Constructing Penrose-like tilings from a single prototile and the implications for quasicrystals

Abstract

We present two sets of rules for constructing quasiperiodic tilings that suggest a simpler structural model of quasicrystals and a more plausible explanation of why quasicrystals form. First, we show that quasiperiodic tilings can be constructed from a single prototile with matching rules which constrain the way that neighbors can overlap. Second, we show that maximizing the density of a certain cluster of fat and thin tiles can force a Penrose tiling without imposing the usual Penrose matching rules.