Great Stellated Dodecahedron
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Great Stellated Dodecahedron
Great stellated dodecahedron
Great stellated dodecahedron.png
Type Kepler-Poinsot polyhedron
Stellation core regular dodecahedron
Elements F = 12, E = 30
V = 20 (? = 2)
Faces by sides 12 5
Schläfli symbol {​,3}
Face configuration (35)/2
Wythoff symbol 3 | 2 ​
Coxeter diagram CDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node 1.png
Symmetry group Ih, H3, [5,3], (*532)
References U52, C68, W22
Properties Regular nonconvex
Great stellated dodecahedron vertfig.png
(​)3
(Vertex figure)
Great icosahedron.png
Great icosahedron
(dual polyhedron)
3D model of a great stellated dodecahedron

In geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedron, with Schläfli symbol {​,3}. It is one of four nonconvex regular polyhedra.

It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at each vertex.

It shares its vertex arrangement, although not its vertex figure or vertex configuration, with the regular dodecahedron, as well as being a stellation of a (smaller) dodecahedron. It is the only dodecahedral stellation with this property, apart from the dodecahedron itself. Its dual, the great icosahedron, is related in a similar fashion to the icosahedron. It is the only regular star polyhedron with a completely unique edge arrangement not shared by any other regular 3-polytope.

Shaving the triangular pyramids off results in an icosahedron.

If the pentagrammic faces are broken into triangles, it is topologically related to the triakis icosahedron, with the same face connectivity, but much taller isosceles triangle faces. If the triangles are instead made to invert themselves and excavate the central icosahedron, the result is a great dodecahedron.

The great stellated dodecahedron can be constructed analogously to the pentagram, its two-dimensional analogue, by attempting to stellate the n-dimensional pentagonal polytope which has pentagonal polytope faces and simplex vertex figures until it can no longer be stellated; that is, it is its final stellation.

Images

Transparent model Tiling
GreatStellatedDodecahedron.jpg
Transparent great stellated dodecahedron (Animation)
Great stellated dodecahedron tiling.png
This polyhedron can be made as spherical tiling with a density of 7. (One spherical pentagram face is shown above, outlined in blue, filled in yellow)
Net Stellation facets

A net of a great stellated dodecahedron (surface geometry); twenty isosceles triangular pyramids, arranged like the faces of an icosahedron.
Third stellation of dodecahedron facets.svg
It can be constructed as the third of three stellations of the dodecahedron, and referenced as Wenninger model [W22].
Geometric Net of a Great Stellated Dodecahedron
Complete net of a great stellated dodecahedron.

Related polyhedra

Animated truncation sequence from {​, 3} to {3, ​}

A truncation process applied to the great stellated dodecahedron produces a series of uniform polyhedra. Truncating edges down to points produces the great icosidodecahedron as a rectified great stellated dodecahedron. The process completes as a birectification, reducing the original faces down to points, and producing the great icosahedron.

The truncated great stellated dodecahedron is a degenerate polyhedron, with 20 triangular faces from the truncated vertices, and 12 (hidden) pentagonal faces as truncations of the original pentagram faces, the latter forming a great dodecahedron inscribed within and sharing the edges of the icosahedron.

References

  • Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9.

External links


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

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