Kepler-Poinsot solid

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A Kepler solid (also called Kepler-Poinsot solid) is a regular non-convex 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).

The faces are only partly at the surface of the solid, and the exposed parts are, if at all, only connected at points. If the parts are counted as separate faces, the solid is no longer regular.

Image:Kepler poinsot solids.gif

There are four different Kepler solids:

Small stellated dodecahedron: 12 pentagram faces, 12 vertices, 30 edges
Great stellated dodecahedron: 12 pentagram faces, 20 vertices, 30 edges
Great icosahedron: 20 triangle faces, 12 vertices, 30 edges
Great dodecahedron: 12 pentagon faces, 12 vertices, 30 edges

The first two are stellations; that is, their faces are complex. The second two have convex faces, but each pair of faces which meet at a vertex in fact does so in two and each face intersects several other faces, actually crossing through them.

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. Wenzel 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.

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

doesn't always hold.

This depends on how we look upon the polyhedron. Consider e.g. the small stellated dodecahedron [1]. It consists of a dodecahedron with a pentagonal pyramid on each of its 12 faces. Thus the 12 faces are extended to pentagrams with the central pentagon inside the solid. The outside part of each face consists of five triangles which are only connected by five points. If we count these separately there are 60 faces (but they are only isosceles triangles, not regular polygons). Similarly each edge can also be counted as three (but then they are of two kinds). Also the "five points" just mentioned, together are 20 points that can be counted as vertices, so that we have a total of 32 vertices (again two kinds). Now the Euler relation holds: 60 - 90 + 32 = 2.

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Trivia

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

Artist M.C. Escher's interest in geometric forms often led to works based on or including regular solids; Gravitation is based on a small stellated dodecahedron.

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