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Lattice packing of spheres and the Wulff-Shape

Part of: Geometry of numbers

Published online by Cambridge University Press:  26 February 2010

J. M. Wills
J. M. Wills
Affiliation:
Fachbereich Mathematik, Universität Siegen, Hölderlinstrasse 3, D-57068 Siegen, Germany.
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Abstract

The shape of large densest sphere packings in a lattice L ⊂ Ed (d ≥ 2), measured by parametric density, tends asymptotically not to a sphere but to a polytope, the Wulff-shape, which depends only on L and the parameter. This is proved via the density deviation, derived from parametric density and diophantine approximation. In crystallography the Wulff-shape describes the shape of ideal crystals. So the result further indicates that the shape of ideal crystals can be described by dense lattice packings of spheres in E3.

MSC classification

Secondary: 11H31: Lattice packing and covering

Information

Type
Research Article
Information
Mathematika , Volume 43 , Issue 2 , December 1996 , pp. 229 - 236
Copyright
Copyright © University College London 1996

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