The Kepler-Poinsot polyhedra are four regular polyhedra which, unlike the Platonic solids, contain intersecting facial planes. In addition, two of the four Kepler-Poinsot polyhedra are constructed using regular polygrammic as opposed to regular polygonal faces. The following table summarizes these polyhedra the regular polygons (or polygrams) from which they are constructed.

polyhedron | face count | face type |

great dodecahedron | 12 | |

great icosahedron | 20 | |

small stellated dodecahedron | 12 | |

great stellated dodecahedra | 12 |

A list of the Kepler-Poinsot polyhedra as implemented in the Wolfram
Language can be given by `PolyhedronData`[`"KeplerPoinsot"`].

While the external appearances of the Kepler-Poinsot polyhedra are visually indistinguishable from solids created by the augmentation of dodecahedra
and icosahedra, they are actually icosahedron
and dodecahedron stellations that contain
portions of faces internally. As a result, the term "Kepler-Poinsot *polyhedra*"
is preferable to the more common term "Kepler-Poinsot *solids.*"

The names of the polyhedra probably originated with Arthur Cayley, who first used them in 1859. Cauchy (1813) proved that these four exhaust all possibilities for
regular star polyhedra (Ball and Coxeter 1987). While these polyhedra were unknown
to the ancients, the small stellated dodecahedron
appeared ca. 1430 as a mosaic by Paolo Uccello on the floor of San Marco cathedral,
Venice (Muraro 1955). The great stellated
dodecahedron was published by Wenzel Jamnitzer in 1568. Kepler rediscovered these
two (Kepler used the term "urchin" for the small stellated dodecahedron)
and described them in his work *Harmonice Mundi* in 1619. The two known polyhedra,
great dodecahedron, and great
icosahedron were subsequently (re)discovered by Poinsot in 1809. As shown by
Cauchy, they are stellated forms of the dodecahedron
and icosahedron.

A table listing these solids, their duals, and compounds is given below. Like the five Platonic solids, duals of the Kepler-Poinsot polyhedra are themselves Kepler-Poinsot polyhedra (Wenninger 1983, pp. 39 and 43-45).

solid | uniform polyhedron | Schläfli symbol | Wythoff symbol | point group | |

1 | great dodecahedron | ||||

2 | great icosahedron | ||||

3 | great stellated dodecahedron | ||||

4 | small stellated dodecahedron |

The polyhedra and fail to satisfy the polyhedral formula

where is the number of vertices, the number of edges, and the number of faces, despite the fact that the formula holds for all ordinary polyhedra (Ball and Coxeter 1987). This unexpected result led none less than Schläfli (1860) to erroneously conclude that they could not exist.

In four dimensions, there are 10 Kepler-Poinsot polyhedra, and in dimensions with , there are none. In four dimensions, nine of the polyhedra have the same polyhedron vertices as , and the tenth has the same as . Their Schläfli symbols are , , , , , , , , , and .

Coxeter *et al. *(1954) have investigated star "Archimedean" polyhedra.