On the fractal nature of Penrose tiling

Abstract

An earliest preoccupation of man has been to find ways of partitioning infinite space into regions having a finite number of distinct shapes and yielding beautiful patterns called tiling. Archaeological edifices, everyday objects of use like baskets, carpets, textiles, etc. and many biological systems such as beehives, onion peels and spider webs also exhibit a variety of tiling. Escher’s classical paintings have not only given a new dimension to the artistic value of tiling but also aroused the curiosity of mathematicians. The generation of aperiodic tiling with five-fold rotational symmetry by Penrose in 1974 and the more recent production of decorated pentagonal tiles by Rosemary Grazebrook have heightened the interest in the subject among artists, engineers, biologists, crystall ographers and mathematicians1–5. In spite of its long history, the subject of tiling is still evolving. In this communication, we propose a novel algorithm for the growth of a Penrose tiling and relate it to the equally fascinating subject of fractal geometry pioneered by Mandelbrot6. The algorithm resembles those for generation of fractal objects such as Koch’s recursion curve, Peano curve, etc. and enables consideration of the tiling as cluster growth as well. Thus it clearly demonstrates the dual nature of a Penrose tiling as a natural and a nonrandom fractal.