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Lattice packing of spheres and the Wulff‐Shape
Author(s) -
Wills J. M.
Publication year - 1996
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300011736
Subject(s) - spheres , mathematics , sphere packing , polytope , lattice (music) , ideal (ethics) , parametric statistics , diophantine equation , circle packing , geometry , condensed matter physics , combinatorics , mathematical analysis , physics , philosophy , statistics , epistemology , astronomy , acoustics
The shape of large densest sphere packings in a lattice L ⊂ E d (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 E 3 .