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' 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 Mahematical Pastimes, dating back to at least 1921 (the earliest
discussion I can find), and H. M Cundy and R. A. 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: § This beautiful tessellation… (Gardner) § … special aesthetic appeal (Schattschneider) § … the tessellation is particularly pleasing to the eye (Macmillan) § The beautiful Cairo tessellation (Martin) § One particularly elegant tiling of the plane by pentagons (Singer) § The tessellation…is one of the most remarkable (MacMahon) 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:
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 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.
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 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 |