Great dodecahedron

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Great dodecahedron

Type	Kepler-Poinsot polyhedron
Stellation core	dodecahedron
Elements	F = 12, E = 30 V = 12 (χ = -6)
Faces by sides	12{5}
Schläfli symbol	{5,⁵/₂}
Wythoff symbol	⁵/₂ \| 2 5
Coxeter-Dynkin
Symmetry group	I_h, [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.

[edit] 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
Net for surface geometry	It can also be constructed as the second of three stellations of the dodecahedron, and referenced as Wenninger model [W21].

[edit] Related polyhedra

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.

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.

The truncated small stellated dodecahedron looks like a dodecahedron on the surface, but it has 24 pentagonal faces: 12 as the truncation facets of the former vertices, and 12 more (coinciding with the first set) as truncated pentagrams.