remote). Although quite why he should describe the Taj Mahal as ‘Moorish’ is yet another minor mystery, but that’s by the bye. As such, there is no documentary evidence for this ‘sighting’ whatsoever in the form of a picture, or indeed any other indirect references to ‘Moorish buildings’ (as with mosques, which come to mind).

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, 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. The background to the Taj Mahal assertion above is that Gardner made a series of notes on his source material for his columns, taken from various mathematics journals. 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 by Gardner to the Taj Mahal by means of underlining.
  • A letter [2] 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 [5] 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. 

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 specific detail.


4. ‘Traditional pattern of Islamic art’

… a traditional Islamic tessellation of pentagonal tiles. R. H. Macmillan [7]

No. There is no evidence whatsoever of the in situ tiling, or come to that, its commonly quoted variations of equilateral or dual of 3, 3, 4, 3, 4 being of a ‘traditional’ Islamic design, of either the plane or as in a three-dimensional sense, as with mosques. That said, there is indeed one instance of a minaret design, but this is of a more modern day occurrence, in the UK, and cannot possibly be what Macmillan was referring to.


5. ‘An old/ancient tessellation’

The Cairo tessellation is an ancient pattern used in Moorish and Mideastern art and architecture for centuries. John.Szinger [9]

No (ancient), No (Moorish and Mideastern art and architecture) and No (centuries)! Three fallacies in one sentence! The tiling is not ancient/centuries old. Descriptions such as ‘ancient’ and of ‘centuries old’, is a complete nonsense; the Cairo tiling is relatively modern, within living memory, of which although a little uncertain, is at the earliest of the late 1950s, and more likely the early 1960s.


6.  ‘Seen in a mosque’

…it appears in a mosque there. Sue Johnston-Wilder and ‎John Mason [6]

No. There is no evidence whatsoever of the tiling appearing in a mosque. Likely this false account is derived from Gardner’s ‘Moorish buildings’ quote.

As ever, I stand to be corrected. Although I find it highly unlikely, if anyone has seen the tiling in any of the above contexts I would indeed be indebted to any reader who could draw my attention to this.


[1] Coxeter, H. S. M. Regular Polytopes. Dover Publications Inc., New York 1963. The first edition is 1947, the second edition is 1963. Only the 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] ————. Letter by H. S. M. Coxeter to Martin Gardner, not dated, c. 1975

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

[4] Gardner, M. ‘On tessellating the plane with convex polygon tiles’. Scientific American June 1975 pp. 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.

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

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

[7] Macmillan, R. H. ‘Pyramids and Pavements: some thoughts from Cairo’. Mathematical Gazette. 63 pp. 251-255, 1979

[8] Schattschneider, Doris. 'Tiling the Plane with Congruent Pentagons', Mathematics Magazine January 1978, 29-44, and specifically pp. 30-31:

[9] Szinger, John. Web page:

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