Small stellated dodecahedron

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Small stellated dodecahedron
Small stellated dodecahedron.png
Rank3
TypeRegular
SpaceSpherical
Notation
Bowers style acronymSissid
Coxeter diagramx5/2o5o (CDel node 1.pngCDel 5.pngCDel rat.pngCDel 2x.pngCDel node.pngCDel 5.pngCDel node.png)
Schläfli symbol
[1]
Elements
Faces12 pentagrams
Edges30
Vertices12
Vertex figurePentagon, edge length (5–1)/2
Small stellated dodecahedron vertfig.png
Measures (edge length 1)
Circumradius
Edge radius
Inradius
Volume
Dihedral angle
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[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 (CDel node 1.pngCDel 5.pngCDel node.pngCDel 5-2.pngCDel node.png)
Great dodecahedron.png
Truncated great dodecahedron tigid t{5,5/2} x5x5/2o (CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 5-2.pngCDel node.png)
Great truncated dodecahedron.png
Dodecadodecahedron did r{5,5/2} o5x5/2o (CDel node.pngCDel 5.pngCDel node 1.pngCDel 5-2.pngCDel node.png)
Dodecadodecahedron.png
Truncated small stellated dodecahedron (degenerate, triple cover of doe) t{5/2,5} o5x5/2x (CDel node.pngCDel 5.pngCDel node 1.pngCDel 5-2.pngCDel node 1.png)
Dodecahedron.png
Small stellated dodecahedron sissid {5/2,5} o5o5/2x (CDel node.pngCDel 5.pngCDel node.pngCDel 5-2.pngCDel node 1.png)
Small stellated dodecahedron.png
Rhombidodecadodecahedron raded rr{5,5/2} x5o5/2x (CDel node 1.pngCDel 5.pngCDel node.pngCDel 5-2.pngCDel node 1.png)
Rhombidodecadodecahedron.png
Truncated dodecadodecahedron (degenerate, sird+12(10/2)) tr{5,5/2} x5x5/2x (CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 5-2.pngCDel node 1.png)
Snub dodecadodecahedron siddid sr{5,5/2} s5s5/2s (CDel node h.pngCDel 5.pngCDel node h.pngCDel 5-2.pngCDel node h.png)
Snub dodecadodecahedron.png

External links[edit | edit source]

References[edit | edit source]

Bibliography[edit | edit source]

  • McMullen, Peter; Schulte, Egon (1997). "Regular Polytopes in Ordinary Space" (PDF). Discrete Computational Geometry (47): 449–478. doi:10.1007/PL00009304.