# The Kepler-Poinsot Polyhedra

If we do not require polyhedra to be convex, we can find four more regular solids. As in the Platonic solids, these solids have identical regular polygons for all their faces, and the same number of faces meet at each vertex. What is new is that we allow for a notion of "going around twice" which results in faces which intersect each other.

In the great stellated dodecahedron and the small stellated dodecahedron, the faces are pentagrams.  It is easier to see which parts of the exterior belong to which pentagram if you look at a six-colored model of the great stellated dodecahedron and a six-colored model of the small stellated dodecahedron.  The center of each pentagram is hidden inside the polyhedron. These two polyhedra were described by Johannes Kepler in 1619, and he deserves credit for first understanding them mathematically, but a 16th century drawing by Jamnitzer is very similar to the former and a 15th century mosaic attributed to Uccello illustrates the latter.

These two polyhedra have three and five pentagrams, respectively, meeting at each vertex. Because the faces intersect each other, parts of each face are hidden by other faces, and you need to see that the visible portions of the faces are not the complete faces.

In the great icosahedron and great dodecahedron (described by Louis Poinsot in 1809, although Jamnitzer made a picture of the great dodecahedron in 1568) the faces (20 triangles and 12 pentagons, respectively) which meet at each vertex "go around twice" and intersect each other, in a manner which is the 3D analog to what happens in 2D with a pentagram.

If you slice the polyhedron near a vertex, you'll see a pentagram cross section as the vertex figure. For example, this cutaway view of the great dodecahedron shows how the five pentagons which meet at a vertex pass through each other in the manner of a pentagram. Study the virtual models to see this. To emphasize that these polyhedra are made of large convex faces, it helps to look at a five-color model of the great icosahedron and a six-color model of the great dodecahedron.

Together, the Platonic solids and these Kepler-Poinsot polyhedra form the set of 9 regular polyhedra. Cauchy first proved that no other polyhedra can exist with identical regular faces and identical regular vertices.

Here are some relationships you should observe:

Each of these polyhedra is related to the dodecahedron or icosahedron in another way as well. If you travel to their centermost region you will find either a dodecahedron or an icosahedron.

Exercise: Which of these four polyhedra have a dodecahedron and which an icosahedron as their innermost region? Imagine how the planes intersect, then travel inside to find out. (Note: be sure to find the centermost region.)