A simpler approach to Penrose tiling with implications for quasicrystal formation

Abstract

QUASICRYSTALS¹have a quasiperiodic atomic structure with symmetries (such as fivefold) that are forbidden to ordinary crystals^2,3. Why do atoms form this complex pattern rather than a regularly repeating crystal? An influential model of quasicrystal structure has been the Penrose tiling⁴, in which two types of tile are laid down according to 'matching rules' that force a fivefold-symmetric quasiperiodic pattern. In physical terms, it has been suggested¹ that atoms form two or more clusters analogous to the tiles, with interactions that mimic the matching rules. Here we show that this complex picture can be simplified. We present proof of the claim⁵ that a quasiperiodic tiling can be forced using only a single type of tile, and furthermore we show that matching rules can be discarded. Instead, maximizing the density of a chosen cluster of tiles suffices to produce a quasiperiodic tiling. If one imagines the tile cluster to represent some energetically preferred atomic cluster, then minimizing the free energy would naturally maximize the cluster density⁶. This provides a simple, physically motivated explanation of why quasicrystals form.