LATTICE PACKINGS OF SPHERES AND THE WULFF-SHAPE

The shape of large densest sphere packings in a lattice L subset of E( d) (d greater than or equal to 2), measured by parametric density, ten ds 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 dens ity deviation, derived from parametric density and diophantine approxi mation. In crystallography the Wulff-shape describes the shape of idea l crystals. So the result further indicates that the shape of ideal cr ystals can be described by dense lattice packings of spheres in E(3).