Great dodecahedron: Difference between revisions

Great dodecahedron
Type	Kepler–Poinsot polyhedron
Stellation core	regular dodecahedron
Elements	F = 12, E = 30 V = 12 (χ = -6)
Faces by sides	12{5}
Schläfli symbol	{5,5⁄2}
Face configuration	V(5⁄2)5
Wythoff symbol	5⁄2 | 2 5
Coxeter diagram
Symmetry group	Ih, H3, [5,3], (*532)
References	U35, C44, W21
Properties	Regular nonconvex
(55)/2 (Vertex figure)	Small stellated dodecahedron (dual polyhedron)

In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol {5,5/2} and Coxeter–Dynkin diagram of . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagons), with five pentagons meeting at each vertex, intersecting each other making a pentagrammic path.

The discovery of the great dodecahedron is sometimes credited to Louis Poinsot in 1810, though there is a drawing of something very similar to a great dodecahedron in the 1568 book Perspectiva Corporum Regularium by Wenzel Jamnitzer.

Transparent model	Spherical tiling
(With animation)	This polyhedron represents a spherical tiling with a density of 3. (One spherical pentagon face is shown above in yellow)
Net	Stellation
× 20 Net for surface geometry; twenty isosceles triangular pyramids, arranged like the faces of an icosahedron	It can also be constructed as the second of three stellations of the dodecahedron, and referenced as Wenninger model [W21].

It shares the same edge arrangement as the convex regular icosahedron.

If the great dodecahedron is considered as a properly intersected surface geometry, it has the same topology as a triakis icosahedron with concave pyramids rather than convex ones. The excavated dodecahedron can be seen as the same process applied to a regular dodecahedron.

A truncation process applied to the great dodecahedron produces a series of nonconvex uniform polyhedra. Truncating edges down to points produces the dodecadodecahedron as a rectified great dodecahedron. The process completes as a birectification, reducing the original faces down to points, and producing the small stellated dodecahedron.

Name	Small stellated dodecahedron	Dodecadodecahedron	Truncated great dodecahedron	Great dodecahedron
Coxeter-Dynkin diagram
Picture

• This shape was the basis for the Rubik's Cube-like Alexander's Star puzzle.
• The great dodecahedron provides an easy mnemonic for the binary Golay code^[1]

• Compound of small stellated dodecahedron and great dodecahedron

• Weisstein, Eric W., "Great dodecahedron" ("Uniform polyhedron") at MathWorld.
• Weisstein, Eric W. "Three dodecahedron stellations". MathWorld.
• Uniform polyhedra and duals
• Metal sculpture of Great Dodecahedron

Stellations of the dodecahedron
Platonic solid	Kepler–Poinsot solids
Dodecahedron	Small stellated dodecahedron	Great dodecahedron	Great stellated dodecahedron