Computer assisted sphere packing in higher dimensions

A computer was used to help study the packing of equal spheres in dimension four and higher. A candidate of the densest packing in 4-space is described. The configuration of 24 spheres touching a central sphere in this packing is shown to be rigid, unlike the analog in 3-space, in which the spheres can slide past each other. A system for interactively manipulating and visualizing such configurations is described. The Voronoi cell for a sphere is the set of points closer to its center than to any other sphere center in the packing. The packing density is the ratio of a sphere´s volume to the average of the volumes of the Voronoi cells. A method of constructing Voronoi cells and computing their volumes that works in any dimension is presented. Examples of Voronoi cell volumes are given