Cyclotomic integers, nondiscrete tessellations, and quasicrystals

Abstract

Physical quasicrystals have been observed with decagonal, octagonal, dodecagonal or icosahedral symmetry. The first three cases are represented by plannar patterns of points, arising by orthogonal projection from the lattice of centres of close-packed balls in 4 dimensions, with each ball kissing 20 or 24 others. Such a pattern of points in the plane may be regarded as the set of vertices of a nondiscrete regular tessellation of overlapping pentagons, octagons or dodecagons. When interpreted in the Argand plane as a set of complex numbers, it is the ring of algebraic integers in the field generated by the n th roots of unity, where n = 5, 8 or 12.