Английская Википедия:Great dodecahedron
Шаблон:Short description Шаблон:Reg polyhedra db
In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol Шаблон:Math and Coxeter–Dynkin diagram of Шаблон:CDD. 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 Шаблон:Math-pentagonal polytope faces of the core Шаблон:Math-polytope (pentagons for the great dodecahedron, and line segments for the pentagram) until the figure again closes.
Images
| Transparent model | Spherical tiling |
|---|---|
| Файл:GreatDodecahedron.jpg (With animation) |
Файл:Great dodecahedron tiling.svg This polyhedron represents a spherical tiling with a density of 3. (One spherical pentagon face is shown above in yellow) |
| Net | Stellation |
| Шаблон:Nowrap Net for surface geometry; twenty isosceles triangular pyramids, arranged like the faces of an icosahedron |
Файл:Second stellation of dodecahedron facets.svg It can also be constructed as the second of three stellations of the dodecahedron, and referenced as Wenninger model [W21]. |
Formulas
For a great dodecahedron with edge length E,
<math display=block>\text{Circumradius} = {\tfrac{E}{4}}\Bigl(\sqrt{10+2+\sqrt{5}}\Bigr)</math>
<math display=block>\text{Surface Area} = 15\Bigl(\sqrt{5-2\sqrt{5}}\Bigr)E^2</math>
<math display=block>\text{Volume} = {\tfrac{5}{4}}(\sqrt{5}-1)E^3</math>
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.
Шаблон:Dodecahedron stellations
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
External links
- Шаблон:Mathworld2
- Шаблон:Mathworld
- Uniform polyhedra and duals
- Metal sculpture of Great Dodecahedron
Шаблон:Star polyhedron navigator
- ↑ * Baez, John "Golay code," Visual Insight, December 1, 2015.
