A Toroidal Honeycomb of 32 Close Packed Spheres

Some shapes can fill space (form a “honeycomb”) with an infinite number of copies of themselves, using only coordinate translations (no rotations). The  obvious and most commonly seen one is a lattice of cubes. Another is based on the rhombic dodecahedron, which is the polyhedron corresponding to a sphere in fcc close packing (described in my last post, and seen below). You can connect as many of these shapes as you want, and they will fill space without gaps.

Rhombic Dodecahedral Honeycomb

If a given polyhedron can fill space, that does not necessarily hold for a similarly shaped collection of spheres (and vice versa). Consider the tetrahedron. As a convex shape it cannot fill space by itself, however, a tetrahedral configuration of spheres can. This is because the shared edges and faces which would normally be infinitely thin in a polyhedral lattice will need to take up space in a close packing. After all, the “edges” in a close packing lattice are made up of spheres, not lines.

32-ball configuration

This configuration of 32 balls is made up of 8 copies of a tetrahedral ball configuration. It’s an interesting object, looking sort of like a cube but not a completely symmetric one. It also has the fcc close packing. This configuration fills space, and if you look closely you can see a Vector Equilibrium embedded in there.

The opposite faces interlock to ensure a close packing, as the following figure shows. This object is made from 8 of the 32-sphere units for a total of 256 spheres.

Another neat thing you can do with this object is to create a toroidal lattice by joining the opposite faces (you’ll have to imagine you’re doing this in another dimension, since you can’t fold the cube space onto itself in 3D). You might have experienced a similar torus world while playing a 2D or 3D video game where objects going off the screen will reappear on the opposite side. That becomes very useful when you want to run a cellular automata (CA) or another simulation in a closed universe. You can make a torus world like this of any size that you want. We’ll revisit this idea later when we explore a CA built around the fcc lattice.

This 32 sphere object, when connected as a torus, is related to the generalized quaternion group. This is also true of any larger toroidal environments made up of these.