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)

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.

Related polyhedra

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.

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 Great dodecahedron Great stellated dodecahedron Stellations of the dodecahedron Platonic solid Kepler–Poinsot solids        Name Small stellated dodecahedron Dodecadodecahedron Truncated
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Coxeter-Dynkin
diagram

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