Great dodecahedron

Short description: Kepler-Poinsot polyhedron

Template:Reg star polyhedron stat table File:Great dodecahedron(full).stl

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), intersecting each other making a pentagrammic path, with five pentagons meeting at each vertex.

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.

The great dodecahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the (n – 1)-pentagonal polytope faces of the core n-polytope (pentagons for the great dodecahedron, and line segments for the pentagram) until the figure again closes.

Images

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].

Related polyhedra

Animated truncation sequence from {5/2, 5} to {5, 5/2}

It shares the same edge arrangement as the convex regular icosahedron; the compound with both is the small complex icosidodecahedron.

If only the visible surface is considered, 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, although this result is not regular.

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

Usage

• 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

See also

• Compound of small stellated dodecahedron and great dodecahedron

External links

• Eric W. Weisstein, Great dodecahedron (Uniform polyhedron) at MathWorld.
• Weisstein, Eric W.. "Three dodecahedron stellations". http://mathworld.wolfram.com/DodecahedronStellations.html. 
• Uniform polyhedra and duals
• Metal sculpture of Great Dodecahedron