A trick of the tiles

摘要

Geometry in Western art predominantly involves space and proportion. But in other cultures, most notably Islamic, Chinese and Japanese, artistic geometry flowered most conspicuously in flat patterns, above all in the invention of striking tessellations in tiling, mosaics and textile designs. The trick was to invent a repeated geometric pattern of considerable complexity without generating 'gaps' that had to be arbitrarily filled. Working with straight-edge and compass, over the centuries geometer-artisans accumulated an astonishing array of periodic patterns, including many of the types later defined in mathematical theory. The basic principles involved in such tessellations — that they are possible only with one-, two-, three-, four- or six-fold symmetries — held good until 1974, when the mathematician Roger Penrose from the University of Oxford unveiled a 'gap-free' tiling based on five-fold symmetry.