Small stellated dodecahedron

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Small stellated dodecahedron
Small stellated dodecahedron.png
Rank3
TypeRegular
SpaceSpherical
Bowers style acronymSissid
Info
Coxeter diagramx5/2o5o
Schläfli symbol{5/2,5}
SymmetryH3, order 120
ArmyIke
RegimentSissid
Elements
Vertex figurePentagon, edge length (5–1)/2
Faces12 pentagrams
Edges30
Vertices12
Measures (edge length 1)
Circumradius
Edge radius
Inradius
Volume
Dihedral angle
Central density3
Euler characteristic-6
Number of pieces60
Level of complexity3
Related polytopes
DualGreat dodecahedron
ConjugateGreat dodecahedron
Convex coreDodecahedron
Properties
ConvexNo
OrientableYes
NatureTame

The small stellated dodecahedron, or sissid, is one of the four Kepler-Poinsot solids. It has 12 pentagrams as faces, joining 5 to a vertex.

It is the first stellation of a dodecahedron, from which its name is derived.

Vertex coordinates[edit | edit source]

The vertices of a small stellated dodecahedron of edge length 1, centered at the origin, are all cyclic permutations of

In vertex figures[edit | edit source]

The small stellated dodecahedron appears as a vertex figure of two Schläfli–Hess polychora.

Name Picture Schläfli symbol Edge length
Great faceted hexacosichoron
Gishi.png
{3,5/2,5}
Great hecatonicosachoron
Schlegel wireframe 600-cell vertex-centered.png
{5,5/2,5}

Related polyhedra[edit | edit source]

The small stellated dodecahedron is the colonel of a two-member regiment that also includes the great icosahedron.

Two uniform polyhedron compounds are composed of small stellated dodecahedra:

o5o5/2o truncations
Name OBSA Schläfli symbol CD diagram Picture
Great dodecahedron gad {5,5/2} x5o5/2o
Great dodecahedron.png
Truncated great dodecahedron tigid t{5,5/2} x5x5/2o
Great truncated dodecahedron.png
Dodecadodecahedron did r{5,5/2} o5x5/2o
Dodecadodecahedron.png
Truncated small stellated dodecahedron (degenerate, triple cover of doe) t{5/2,5} o5x5/2x
Dodecahedron.png
Small stellated dodecahedron sissid {5/2,5} o5o5/2x
Small stellated dodecahedron.png
Rhombidodecadodecahedron raded rr{5,5/2} x5o5/2x
Rhombidodecadodecahedron.png
Truncated dodecadodecahedron (degenerate, sird+12(10/2)) tr{5,5/2} x5x5/2x
Snub dodecadodecahedron siddid sr{5,5/2} s5s5/2s
Snub dodecadodecahedron.png

External links[edit | edit source]