Symmetries of Icosahedral Quasicrystals

Abstract

The icosahedral group A(5) as the point symmetry group of quasicrystals is by now well established for a variety of materials. Quasicrystals have been grown up to macroscopic scale and display polyhedral shapes with this symmetry, compare the review by Guyot, Kramer and de Boissieu 1990 [1]. The non-crystallographic icosahedral point group, considered in crystallography for local sites only, is defined more precisely through its three-dimensional (3D) faithful representation, the symmetry group of the regular icosahedron and dodecahedron. This representation requires, for the embedding into a periodic lattice, at least the dimension 6D. The embedding into a lattice and the study of 3D sections through the embedding periodic structure is one of the main theoretical tools for the analysis of ideal icosahedral quasicrystals. In this survey we shall develop some aspects of this embedding for centered hypercubic 6D lattices. The primitive hypercubic lattice has been studied before, but will be considered for comparison. A detailed analysis is given for the root lattice D 6. We hope to show that the 3D sections of this lattice display a rich geometric structure which we expect to encounter in the geometry and physics of the corresponding quasicrystals.