On the fractal nature of Penrose tiling

An earliest preoccupation of man has been to find ways of partitioning infi nite space into regions having a finite number of distinct shapes and yield ing beautiful patterns called tiling. Archaeological edifices, everyday obj ects of use like baskets, carpets, textiles, etc. and many biological syste ms 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 mathematici ans. The generation of aperiodic tiling with five-fold rotational symmetry by Penrose in 1974 and the more recent production of decorated pentagonal t iles by Rosemary Grazebrook have heightened the interest in the subject amo ng artists, engineers, biologists, crystallographers and mathematicians(1-5 ). In spite of its long history, the subject of tiling is still evolving. I n this communication, we propose a novel algorithm for the growth of a Penr ose tiling and relate it to the equally fascinating: subject of fractal geo metry pioneered by Mandelbrot(6). The algorithm resembles those for generat ion of fractal objects such as Koch's recursion curve, Peano curve, etc. an d enables consideration of the tiling as cluster growth as well. Thus it cl early demonstrates the dual nature of a Penrose tiling as a natural and a n onrandom fractal.