Kepler-Poinsot Solids: The Four Regular Polyhedra Nobody Expects

The last four regular polyhedra

Beyond the five convex Platonic solids there are exactly four more shapes that satisfy the strict definition of regular polyhedron — every face is a congruent regular polygon, every vertex is surrounded identically — but only if you allow faces and vertex figures to self-intersect. These are the Kepler-Poinsot solids: the small stellated dodecahedron and great dodecahedron, found by Johannes Kepler in 1619, and the great stellated dodecahedron and great icosahedron, added by Louis Poinsot in 1809. Together with the five Platonic solids they complete the full list of regular polyhedra — Cauchy proved in 1813 that no others exist.

Stellation: extending faces until they meet again

Kepler built his two solids by stellation — extending the faces of a Platonic solid outward as flat planes until they intersect again far from the original body, forming star-shaped points. Stellate the regular dodecahedron's twelve pentagonal faces outward and they meet to form twelve intersecting pentagrams (five-pointed stars) — the small stellated dodecahedron. Stellate further, using a different extension rule, and you get the great stellated dodecahedron. Because a regular pentagon's diagonal divides its side in golden-ratio proportion, every one of these solids is built entirely from φ-scaled measurements — the same constant that governs the golden ratio spiral turns out to be the geometric skeleton of every star polyhedron.

dodecahedron (12 pentagon faces)  --stellate-->  small stellated dodecahedron (12 pentagram faces)
icosahedron (20 triangle faces)   --facet----->  great dodecahedron (12 pentagon faces, self-intersecting)
                                    --stellate-->  great stellated dodecahedron (12 pentagram faces)
                                    --facet----->  great icosahedron (20 triangle faces, self-intersecting)

Facetting: the other half of the pair

The great dodecahedron and great icosahedron are not stellations at all — they are built by facetting, the dual operation: keep the original vertices of an icosahedron in place but join them with a different, larger set of faces that pass through the solid's interior. The great dodecahedron uses the icosahedron's twelve vertices as the centres of twelve intersecting pentagons; the great icosahedron reuses the dodecahedron's twelve vertices with twenty large intersecting triangles. Stellation extends faces outward from an existing solid; facetting reconnects existing vertices with new, self-crossing faces — two structurally different routes that happen to land on the same short list of four shapes.

Why V − E + F ≠ 2 here

None of the four Kepler-Poinsot solids satisfies the ordinary Euler formula V − E + F = 2 from the companion article on that topic. The small stellated dodecahedron, for instance, has 12 vertices, 30 edges and 12 faces: 12 − 30 + 12 = −6. The reason is that the simple planar-graph proof of V − E + F = 2 assumes the polyhedron's surface can be flattened without any faces crossing each other — true for every convex solid, false here, since the pentagram faces pass through the body and intersect one another. The correct generalisation replaces the plain face count with a density d, the number of times the faces wind around the centre (d = 3 for the small stellated dodecahedron), giving the corrected relation V − E + dF = 2 for these self-intersecting solids.

The duality that pairs them up

Like the Platonic solids, the four Kepler-Poinsot solids come in dual pairs: swap each face for a vertex and each vertex for a face and one solid becomes another. The small stellated dodecahedron is dual to the great dodecahedron, and the great stellated dodecahedron is dual to the great icosahedron. This mirrors exactly how the cube and octahedron, or the dodecahedron and icosahedron, are Platonic duals of each other — the same duality operation, just applied to a non-convex, self-intersecting starting shape.

Frequently asked questions

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