Small stellated dodecahedron

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Small stellated dodecahedron
Small stellated dodecahedron.png
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 CDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node 1.png
Symmetry group Ih, H3, [5,3], (*532)
References U34, C43, W20
Properties Regular nonconvex
Small stellated dodecahedron vertfig.png
(5/2)5
(Vertex figure)
Great dodecahedron.png
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[edit]

Transparent model Handmade models
SmallStellatedDodecahedron.jpg
(See also: animated)
Small Stellated Dodecahedron 1.jpg Small Stellated Dodecahedron 2.jpg
Spherical tiling Stellation net
Small stellated dodecahedron tiling.png
This polyhedron also represents a spherical tiling with a density of 3. (One spherical pentagram face, outlined in blue, filled in yellow)
First stellation of dodecahedron facets.svg
It can also be constructed as the first of three stellations of the dodecahedron, and referenced as Wenninger model [W20].
Small Stellated Dodecahedron Net.svg
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[edit]

Floor mosaic by Paolo Uccello, 1430

Related polyhedra[edit]

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
CDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png
Picture Small stellated dodecahedron.png Dodecahedron.png Dodecadodecahedron.png Great truncated dodecahedron.png Great dodecahedron.png

References[edit]

See also[edit]

External links[edit]

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