The densest and the least dense packings of equal spheres

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Abstract
Of the results which have so far been achieved by study of the packing of equal spheres one of the most remarkable, both by reason of its simplicity and of its fundamental importance, was that announced by Barlow in 1883. He called attention to the fact that equal spheres can be most densely packed in two ways, one possessing cubic symmetry and the other hexagonal (fig. 1).Already in 1862 Tait had investigated the piling of marbles of equal size and had noticed that ‘there are two obvious ways of constructing the layers, and two of applying layer to layer’: nevertheless his two densest arrangements are in fact identical. He saw that the cubic structure could be begun either upon a square base or a triangular base but failed to perceive the possibility of the hexagonal arrangement.

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