QUASI-CRYSTALS AND ICOSIANS

A family of quasicrystals of dimensions 1, 2, 3, 4 governed by the root lattice E8 is constructed. The use of the icosian ring, found in the quaternions with coefficients in Q(square-root 5), allows simultaneous interpretation of the construction both in physical space and as a result of the standard 'cut-and-projection' method in double dimension. Icosians are seen to provide a natural co-ordination scheme for these quasicrystals. Nested sequences of quasicrystals form systems whose symmetries are all derivable from inflational and reflective symmetries directly related to the arithmetic of the icosians. The use of Coxeter diagrams clarifies the amazing relationship of E8 and quasicrystal symmetries and leads to the fundamental chain E8 superset-of D6 superset-of A4 superset-of A1 x A1 that underlies five-fold symmetry in quasicrystals. Decomposition of quasicrystals into concentric shells and a counting formula for the cardinalities of these shells is discussed.