In the course of implementing a new computer graphics algorithm, I recently found myself immersed in the wonderful world of polyhedra. In particular, I got to play with two of my favorite polyhedra — the cube and the rhombic dodecahedron.
The first one everybody knows about. The second one is a little more obscure, but very cool. It’s sort of if a cube and an octahedron decided to get together and have a party.
One thing that both the cube and the rhombic dodecahedron have in common is that they both tile 3D space: That is, if you start with either a cube or a rhombic dodecahedron, you can fill all of space by putting copies of that shape together like an infinite set of identical blocks.
The rhombic dodecahedron has 14 vertices. Eight of them are the vertices of a cube, the other six are the vertices of an octahedron.
Here’s an interesting thing: If you push in on the six vertices of the octahedron part, the rhombic dodecahedron collapses down into a cube. This cube has exactly half the volume of the rhombic dodecahedron you started with.
If all of this sounds hard to follow, you can just look at the picture. Suddenly it all becomes crystal clear. And also rather beautiful.