## Walls and Connections

The cubical lattice is a seemingly simple way to arrange spheres in space. By connecting spheres that are closest to each other, we get a line configuration I have also written about before.

Let’s increase the complexity by adding another copy of the same configuration, shifted by 1/2 of a unit step in all coordinate directions. This is sometimes called the body-centered cubical Bravais lattice.

We can also recognize here the two skeletal graphs of the two components of the complement of the Schwarz P minimal surface. This means that the P surface will separate the yellow and the red lattices.

Now we would like to connect the two separate systems of spheres with each other. Note that each yellow sphere is surrounded by 8 red spheres (and vice vera), at the vertices of a cube centered at the yellow sphere. This suggests to connect the yellow center to just four of these red neighbors, by choosing the vertices of a tetrahedron, as to obtain a 4-valent graph. Like so:

While this is still simple, it starts to look confusing. The new skeleton has again two components, and again they can be separated by a classical minimal surface, the Diamond surface of Hermann Amandus Schwarz.

All this should remind us of the Laves graphs, which are skeletal graphs of the gyroid.

You can see that these skeletal graphs have girth 6. Below is a larger piece of the D-surface. Everything here is triply periodic and very symmetric. In contrast to the Laves graph, these here have no chirality.

Next week, we will decorate these skeletons a little.