Kepler–Poinsot polyhedron

In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra. They may be obtained by stellating and faceting the regular convex dodecahedron and icosahedron, and differ from these in having regular pentagrammic faces or vertex figures. They can all be seen as three-dimensional analogues of the pentagram in one way or another.

Characteristics The Kepler–Poinsot polyhedra are the regular star polyhedra, obtained by extending both regular icosahedron and regular dodecahedron, an operation named stellation. This operation results in four different polyhedra:

Great dodecahedron: constructed from attaching twelve pentagonal pyramids (with regular polygonal faces) onto the face of a regular dodecahedron, and attached again with thirty wedges. However, this can be constructed alternatively by removing its polygonal faces without changing or creating new vertices of a regular icosahedron. Small stellated dodecahedron: attaching twelve pentagonal pyramids onto a regular dodecahedron's faces. Topologically, this shares the same surface as the pentakis dodecahedron. Great icosahedron; and Great stellated dodecahedron: constructed from a great dodecahedron with twenty asymmetric triangular bipyramids, attaching to the hollow between the wedges.

John Conway introduces operators for the Kepler–Poinsot polyhedra known as greatenings—(g), maintaining the type of faces, shifting and resizing them into parallel planes—and stellations—(s), changing pentagonal faces into pentagrams—of the convex solids. In his naming convention, the small stellated dodecahedron is just the stellated dodecahedron. By the construction above, these figures have pentagrams (star pentagons) as faces or vertex figures. The dual polyhedron of a great dodecahedron is the small stellated dodecahedron, and the dual of a great icosahedron is the great stellated dodecahedron. The four share the symmetry as both regular icosahedron and regular dodecahedron, the icosahedral symmetry.

Euler characteristic A Kepler–Poinsot polyhedron covers its circumscribed sphere more than once, with the centers of faces acting as winding points in the figures which have 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

χ = V − E + F = 2 {\displaystyle \chi =V-E+F=2\ }

does not always hold. Schläfli held that all polyhedra must have χ = 2, and he rejected the small stellated dodecahedron and great dodecahedron as proper polyhedra. This view was never widely held. A modified form of Euler's formula, using density (D) of the vertex figures (dv) and faces (df) was given by Arthur Cayley, and holds both for convex polyhedra (where the correction factors are all 1), and the Kepler–Poinsot polyhedra:

d v V − E + d f F = 2 D , {\displaystyle d_{v}V-E+d_{f}F=2D,}

and by this calculation, the density of the great icosahedron and the great stellated dodecahedron are 7, whereas the great dodecahedron and the small stellated dodecahedron are 3.

Duality and Petrie polygons The Kepler–Poinsot polyhedra exist in dual pairs. Duals have the same Petrie polygon, or more precisely, Petrie polygons with the same two-dimensional projection. The following images show the two dual compounds with the same edge radius. They also show that the Petrie polygons are skew. Two relationships described in the article below are also easily seen in the images: That the violet edges are the same, and that the green faces lie in the same planes.

Summary

History

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