Grammars for icosahedral Danzer tilings

Since the definition of quasiperiodicity is intimately connected to the indexing of a Fourier transform, for the case of an icosahedral solid, the step necessary to prove, using diffraction, that an object is quasiperiodic, is described. Various coordinate systems are discussed and reasons are given for choosing one aligned with a set of three orthogonal two-fold axes. Based on this coordinate system, the main crystallographic projections are presented and several analyzed single-crystal electron diffraction patterns are demonstrated. The extinction rules for three of the five icosahedral Bravais quasilattices are compared, and some simple relationships with the six-dimensional cut and projection crystallography are derived. This analysis leads to a simple application for indexing powder diffraction patterns.