Discrete & Computational Geometry

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

Quasicrystals and the Wulff-Shape

  • K. BöröczkyJr.
  • U. Schnell


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.


Surface Energy Packing Density Direct Relation Linear Density Isoperimetric Inequality 
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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 HU
  2. 2.Mathematisches Institut, Universität Siegen, D-57068 Siegen, Germany schnell@mathematik.uni-siegen.deDE

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