Great stellated dodecahedron

Great stellated dodecahedron
File:Great stellated dodecahedron.png
Type	Kepler–Poinsot polyhedron
Stellation core	regular dodecahedron
Elements	F = 12, E = 30 V = 20 (χ = 2)
Faces by sides	12 { 5⁄2 }
Schläfli symbol	{5⁄2,3}
Face configuration	V(35)/2
Wythoff symbol	3 | 2 5⁄2
Coxeter diagram	File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel rat.pngFile:CDel d2.pngFile:CDel node 1.png
Symmetry group	Ih, H3, [5,3], (*532)
References	U52, C68, W22
Properties	Regular nonconvex
File:Great stellated dodecahedron vertfig.png (5⁄2)3 (Vertex figure)	File:Great icosahedron.png Great icosahedron (dual polyhedron)

File:Great stellated dodecahedron.stl

3D model of a 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.

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.

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
File:GreatStellatedDodecahedron.jpg Transparent great stellated dodecahedron (Animation)	File:Great stellated dodecahedron tiling.svg 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
File:Great stellated dodecahedron net.png × 20 A net of a great stellated dodecahedron (surface geometry); twenty isosceles triangular pyramids, arranged like the faces of an icosahedron.	File: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 the surface geometry of a great stellated dodecahedron. Making a net with the actual pentagrams that make up the polyhedron would self intersect even if layed out flat.

Formulas

For a great stellated dodecahedron with edge length E (where E represents the length of any edge of the internal icosahedron),

$$\text{Inradius}=\frac{\text{E}(\sqrt{5}-1)}{2}$$

$$\text{Midradius}=\frac{\text{E}(1+\sqrt{5})}{4}$$

$$\text{Circumradius}=\frac{\text{E}(3+\sqrt{5})\sqrt{3}}{4}$$

$$\text{Surface Area}=15\sqrt{5+2\sqrt{5}}{\text{E}}^2$$

$$\text{Volume}=\frac{5(3+\sqrt{5}){\text{E}}^3}{4}$$

Related polyhedra

File:Great stellated dodecahedron truncations.gif

Animated truncation sequence from {5⁄2, 3} to {3, 5⁄2}

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.

Stellations of the dodecahedron
Platonic solid	Kepler–Poinsot solids
Dodecahedron	Small stellated dodecahedron	Great dodecahedron	Great stellated dodecahedron
File:Zeroth stellation of dodecahedron.svg	File:First stellation of dodecahedron.svg	File:Second stellation of dodecahedron.svg	File:Third stellation of dodecahedron.svg
File:Zeroth stellation of dodecahedron facets.svg	File:First stellation of dodecahedron facets.svg	File:Second stellation of dodecahedron facets.svg	File:Third stellation of dodecahedron facets.svg

Name	Great stellated dodecahedron	Truncated great stellated dodecahedron	Great icosidodecahedron	Truncated great icosahedron	Great icosahedron
Coxeter-Dynkin diagram	File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel rat.pngFile:CDel d2.pngFile:CDel node 1.png	File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel rat.pngFile:CDel d2.pngFile:CDel node 1.png	File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel rat.pngFile:CDel d2.pngFile:CDel node.png	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel rat.pngFile:CDel d2.pngFile:CDel node.png	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel rat.pngFile:CDel d2.pngFile:CDel node.png
Picture	File:Great stellated dodecahedron.png	File:Icosahedron.png	File:Great icosidodecahedron.png	File:Great truncated icosahedron.png	File:Great icosahedron.png

External links

• Weisstein, Eric W., "Great stellated dodecahedron" ("Uniform polyhedron") at MathWorld.
• Uniform polyhedra and duals