3D lattice of a tetrahedron. What is it called?

Question

I recently stumbled on this image and have been looking for a name for it:

_{(source: tetraktys.de)}

It’s not a Seirpinski pyramid because it doesn’t become fractal, it’s just a subdivision of the original pyramid. Any ideas?

Answer 1

This is the tetrahedral-octahedral honeycomb. A honeycomb is the three-dimensional analog of a tiling.

It's also called the face-centered cubic lattice because the vertices lie at the vertices of a simple cubic lattice, plus also at the centers of the faces of the cubes. (But not at the centers of the cubes themselves.) This ties in with Will Jagy's comment about the vertices being at points where $x+y+z$ is an even integer.

If you stack up spheres in the obvious way (the way they make pyramids of oranges and apples in the supermarket) the centers of the spheres lie at the vertices of this honeycomb.