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
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
Symmetry group	Ih, H3, [5,3], (*532)
References	U52, C68, W22
Properties	Regular nonconvex
(5⁄2)3 (Vertex figure)	Great icosahedron (dual polyhedron)

3D model of a great stellated dodecahedron

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
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.

Formulas

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

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

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

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

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

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

Related polyhedra

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

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.

• Coxeter, Harold (1954). "Uniform Polyhedra". Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences. 246 (916). Royal Society: 401–450. Bibcode:1954RSPTA.246..401C. doi:10.1098/rsta.1954.0003. JSTOR 91532. S2CID 202575183.

External links

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