# Small stellated dodecahedron

(Redirected from Sissid)
Small stellated dodecahedron Rank3
TypeRegular
SpaceSpherical
Notation
Bowers style acronymSissid
Coxeter diagramx5/2o5o (       )
Schläfli symbol$\{5/2,5\}$ $\{5,5\mid 3\}$ Elements
Faces12 pentagrams
Edges30
Vertices12
Vertex figurePentagon, edge length (5–1)/2 Measures (edge length 1)
Circumradius$\sqrt{\frac{5-\sqrt5}{8}} ≈ 0.58779$ Edge radius$\frac{\sqrt5-1}{4} ≈ 0.30902$ Inradius$\sqrt{\frac{5-\sqrt5}{40}} ≈ 0.26287$ Volume$\frac{3\sqrt5-5}{4} ≈ 0.42705$ Dihedral angle$\arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505^\circ$ Central density3
Number of pieces60
Level of complexity3
Related polytopes
ArmyIke
RegimentSissid
DualGreat dodecahedron
Petrie dualPetrial small stellated dodecahedron
ConjugateGreat dodecahedron
Convex coreDodecahedron
Abstract properties
Flag count120
Euler characteristic-6
Schläfli type{5,5}
Topological properties
SurfaceBring's surface
OrientableYes
Genus4
Properties
SymmetryH3, order 120
ConvexNo
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

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

• $\left(0,\,±\frac{1}{2},\,±\frac{\sqrt{5}-1}{4}\right).$ ## In vertex figures

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

Name Picture Schläfli symbol Edge length
Great faceted hexacosichoron {3,5/2,5} $1$ Great hecatonicosachoron {5,5/2,5} $\frac{1+\sqrt{5}}{2}$ ## Related polyhedra

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 (     )
Truncated great dodecahedron tigid t{5,5/2} x5x5/2o (     )
Dodecadodecahedron did r{5,5/2} o5x5/2o (     )
Truncated small stellated dodecahedron (degenerate, triple cover of doe) t{5/2,5} o5x5/2x (     )
Small stellated dodecahedron sissid {5/2,5} o5o5/2x (     )
Rhombidodecadodecahedron raded rr{5,5/2} x5o5/2x (     )
Truncated dodecadodecahedron (degenerate, sird+12(10/2)) tr{5,5/2} x5x5/2x (     )
Snub dodecadodecahedron siddid sr{5,5/2} s5s5/2s (     )