Kepler solid

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

The Kepler solids were defined by Johannes Kepler in 1619, when he noticed that the stellated dodecahedrons (there are two, a greater and a lesser) were composed of "hidden" dodecadrons (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 lesser stellated dodecadron 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 greater icosahedron and greater 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 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.


References: http://www.georgehart.com/virtual-polyhedra/kepler-poinsot-info.html