Symmetry and Topology of Icosahedral Crystals

Abstract

The symmetry of icosahedral crystals, discovered by Shechtman et al., will be described in terms of six-dimensional space groups. The positions of atoms correspond to arrangements of hypersurfaces in six-dimensional lattices. There are six continuous symmetries: Three displacements and three phase-shifts. The icosahedral symmetry imposes topological restrictions on the arrangements of atoms.