# Kepler’s Rhombic Polyhedra

Johannes Kepler was an interesting character. He was interested in a lot more than just astronomy. He’s famous among aficionados of polyhedra for organizing all of the new ones that Renaissance artists Wenzel Jamnitzer and Paolo Uccello had been using to show off their new high-tech invention – perspective.

These are two of my favorite Kepler polyhedra:

The one on the left is called the rhombic dodecahedron. The one on the right is called the rhombic triacontahedron. They both have faces which are rhombuses. In these models, I’ve added the diagonals of the rhombuses, because they highlight their relationship to other, more familiar polyhedra.

Here’s a closer look at the dodecahedron.

The ratio of the lengths of the diagonals of these rhombuses are √2 for the dodecahedron. Although the 12 faces are identical, the vertices are not. It has 6 vertices of valence 4 and 8 vertices of valence 3. If you look closely at all of the short diagonals, you’ll notice that they’re the edges of a cube. And if you look at the long diagonals, you’ll notice that they’re the edges of an octahedron.

And here’s the triacontahedron.

The ratio of the lengths of the diagonals of these rhombuses are φ. This one has 12 vertices of valence 5 and 20 vertices of valence 3. It’s a little harder to see here, but the diagonals are again the edges of other polyhedra. The short diagonals are the edges of a dodecahedron, and the long diagonals are the edges of an icosahedron.

I printed these in the new Formlabs grey resin. If you’d like to print them yourself, you can get them from my Pinshape account here and here.