Open Packing of Spheres

Abstract

MR. S. E. ACKERMANN'S axiom¹ that spherical lumps of coal of equal size have the same bulk density irrespective of the actual size is surely applicable to any symmetrical arrangement of such spheres, including the closest packing. The arrangement of spheres which he uses in his first calculation is symmetrical and satisfies the axiom, but is not a stable arrangement which would be attained by pouring the spheres into a box. His statement that the closest packing gives an increase in density as the size of sphere is decreased appears to be derived from an incorrect conception of what constitutes the closest packing. This is, of course, the arrangement in which each sphere touches twelve others (not eight), lines joining the centres forming a system of regular tetrahedra, and is the stable one which is taken up automatically.