- 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.
- The great dodecahedron has the same vertices and edges as the icosahedron.
- The small stellated dodecahedron has the same vertices and edges as the great icosahedron. These both have the same vertices (but not edges) as the icosahedron.
- The great stellated dodecahedron has the same vertices (but not edges) as the dodecahedron.
- All four have the same symmetry axes and symmetry planes as the icosahedron and dodecahedron.

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:

**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.)*

**Answer: ***Find out when you read about stellations.*

If you are considering making paper models of these polyhedra, be sure to consider a number of nice colorings which are possible.