What has the Penrose tiling to do with the icosahedral phases? Geometrical aspects of the icosahedral quasicrystal problem

Abstract

SUMMARY The Penrose tiling is indeed a geometrical model appropriate for the description of the phenomena of icosahedral quasicrystals, although it is probably a related example, rather than the central theory itself, but many aspects of the theory remain to be developed. The transition from X‐ray crystal structure analysis (XRCSA) to high resolution electron microscopy (HREM) as the paradigmatic method for the investigation of crystal structures has facilitated this development and promises other advances from classical crystallography.

The Equivalence Between Unit-Cell Twinning and Tiling in Icosahedral Quasicrystals

“…Intergrowth in QCs was suggested before already, e.g. by Mackay [32] and in one of the papers [33], where the QC structure was considered an interpenetrating incommensurately modulated structure, described as a three-dimensional section through a higher-dimensional space [33][34][35][36]. The related periodicities were described as structural modulations, induced by interacting intergrown subsystems [33].…”

A Game of Life on Penrose tilings

“…Quasiperiodic tilings may be thought of as "toy models" of quasicrystals -physical substances which, like quasiperiodic tilings, exhibit order but not periodicity. Penrose tilings, in particular, are a geometric model for icosahedral quasicrystals [9]. This investigation is motivated by the promise of new media for hosting computation -a topic which gives rise to the question of how to "compute" in non-periodically structured substrates.…”