THE ROOT LATTICE D4 AND PLANAR QUASILATTICES WITH OCTAGONAL AND DODECAGONAL SYMMETRY

Abstract

Quasiperiodic patterns with eight- and twelvefold symmetry are presented which share the root lattice D4, i.e., the 4-D face-centered hypercubic lattice, for their minimal embedding in four-space. We derive the patterns by means of the dualization method and investigate key properties like vertex configurations, local deflation/inflation symmetries and kinematic diffraction. The generalized point symmetries (and thus the Laue group) of these patterns are the dihedral groups d8 and d12, respectively, which share a common subgroup, d4. We introduce a contiunous one-parameter rotation between the two phases which leaves this subgroup invariant. This should prove useful for modelling alloys like V 15 Ni 10 Si where quasicrystalline phases with eight- and twelvefold symmetry coexist.