When we stack cannon balls such that each one touches all the others immediately adjacent to it we can place them in several different configurations. One is to place four in a square pattern and a fifth on top in the middle of the four. We can repeat the pattern to the sides and in additional layers. This produces a square pyramid. Another is to place three cannon balls in a triangle and a fourth on top in the middle of the three. We can also repeat this pattern to the side and in additional layers. This makes a triangular pyramid.

I find the triangular pyramid more interesting than the square pyramid. If we imagine lines connecting the centers of the first four cannon balls we recognize a tetrahedron, a shape with four vertices and four faces which are equilateral triangles . We can produce a larger tetrahedron by adding more cannon balls so that the number of the layers is equal to the number of cannon balls along the edges.

The interesting part, to me, is that if we had placed a canon ball just below ground level and then placed three in a triangle around that one and a fifth one on top of the three we could imagine a shape with five vertices and all six faces would be equilateral triangles. This is called a triangular bi-pyramid.

We should recognize that a triangular bi-pyramid tessellates a volume. If we consider that the vertices imagined in the three cannon balls at ground level form a plane and all six of the other faces form planes we have the basis for a seven-plane geometry of a volume in the same fashion that tessellation of a volume by the cube provides the basis for a three dimensional geometry.

Enjoy