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Discrete & Computational Geometry

, Volume 21, Issue 3, pp 421–436 | Cite as

Quasicrystals and the Wulff-Shape

  • K. BöröczkyJr.
  • U. Schnell

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.

Keywords

Surface Energy Packing Density Direct Relation Linear Density Isoperimetric Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer-Verlag New York Inc. 1999

Authors and Affiliations

  • K. BöröczkyJr.
    • 1
  • U. Schnell
    • 2
  1. 1.Mathematical Institute of the Hungarian Academy of Sciences, Pf. 127, H-1364 Budapest, Hungary carlos@cs.elte.hu HU
  2. 2.Mathematisches Institut, Universität Siegen, D-57068 Siegen, Germany schnell@mathematik.uni-siegen.deDE

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