Great dodecahedron: Difference between revisions
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In [[geometry]], the '''great dodecahedron''' is a [[Kepler-Poinsot polyhedra|Kepler-Poinsot polyhedron]]</sub>, with [[Schläfli symbol]] {5,5/2} and [[Coxeter-Dynkin diagram]] of {{CDD|node_1|5|node|5|rat|d2|node}}. It is one of four [[nonconvex]] [[List_of_regular_polytopes#Non-convex_2|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 [[pentagram]]mic path. |
In [[geometry]], the '''great dodecahedron''' is a [[Kepler-Poinsot polyhedra|Kepler-Poinsot polyhedron]]</sub>, with [[Schläfli symbol]] {5,5/2} and [[Coxeter-Dynkin diagram]] of {{CDD|node_1|5|node|5|rat|d2|node}}. It is one of four [[nonconvex]] [[List_of_regular_polytopes#Non-convex_2|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 [[pentagram]]mic path. |
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Revision as of 18:23, 8 December 2011
| Great dodecahedron | |
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| 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 |     
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| Symmetry group | Ih, H3, [5,3], (*532) |
| References | U35, C44, W21 |
| Properties | Regular nonconvex |
 (55)/2 (Vertex figure) |
 Small stellated dodecahedron (dual polyhedron) |
i HATE HILDA SHE IS AN APPLE
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.
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]. |
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.
| Name | Small stellated dodecahedron | Truncated small stellated dodecahedron | Dodecadodecahedron | Truncated great dodecahedron |
Great dodecahedron |
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Usage
- This shape was the basis for the Rubik's Cube-like Alexander's Star puzzle.
See also
External links
- Weisstein, Eric W., "Great dodecahedron" ("Uniform polyhedron") at MathWorld.
- Uniform polyhedra and duals
- Metal sculpture of Great Dodecahedron
| Stellations of the dodecahedron | ||||||
| Platonic solid | Kepler–Poinsot solids | |||||
| Dodecahedron | Small stellated dodecahedron | Great dodecahedron | Great stellated dodecahedron | |||
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