A Kepler solid is a regular nonconvex polyhedron, all the faces of which are identical regular polygons and which has the same number of faces meeting at all its vertices. (compare to Platonic solids). There are four:

• great stellated dodecahedron - 12 faces, 12 vertices, 30 edges
• small stellated dodecahedron - 12 faces, 12 vertices, 30 edges
• great dodecahedron - 12 faces, 20 vertices, 30 edges
• great icosahedron - 20 faces, 12 vertices, 30 edges

The first two are stellations, that is, their faces are concave. The second two have convex faces, but each pair of faces which meet at a vertex in fact does so in two.

The Kepler solids were defined by Johannes Kepler in 1619, when he noticed that the stellated dodecahedra (there are two, the great and the small) were composed of "hidden" dodecahedra (with pentagonal faces) that have faces composed of triangles, and thus look like stylized stars. Wentzel Jamnitzer actually found the great stellated dodecahedron and the great dodecahedron in the 1500s, and Paolo Uccello discovered and drew the small stellated dodecahedron in the 1400s. Kepler's contribution was in recognizing that they fit the definition of regular solids, even though they were concave rather than convex, as the traditional Platonic solids were.

There are only four Kepler solids. The other two are the great icosahedron and great dodecahedron which were described by Louis Poinsot in 1809. Some people call these the two Poinsot solids.

A Kepler solid covers its circumscribed sphere more than once, with the centers of faces acting as winding points in the solids with pentagrammic faces and the vertices in the others. Because of this, they are not necessarily topologically equivalent to the sphere as Platonic solids are, and in particular the Euler relation

V − E + F = 2

may not hold.

Trivia

A cutaway view of the greater dodecahedron was used for the 1980s puzzle game Alexander's Star .

External links

• Paper models of Kepler-Poinsot polyhedra
• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
• http://www.georgehart.com/virtual-polyhedra/kepler-poinsot-info.html