Great Stellated Dodecahedron

The great stellated dodecahedron is one of the Kepler-Poinsot polyhedra. It is also the uniform polyhedron with Maeder index 52 (Maeder 1997), Wenninger index 22 (Wenninger 1989), Coxeter index 68 (Coxeter et al. 1954), and Har'El index 57 (Har'El 1993). It is the third dodecahedron stellation (Wenninger 1989). The great stellated dodecahedron has Schläfli symbol and Wythoff symbol . It has 12 pentagrammic faces.

Its dual is the great icosahedron.

The great stellated dodecahedron was published by Wenzel Jamnitzer in 1568. It was rediscovered by Kepler (and published in his work Harmonice Mundi in 1619), and again by Poinsot in 1809.

The great stellated dodecahedron is implemented in the Wolfram Language as UniformPolyhedron["GreatStellatedDodecahedron"]. Precomputed properties are available as PolyhedronData["GreatStellatedDodecahedron", prop].

The great stellated dodecahedron can be constructed from a unit dodecahedron by selecting the 144 sets of five coplanar vertices, then discarding sets whose edges correspond to the edges of the original dodecahedron. This gives 12 pentagrams of edge length , where is the golden ratio. Rescaling to give the pentagrams unit edge lengths, the circumradius of the great stellated dodecahedron is

			(1)
			(2)

where is the golden ratio.

The skeleton of the great stellated dodecahedron is isomorphic to the dodecahedral graph.

Another way to construct a great stellated dodecahedron via augmentation is to make 20 triangular pyramids with side length (the golden ratio) times the base, as illustrated above, and attach them to the sides of an icosahedron. The height of these pyramids is then .

Cumulating a unit dodecahedron to construct a great stellated dodecahedron produces a solid with edge lengths

			(3)
			(4)

The surface area and volume of such a great stellated dodecahedron are

			(5)
			(6)

The convex hull of the great stellated dodecahedron is a regular dodecahedron and the dual of the dodecahedron is the icosahedron, so the dual of the great stellated dodecahedron (i.e., the great icosahedron) is one of the icosahedron stellations (Wenninger 1983, p. 40)