the four non convex regular polyhedra from Kepler and Poinsot

Kepler discovered that twelve stellated pentagons can be assembled along thirty edges, building two new regular non convex polyhedra: the small stellated dodecahedron and the great stellated dodecahedron.
The sixty visible parts of their faces are golden triangles (isosceles, with one angle of 36°).

On each of the four polyhedra two opposite faces have been highlighted (dark color).

Noting that the vertices of a regular icosahedron are also the vertices of twelve regular pentagons or of twenty equilateral triangles, Poinsot discovered two more regular non convex polyhedra: the great dodecahedron (Poinsot star) and the great icosahedron.
These duals of the two stellated dodecahedra complete the list of the regular polyhedra (Cauchy proved that there are no other finite regular star polyhedra).

The convex hull of the small stellated dodecahedron is a regular icosahedron; its convex kernel is a regular dodecahedron. It's the opposite for the great stellated dodecahedron.
The two Kepler's polyhedra can thus be built by assembling a regular pyramid (with pentagonal or triangular base) on each face of a dodecahedron or an icosahedron.

To make your own models you will find nets on the site by Xavier Hubaut (in French).

53 non convex semiregular polyhedra have been counted (the equivalent of the Archimedes' polyhedra for the convex polyhedra).

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convex polyhedra - non convex polyhedra - interesting polyhedra - related subjects February 1999
updated 21-08-2005