Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-05T06:58:23.582Z Has data issue: false hasContentIssue false

Lattice packing of spheres and the Wulff-Shape

Published online by Cambridge University Press:  26 February 2010

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

Abstract

The shape of large densest sphere packings in a lattice LEd (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

Type
Research Article
Copyright
Copyright © University College London 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BHW1.Betke, U.Henk, M. and Wills, J. M.. Finite and infinite packings, J. reine angew. Math., 453 (1994). 165191.Google Scholar
BHW2.Betke, U.Henk, M. and Wills, J. M.. Sausages are good packings. Discrete Comp. Geom. 13 (1995), 297311.Google Scholar
BS.Böröczky, K. Jr and Schnell, U.. Asymptotic shape of finite packings. To appear.Google Scholar
CS.Conway, J. H. and Sloane, N. J. A.. Sphere packings, lattices and groups (Springer, New York, 1988).Google Scholar
D.Dinghas, A.. Über einen geometrischen Satz von Wulff über die Gleichgewichtsform von Kristallen. Z. Kristallogr., 105 (1943), 304314.Google Scholar
E.Engel, P.. Geometric crystallography, Ch. 3.7 In Gruber, P. M. and Wills, J. M., editors. Handbook of Convex Geometry (North Holland, Amsterdam, 1993).Google Scholar
GL.Gruber, P. M. and Lekkerkerker, C. G.. Geometry of Numbers (North Holland, Amsterdam, 1987).Google Scholar
GW.Gritzmann, P. and Wills, J. M.. Finite packing and covering, Ch. 3.4 In Gruber, P. M. and Wills, J. M., editors, Handbook of Convex Geometry (North Holland, Amsterdam, 1993).Google Scholar
K.Koksma, J. F.. Diophantische Approximationen (Springer, Berlin, 1936) or (Chelsea Publ. Comp., New York).Google Scholar
L.Laue, M. v.. Der Wulffsche Satz für die Gleichgewichtsform von Kristallen. Z. Kristallogr., 105 (1943), 124133.Google Scholar
S.Schnell, U.. Parametric density, Wulff-shape and crystals. Submitted.Google Scholar
Sch.Schneider, R.. Convex Bodies: The Brunn-Minkowski Theory (Cambridge University Press, Cambridge, 1993).CrossRefGoogle Scholar
W1.Wills, J. M.. Finite sphere packings and sphere coverings. Rend. Semin. Mat., Messina, Ser. II, 2 (1993), 9197.Google Scholar
W2.Wills, J. M.. On large lattice packings of spheres. Geometriae Dedicata, 63 (1996). To appear.Google Scholar
Wn.Willson, S. J.. A semigroup on the space of compact convex bodies. SIAM J. Math. Anal., 11 (1980), 448457.Google Scholar