Quasicrystals: Atomic coverings and windows are dual projects

TL;DR

This paper explores the duality between window and covering approaches in quasicrystals, analyzing cluster structures and their associated windows in various tiling models, revealing their geometric and atomic configurations.

Contribution

It demonstrates the dual relationship between D- and V-clusters in quasicrystals and analyzes their geometric properties and windows in different tiling models.

Findings

D- and V-clusters are dual projections of Voronoi and Delone cells.

Decagonal V-clusters in Penrose tiling relate to decagon coverings.

Icosahedral V-clusters are Kepler triacontahedra, D-clusters include icosahedra and dodecahedra.

Abstract

In the window approach to quasicrystals, the atomic position space E_parallel is embedded into a space E^n = E_parallel + E_perp. Windows are attached to points of a lattice Lambda \in E^n. For standard 5fold and icosahedral tiling models, the windows are perpendicular projections of dual Voronoi and Delone cells from Lambda. Their cuts by the position space E_parallel mark tiles and atomic positions. In the alternative covering approach, the position space is covered by overlapping copies of a quasi-unit cell which carries a fixed atomic configuration. The covering and window approach to quasicrystals are shown to be dual projects: D- and V- clusters are defined as projections to position space E_parallel of Delone or Voronoi cells. Decagonal V-clusters in the Penrose tiling, related to the decagon covering, and two types of pentagonal D-clusters in the triangle tiling of 5fold point symmetry with their windows are analyzed. They are linked, cover position space and have definite windows. For functions compatible with the tilings they form domains of definition. For icosahedral tilings the V-clusters are Kepler triacontahedra, the D-clusters are two icosahedra and one dodecahedron.