Hales, T.C., The sphere packing problem, Journal of Computational and Applied Mathematics 44 (1992) 41-76. The sphere packing problem asks whether any packing of spheres of equal radius in three dimensions has density exceeding that of the face-centered-cubic lattice packing (of density IT/V%). This paper sketches a solution to this problem. Keywords: Sphere packing; Delaunay triangulation; packing and covering; spherical geometry; Hilbert's problems; Voronoi cells. We begin with a general discussion of the strategy of the proof that no packing of equal spheres in three dimensions has density exceeding 7~/ m. The density of any packing may be improved by adding spheres as long as there is sufficient room to do so. When there is no longer room to add additional spheres, we say that the packing is saturated. We assume that our packings are saturated. We take our spheres to be of radius 1. Thus in a saturated packing no point of space lies more than distance 2 from a sphere center. The sphere centers are called the packing points. Every saturated packing gives rise to a decomposition of space into simplices called the Delaunay decomposition. The simplices are called Delaunay simplices. Each vertex of a simplex lies at a packing point. Every sphere (abstract sphere, not packing sphere) circumscribing a simplex has the property that none of the packing points lie in the interior of the sphere. In fact, this property is enough to completely determine the Delaunay decomposition except for certain degeneracies. When all of the simplices sharing a common vertex are grouped together, the resulting polytope is called a Delaunay star. Thus each Delaunay star is the union of Delaunay simplices.
When considering three dimensions, the question for optimal node positioning is called the sphere packing problem REF .
The sphere packing problem
Journal of Computational and Applied Mathematics
Mathematics