Locally Optimal 2-Periodic Sphere Packings

The Wulff-shape of large periodic sphere packings

We determine the shape of large densest periodic packings of spheres with diierent radii. The density is measured by parametric density and the density deviation, which measures the average diierence between the density of nite and innnite packings. The asymptotic shapes are the Wull{ shapes, which are polytopes and good models for crystal shapes.

Sphere Packings

This paper is a continuation of the first two parts of this series ([I],[II]). It relies on the formulation of the Kepler conjecture in [F]. The terminology and notation of this paper are consistent with these earlier papers, and we refer to results from them by prefixing the relevant section numbers with I, II, or F. Around each vertex is a modification of the Voronoi cell, called the V -cell ...

Sphere Packings Iii

Thus in the closest pack in three dimensions, the triangular pattern cannot exist without the square, and vice versa. Abstract: An earlier paper describes a program to prove the Kepler conjecture. This paper reduces the third step of that program to a system of inequalities and one exceptional connguration of spheres. Although these inequalities have not yet been rigorously established, they se...

Regular Sphere Packings

A collection of non-overlapping spheres in the space is called a packing. Two spheres are said to be neighbours if they have a boundary point in common. A packing is called k-regular if each sphere has exactly k neighbours. We are concerned with the following question. What is the minimum number of not necessarily congruent spheres which may form a k-regular packing? In general, for which natur...

Sphere Packings Iv