Decagon

Abstract: The unit-cluster approach describing quasicrystalline structures by a single unit with possible ‘fat’ overlaps has mainly been worked out for the ideal case of perfect order. A general concept for corresponding random covering ensembles is still missing. As a natural framework, we introduce an ‘overlap-version’ of the well-known random tiling model in terms of relaxed cluster matching rules. We illustrate our method by a relaxed decagon determining a covering ensemble with positive configurational entropy. We characterize this coverings using equivalent underlying Penrose-type tilings, which form Hexagon-Boat-Star (HBS)-supertilings. Given an HBS-supertiling, we describe the effect of so-called bow-tie flips on corresponding decagon coverings.

Abstract: Infinite sphere packings give information about the structure but not about the shape of large dense sphere packings. For periodic sphere packings a new method was introduced in [W2], [W3], [Sc], and [BB], which gave a direct relation between dense periodic sphere packings and the Wulff-shape, which describes the shape of ideal crystals. In this paper we show for the classical Penrose tiling that dense finite quasiperiodic circle packings also lead to a Wulff-shape. This indicates that the shape of quasicrystals might be explained in terms of a finite packing density. Here we prove an isoperimetric inequality for unions of Penrose rhombs, which shows that the regular decagon is, in a sense, optimal among these sets. Motivated by the analysis of linear densities in the Penrose plane we introduce a surface energy for a class of polygons, which is analogous to the Gibbs—Curie surface energy for periodic crystals. This energy is minimized by the Wulff-shape, which is always a polygon and in certain cases it is the regular decagon, in accordance with the fivefold symmetry of quasicrystals.