Quasicrystals and icosians

A family of quasicrystals of dimensions 1, 2, 3, 4 governed by the root lattice E 8 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 E 8 and quasicrystal symmetries and leads to the fundamental chain E 8 contains/implies D 6 contains/implies A 4 contains/implies A 1 *A 1 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.