Summary:
Close-packings of uniformly-sized spheres with centres on various lattices are described, with volume fractions equal or close to the maximum possible pi / sqrt(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).
Extrapolation and edge-correction techniques for determining the volume fraction of an in nite array of such balls in an in nitely large sphere are indicated. The paper also investigates questions of inaccessibility of part of the space except to spheres of in nitesimal size. Various questions and problems are recorded also.
The study began from the observation that the volume fraction of aggregate in concrete has a volume fraction in the range 60% to 70%. It is known how to locate spheres on a perturbed lattice and, depending on the perturbation, obtain a volume fraction arbitrarily close to pi / sqrt(18). If cubes of irregular size but common orientation are used instead of spheres, then the volume fraction can be made arbitrarily close to 1.0 by choosing suciently small perturbations (Daley, 2000).
tr104.pdf