There’s Just One Way to Stack Cannonballs

Greg Frederick

It may seem that there are at least three ways of stacking cannonballs, i.e., from a triangular base to form a pyramid with three sides; from a square base to form a pyramid with four sides; or as a shallow 3-sided pyramid.

Three pyramids of semi-transparent spheres centered on the vertices of the tetrahedron-octahedron matrix.
The three standard methods of stacking unit-radius spheres: regular tetrahedron (left); one-quarter tetrahedron (center); and one-half octahedron (right).

But really, there’s only one way. All three arrangements coexist in the radial close-packing of spheres.

Fourty-two spheres radially close packed around a central sphere, animated to show that it constitutes each of the three methods of stacking spheres.
All three methods of stacking unit-radius spheres are really just different ways of looking at the single method of the radial close-packing of spheres around a central nucleus.

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