Great dodecahedron
In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol and Coxeter–Dynkin diagram of. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces, 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 -D pentagonal polytope faces of the core nD polytope until the figure again closes.
Images
| Transparent model | Spherical tiling |
This polyhedron represents a spherical tiling with a density of 3. | |
| 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 List of Wenninger polyhedron models#Stellations of dodecahedron|Wenninger model . |
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.
| Name | Small stellated dodecahedron | Dodecadodecahedron | Truncated great dodecahedron | Great dodecahedron |
| Coxeter-Dynkin diagram | ||||
| Picture |
Usage
- This shape was the basis for the Rubik's Cube-like Alexander's Star puzzle.
- The great dodecahedron provides an easy mnemonic for the binary Golay code