Covering Presentation and Coloring of Dual Canonical Tilings

Abstract

In the crystallography of quasicrystals, the space for the structure description is taken as a subspace E‖, embedded irrationally in an N-dimensional space equipped with a lattice Λ. Quasiperiodic dual canonical tilings (\(\mathcal{T}\), Λ), (\(\mathcal{T}\)*, Λ) arise from the geometry and projection of Voronoi and Delone polytopes, respectively, associated with Λ. In a new approach to quasicrystals, tiles are replaced by a few covering clusters. As a typical example, the tiling (\(\mathcal{T}\)*, A4) of fivefold symmetry is covered by two Delone clusters. This construction is called a covering presentation compatible with the tiling. We show that any given tile in the tiling can be uniquely associated with a specific overlap of Delone clusters. Each specific overlap is called a color for this tile. The range of colors and their relative frequencies are explicitly determined.