# 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 as 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

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].

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.
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## In art

Floor mosaic by Paolo Uccello, 1430
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## 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
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## References

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