Great dodecahedron

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)

3D model of a great dodecahedron

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].

Formulas

For a great dodecahedron with edge length E,

$${\text{Circumradius}}={\tfrac {E}{4}}{\Bigl (}{\sqrt {10+2+{\sqrt {5}}}}{\Bigr )}$$

$${\text{Surface Area}}=15{\Bigl (}{\sqrt {5-2{\sqrt {5}}}}{\Bigr )}E^{2}$$

$${\text{Volume}}={\tfrac {5}{4}}({\sqrt {5}}-1)E^{3}$$

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.

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

• 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