The Kepler-Poinsot Polyhedra

A polyhedron is regular if the faces are a single kind of regular polygon and the vertices are all the same. The 5 Platonic Solids are the convex regular polyhedrons. If we remove the constraint of convexity it turns out that there are only four more solids that can be added to the list; these are known as the Kepler-Poinsot Polyhedra.

It was Johann Kepler who, in 1619, first realized that 12 pentagrams can be joined in pairs along their edges in two different ways that result in regular solids. If five pentagrams meet at each vertex, the resulting solid has come to be known as the small stellated dodecahedron.

Small Stellated Dodecahedron

If three pentagrams meet at each vertex, the resulting solid is now named the great stellated dodecahedron (The perhaps surprising reason for these names will be made evident shortly).

Great Stellated Dodecahedron

Two centuries later, in 1809, Louis Poinsot discovered two more non-convex regular solids: the great dodecahedron and the great icosahedron. The twelve faces of the great dodecahedron are pentagons (as with the ordinary dodecahedron), but which intersect each other. Likewise, the faces of the great icosahedron are the 20 triangles of the ordinary icosahedron, but intersecting each other.

The great dodecahedron is a most pleasing and intriguing solid, giving the illusion of a pentagonal star embossed on each pentagon; each star shares each of its arms with another, so that one star disappears as soon as you bring your attention to another!

Great Dodecahedron

It has been proven that these four solids, together with the 5 platonic solids, are the only regular solids possible. Another wonderful truth is that the Kepler solids are the duals of the Poinsot solids!

Great Icosahedron

It was not until 1811 that the French mathematician Augustin Cauchy showed that the Kepler-Poinsot solids are stellated forms of the dodecahedron or the icosahedron. It was this insight that led to the names for these solids that we use today.

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