# Stellations of the Icosahedron

Starting with the icosahedron, it turns out that there are 59 possible stellations. (Here we assuming some reasonable conditions, called Miller's rules, e.g., the resulting object should be finite, it should have the symmetries of the icosahedron, and we don't allow hollow centers that are not connected to the outside surface. We do allow hollow centers if they can be seen, however. And, we also allow disconnected polyhedra.)

A few of these we have seen elsewhere: the great icosahedron is one of the Kepler-Poinsot polyhedra, and the first stellation in the official list is just the icosahedron itself. Some other stellations are familiar compounds: the compound of five octahedra, the compound of five tetrahedra, and the compound of ten tetrahedra. However, most of the 59 are simply beautiful new polyhedra, well worth a look. The one illustrated at left is relatively easy to make out of paper because it requires pieces of only a single shape.

I have rendered each of these models with two different coloring schemes: monochromatic and five colors. For the five-colored versions, each of the 20 icosahedral planes is assigned a color according to this five-coloring of the icosahedron, which in turn comes from the most famous of its stellations, the compound of five tetrahedra. (Travel inside it to see the five-colored icosahedron at its core. Note that two triangles of the same color never touch. There are two ways to do this; the other way is the mirror image of this one.) So in each of the 59 stellations, for each color there are four faces, arranged according to the planes of one of the five tetrahedra in the compound of five tetrahedra. Knowing this should help to make clear the common planes of different facets.

Here are a few of my favorites:

Stellation #4 (solid color) (5 colors) has the special property of having equal faces and equal vertex figures.

And here is the complete list of 59 stellations of the icosahedron.

Exercise: Note that both the shape of the compound of five tetrahedra and the color pattern of the five-colored icosahedron are chiral. If we reflect a five-colored compound of five tetrahedra in a mirror, we are actually reflecting two things, the shape and the coloring. What results if we reflect just one of these two aspects? In other words, imagine assigning colors to the planes of the compound's shape according to the colors of the reflected colored icosahedron.

Answer: If you can't imagine the result, at least determine its symmetry group before looking at the answer.

If you are willing to relax the condition that the stellations have icosahedral symmetry, there are many stellations of the icosahedron with tetrahedral symmetry