Small stellated dodecahedron

Small stellated dodecahedron
Type	Kepler-Poinsot polyhedron
Stellation core	dodecahedron
Elements	F = 12, E = 30 V = 12 (χ = -6)
Faces by sides	12{5/2}
Schläfli symbol	{5/2,5}
Wythoff symbol	5 | 25/2
Coxeter-Dynkin
Symmetry group	Ih, H3, [5,3], (*532)
References	U34, C43, W20
Properties	Regular nonconvex
(5/2)5 (Vertex figure)	Great dodecahedron (dual polyhedron)

In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol {5/2,5}. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex.

It shares the same vertex arrangement as the convex regular icosahedron. It also shares the same edge arrangement with the great icosahedron.

It is considered the first of three stellations of the dodecahedron.

If the pentagrammic faces are considered as 5 triangular faces, it shares the same surface topology as the pentakis dodecahedron, but with much taller isosceles triangle faces.

Images

Transparent model	Handmade models
(See also: animated)
Spherical tiling	Stellation	Net
This polyhedron also represents a spherical tiling with a density of 3. (One spherical pentagram face, outlined in blue, filled in yellow)	It can also be constructed as the first of three stellations of the dodecahedron, and referenced as Wenninger model [W20].	× 12 Small stellated dodecahedra can be constructed out of paper or cardstock by connecting together 12 five-sided isosceles pyramids in the same manner as the pentagons in a regular dodecahedron. With an opaque material, this visually represents the exterior portion of each pentagrammic face.

In art

Floor mosaic by Paolo Uccello, 1430

• It can also be seen in a floor mosaic in St Mark's Basilica, Venice by Paolo Uccello circa 1430.
• It is central to two lithographs by M. C. Escher: Contrast (Order and Chaos) (1950) and Gravitation (1952).

Related polyhedra

Its convex hull is the regular convex icosahedron. It also shares its edges with the great icosahedron.

This polyhedron is the truncation of the great dodecahedron:

The truncated small stellated dodecahedron looks like a dodecahedron on the surface, but it has 24 faces: 12 pentagons from the truncated vertices and 12 overlapping (as truncated pentagrams).

Name	Small stellated dodecahedron	Truncated small stellated dodecahedron	Dodecadodecahedron	Truncated great dodecahedron	Great dodecahedron
Coxeter-Dynkin diagram
Picture

References

• Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9. 
• Coxeter, H. S. M. (1938). The Fifty-Nine Icosahedra. Springer-Verlag, New York, Berlin, Heidelberg. ISBN 0-387-90770-X. 

See also

• Compound of small stellated dodecahedron and great dodecahedron

External links

• Eric W. Weisstein, Small stellated dodecahedron (Uniform polyhedron) at MathWorld
  • Weisstein, Eric W., "DodecahedronStellations", MathWorld.
• Uniform polyhedra and duals
• Bronze sculpture of small stellated dodecahedron

Stellations of the dodecahedron
Platonic solid	Kepler-Poinsot solids
Dodecahedron	Small stellated dodecahedron	Great dodecahedron	Great stellated dodecahedron