Great Dodecahedron
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Great Dodecahedron
Great dodecahedron
Type Kepler-Poinsot polyhedron
Stellation core regular dodecahedron
Elements F = 12, E = 30
V = 12 (? = -6)
Faces by sides 12{5}
Schläfli symbol {5,​}
Face configuration V(​)5
Wythoff symbol ​ | 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), 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.

The great dodecahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the (n-1)-D pentagonal polytope faces of the core nD 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

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.

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

## References

1. ^ * Baez, John "Golay code," Visual Insight, December 1, 2015.

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.