Small stellated dodecahedron

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Small stellated dodecahedron
Rank3
TypeRegular
Notation
Bowers style acronymSissid
Coxeter diagramx5/2o5o ()
Schläfli symbol
  • {5/2,5}
  • {5,5∣3}[1]
Elements
Faces12 pentagrams
Edges30
Vertices12
Vertex figurePentagon, edge length (5–1)/2
Petrie polygons10 skew hexagons
Holes20 triangles
Measures (edge length 1)
Circumradius
Edge radius
Inradius
Volume
Dihedral angle
Central density3
Number of external pieces60
Level of complexity3
Related polytopes
ArmyIke, edge length
RegimentSissid
DualGreat dodecahedron
Petrie dualPetrial small stellated dodecahedron
φ 2 Great icosahedron
κ ?Petrial icosahedron
ConjugateGreat dodecahedron
Convex coreDodecahedron
Abstract & topological properties
Flag count120
Euler characteristic-6
Schläfli type{5,5}
SurfaceBring's surface[2]
OrientableYes
Genus4
SkeletonIcosahedral graph
Properties
SymmetryH3, order 120
ConvexNo
NatureTame
History
Discovered byJohannes Kepler[note 1]
First discovered1613

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.

Small stellated dodecahedra appear as cells in two nonconvex regular polychora, namely the small stellated hecatonicosachoron and grand stellated hecatonicosachoron.

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

  • .

Related polytopes[edit | edit source]

Alternative realizations[edit | edit source]

A fundamental domain of the small stellated dodecahedron in {5,5}.

The small stellated dodecahedron and the great dodecahedron are conjugates. Thus they are both faithful symmetric realizations of the same abstract regular polytope, {5,5∣3}. This abstract polytope is a quotient of the order-5 pentagonal tiling which tessellates Bring's surface. There are in total 6 faithful symmetric realizations of the underlying abstract polytope. The great dodecahedron and the small stellated dodecahedron are the only pure faithfully symmetric realizations, the others are the results of blending those two along with {5,5∣3}/2.

Faithful symmetric realizations of {5,5∣3}
Dimension Components Name
3 Great dodecahedron Great dodecahedron
3 Small stellated dodecahedron Small stellated dodecahedron
6
8
8
11

Compounds[edit | edit source]

Two uniform polyhedron compounds are composed of small stellated dodecahedra:

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
{3,5/2,5}
Great hecatonicosachoron
{5,5/2,5}

External links[edit | edit source]

Notes[edit | edit source]

  1. Earlier authors drew shapes the Small stellated dodecahedron or similar shapes earlier, however Kepler was the first to recognize the Small stellated dodecahedron as regular, and explicitly describe it.

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.
  • Weber, Matthias (2005). "Kepler's small stellated dodecahedron as a Riemann surface" (PDF). Pacific Journal of Mathematics. 220 (1): 167–182. doi:10.2140/pjm.2005.220.167.