Beiträge zur Algebra und Geometrie / Contributions to Algebra and GeometryVol. 40, No. 1, pp. 125-140 (1999)

Periodic Sphere Packings and the Wulff-Shape

Uwe Schnell

Mathematisches Institut, Universität Siegen, D-57068 Siegen, Germany, e-mail:

Abstract: The asymptotic shape of large densest sphere packings in a lattice $L\subset E^d(d\geq 2)$, measured by parametric density, is a polytope, the so-called Wulff-shape of ideal crystals in crystallography. This was shown in [WA] and [WB] by density deviation derived from parametric density. Our main goal is a generalization of this approach from lattice packings to periodic packings of spheres. It turns out that the asymptotic shape of large densest periodic packings is a generalized Wulff-shape which is more complicate than in the lattice case. As an application we determine this Wulff-shape for a type of crystals with two different atoms (NaCl-type) and for a non-lattice crystal (diamond) and for a combination of both (ZnS-type). In all cases the result coincides for suitable parameter with the natural shapes. In particular the asymptotic shape in the case of the ZnS-type is a tetrahedron, i.e. not centrally symmetric. Although the intention of these generalizations is application to crystallography, the results can also be considered as contributions to finite and infinite packings of spheres.

[WA] Wills, J. M.: On large lattice packings of spheres. Geom. Dedicata 63 (1996), to appear.

[WB] Wills. J. M.: Lattice packings of spheres and Wulff-shape. Mathematika 86 (1996), 229-236.

Classification (MSC91): 52C17

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