Packings and Approximate Packings of Spheres

Close-packings of uniformly-sized spheres with centres on various lattices are described, with volume fractions equal or close to the maximum possible = p 18 (this value has long been `known' via Kepler's conjecture, and has been proved). Regular packings with two or three sized spheres can push this volume fraction to beyond 80%. The bulk of the paper studies irregular `packings' of a large sphere by spheres of varying sizes, and attempts to evaluate the in uence of factors in the algorithm specifying how the random packing is constructed, in determining the volume fraction of the resultant random set (meaning, the union of all the spheres).