Simple Regular Sphere Packings in Three Dimensions

Abstract

Introduction. Rogers (1) has defined a packing as a system of equal closed spheres in n-dimensional space in which no two spheres have any inner point in common. If classified in terms of density, regular packings in three dimensions range from 0.056 * to 0.740 . These limits have been established empirically, they are not the proven bounds of density. In the highest density packing each sphere touches twelve others. There is an infinite number of variations of this packing in which the spheres have slightly differing neighbour relationships. There are only two in which the spheres are all equivalent as Barlow discovered in 1883. Interest in the loosest packings came much later; the loosest packings known were discovered by Heesch and Laves in 1933. The packings of extreme density have been fully discussed by Melmore (2). There exists a range of regular packings having densities intermediate between the two extremes. These may be distinguished by density; by coordination number (CN), that is the number of spheres touching every sphere in the pack; and by the shape of the Voronoi polyhedron associated with the lattice formed by the centres of the spheres. The Voronoi polyhedron (VP) is a convex polyhedron associated with each point in the lattice, and these polyhedra pack together to fill space. The planes which form the faces of the polyhedron are the perpendicular bisectors of the lines joining the centre of the reference sphere to the centres of the nearest spheres. Rogers uses this terminology but Coxeter (3) prefers to call this figure the Dirichlet region. He defines the two dimensional case as a polygon whose interior consists of all the points in the plane which are nearer to a particular lattice point than any other lattice point. Regular packings can be represented by point lattices formed by the centres of the spheres and may be divided into two types, simple and nonsimple. The lattice is represented by a unit cell, this is the simplest convenient figure which by means of suitable translatory movements will describe the whole lattice. The unit cell for the simple lattice is the simplest possible: a parallelepiped with lattice points only at the corners. The cell for a non-simple lattice will have lattice points in the interior or on the boundary. It is possible for different parallelepipeds to give the same lattice provided they have the same volume. One particular cell is chosen for the derivation of the range of simple packs, the conditions for this are described below. The unit cell of a simple pack can be defined by the centres of eight adjacent spheres and each of these spheres contributes to eight cells. Each cell contains in effect one sphere.