# 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 {5,52}
Face configuration V(52)5
Wythoff symbol 52 | 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), 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

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

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    