Great stellated dodecahedron

In geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedron, with Schläfli symbol {5/2,3}. It is one of four nonconvex regular polyhedra.

Great stellated dodecahedron
TypeKepler–Poinsot polyhedron
Stellation coreregular dodecahedron
ElementsF = 12, E = 30
V = 20 (χ = 2)
Faces by sides125
Schläfli symbol{5/2,3}
Face configuration(35)/2
Wythoff symbol3 | 25/2
Coxeter diagram
Symmetry groupIh, H3, [5,3], (*532)
ReferencesU52, C68, W22
PropertiesRegular nonconvex

(5/2)3
(Vertex figure)

Great icosahedron
(dual polyhedron)

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

It shares its vertex arrangement 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.

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

Transparent great stellated dodecahedron (Animation)

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
× 20
A net of a great stellated dodecahedron (surface geometry); twenty isosceles triangular pyramids, arranged like the faces of an icosahedron

It can be constructed as the third of three stellations of the dodecahedron, and referenced as Wenninger model [W22].

Complete net of a great stellated dodecahedron.

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.

Name Great
stellated
dodecahedron
Truncated great stellated dodecahedron Great
icosidodecahedron
Truncated
great
icosahedron
Great
icosahedron
Coxeter-Dynkin
diagram
Picture

References

  • Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9.
Stellations of the dodecahedron
Platonic solid Kepler–Poinsot solids
Dodecahedron Small stellated dodecahedron Great dodecahedron Great stellated dodecahedron
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.