Historic Mathematical Facts about The Great Stellated Dodecahedron

Great Stellated Dodecahedron


The great stellated dodecahedron is one of the Kepler-Poinsot solids. It is also uniform polyhedron , Wenninger model , and is the third dodecahedron stellation (Wenninger 1989). Its dual is the great icosahedron. The great stellated dodecahedron has Schläfli symbol and Wythoff symbol . It has 12 pentagrammic faces.
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.
Its circumradius for unit edge length is
(1)
(2)
where is the golden ratio.
The skeleton of the great stellated dodecahedron is isomorphic to the dodecahedral graph.
The great stellated dodecahedron can be constructed from a 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.
Another way to construct a great stellated dodecahedron via cumulation 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 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)
SEE ALSO: Dodecahedron, Dodecahedron Stellations, Great Dodecahedron, Great Icosahedron, Great Stellated Truncated Dodecahedron, Kepler-Poinsot Solid, Small Stellated Dodecahedron, Spiky, Stellation. [Pages Linking Here]