Great Dodecahedron
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Great Dodecahedron
Great dodecahedron
Great dodecahedron.png
Type Kepler-Poinsot polyhedron
Stellation core regular dodecahedron
Elements F = 12, E = 30
V = 12 (? = -6)
Faces by sides 12{5}
Schläfli symbol {5,​}
Face configuration V(​)5
Wythoff symbol ​ | 2 5
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png
Symmetry group Ih, H3, [5,3], (*532)
References U35, C44, W21
Properties Regular nonconvex
Great dodecahedron vertfig.png
(55)/2
(Vertex figure)
Small stellated dodecahedron.png
Small stellated dodecahedron
(dual polyhedron)
3D model of a great dodecahedron

In geometry, the great dodecahedron is a Kepler-Poinsot polyhedron, with Schläfli symbol {5,5/2} and Coxeter-Dynkin diagram of CDel node 1.pngCDel 5.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png. 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.

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)-D pentagonal polytope faces of the core nD polytope (pentagons for the great dodecahedron, and line segments for the pentagram) until the figure again closes.

Images

Transparent model Spherical tiling
GreatDodecahedron.jpg
(With animation)
Great dodecahedron tiling.png
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
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].

Related polyhedra

Animated truncation sequence from {5/2, 5} to {5, 5/2}

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.

Usage

See also

References

  1. ^ * Baez, John "Golay code," Visual Insight, December 1, 2015.

External links

  • Eric W. Weisstein, Great dodecahedron (Uniform polyhedron) at MathWorld.
  • Weisstein, Eric W. "Three dodecahedron stellations". MathWorld.
  • Uniform polyhedra and duals
  • Metal sculpture of Great Dodecahedron

  This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Great_dodecahedron
 



 



 
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