Great stellated dodecahedron

In geometry, the great stellated dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol {5⁄2,3}.

It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at each vertex.

Shaving the triangular pyramids off results in an icosahedron.

If the triangles are instead made to invert themselves and excavate the central icosahedron, the result is a great dodecahedron.

The great stellated dodecahedron can be constructed analogously to the pentagram, its two-dimensional analogue, by attempting to stellate the n-dimensional pentagonal polytope (which has pentagonal polytope faces and simplex vertex figures) until it can no longer be stellated; that is, it is its final stellation.

For a great stellated dodecahedron with edge length E,

A truncation process applied to the great stellated dodecahedron produces a series of uniform polyhedra.

Truncating edges down to points produces the great icosidodecahedron as a rectified great stellated dodecahedron.

The process completes as a birectification, reducing the original faces down to points, and producing the great icosahedron.

The truncated great stellated dodecahedron is a degenerate polyhedron, with 20 triangular faces from the truncated vertices, and 12 (hidden) pentagonal faces as truncations of the original pentagram faces, the latter forming a great dodecahedron inscribed within and sharing the edges of the icosahedron.