Sphere-packings, lattices, and groups
Sphere-packings, lattices, and groups
Numerical recipes in C: the art of scientific computing
Numerical recipes in C: the art of scientific computing
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A computer was used to help study the packing of equal spheres in dimension four and higher. A candidate for 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 analogue 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 is presented, which works in any dimension. Examples of Voronoi cell volumes are given.