A Dense Packing of Hard Spheres with Five-fold Symmetry

Abstract

SUPPOSE a plane of hard spheres is constructed such that the spheres form concentric pentagons with an odd number of balls per pentagon side. A second plane of hard spheres is now constructed such that the spheres form concentric pentagons with an even number of spheres per pentagon side. If this second plane is placed in intimate contact with the first, with their five-fold axes coincident, there results a layer which, within the plane of the layer, can be continuously packed to infinity (Fig. 1). Identical layers can then be stacked one on another, with their five-fold axes coincident, to give an infinite packing along the five-fold axis. An infinite structure can thus be constructed the nucleus of which is a pentagonal dipyramid of seven spheres.