Sphere packing in a cube
Template:Short description In geometry, sphere packing in a cube is a three-dimensional sphere packing problem with the objective of packing spheres inside a cube. It is the three-dimensional equivalent of the circle packing in a square problem in two dimensions. The problem consists of determining the optimal packing of a given number of spheres inside the cube.
Gensane[1] traces the origin of the problem to work of J. Schaer in the mid-1960s.[2] Reviewing Schaer's work, H. S. M. Coxeter writes that he "proves that the arrangements for are what anyone would have guessed".[3] The cases and were resolved in later work of Schaer,[4] and a packing for was proven optimal by Joós.[5] For larger numbers of spheres, all results so far are conjectural.[1] In a 1971 paper, Goldberg found many non-optimal packings for other values of and three that may still be optimal.[6] Gensane improved the rest of Goldberg's packings and found good packings for up to 32 spheres.[1]
Goldberg also conjectured that for numbers of spheres of the form , the optimal packing of spheres in a cube is a form of cubic close-packing. However, omitting as few as two spheres from this number allows a different and tighter packing.[7]