A | B | C | D | E | F | G | H | CH | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
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 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Symmetry group | Ih, H3, , (*532) |
References | U35, C44, W21 |
Properties | Regular nonconvex |
![]() (55)/2 (Vertex figure) |
![]() Small stellated dodecahedron (dual polyhedron) |
![](http://upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Great_dodecahedron%28full%29.stl/220px-Great_dodecahedron%28full%29.stl.png)
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 |
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![]() (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 . |
Formulas
For a great dodecahedron with edge length E,
Related polyhedra
![](http://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Small_stellated_dodecahedron_truncations.gif/240px-Small_stellated_dodecahedron_truncations.gif)
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.
Stellations of the dodecahedron | ||||||
Platonic solid | Kepler–Poinsot solids | |||||
Dodecahedron | Small stellated dodecahedron | Great dodecahedron | Great stellated dodecahedron | |||
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Name | Small stellated dodecahedron | Dodecadodecahedron | Truncated great dodecahedron |
Great dodecahedron |
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Coxeter-Dynkin diagram |
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Picture | ![]() |
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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[1]
See also
References
- ^ * Baez, John "Golay code," Visual Insight, December 1, 2015.
External links
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