Local growth of planar rhombus tilings

Abstract

This paper is motivated by the issue of quasicrystal growth. It describes a simple local algorithm which appears to grow an infinite family of aperiodic tilings. Starting from a seed, tiles are added one by one at randomly chosen sites. A tile is added only if there is only one way to do this so that no forbidden local configuration is created. This algorithm rapidly grows large round-shaped patterns, up to a proportion of missing tiles which can be made arbitrarily small by taking a large enough seed.