| 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 | Ih, [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.
Contents |
| 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]. |
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.
| Name | Small stellated dodecahedron | Truncated small stellated dodecahedron | Dodecadodecahedron | Truncated great dodecahedron |
Great dodecahedron |
|---|---|---|---|---|---|
| Coxeter-Dynkin diagram |
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| Picture |
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| Stellations of the dodecahedron | ||||||
| Platonic solid | Kepler-Poinsot solids | |||||
| Dodecahedron | Small stellated dodecahedron | Great dodecahedron | Great stellated dodecahedron | |||
|---|---|---|---|---|---|---|