Great stellated dodecahedron

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Great stellated dodecahedron
Rank3
TypeRegular
Notation
Bowers style acronymGissid
Coxeter diagramx5/2o3o ()
Schläfli symbol
Elements
Faces12 pentagrams
Edges30
Vertices20
Vertex figureTriangle, edge length (5–1)/2
Petrie polygons6 skew decagrams
Measures (edge length 1)
Circumradius
Edge radius
Inradius
Volume
Dihedral angle
Central density7
Number of external pieces60
Level of complexity3
Related polytopes
ArmyDoe, edge length
RegimentGissid
DualGreat icosahedron
Petrie dualPetrial great stellated dodecahedron
κ ?Petrial dodecahedron
ConjugateDodecahedron
Convex coreDodecahedron
Abstract & topological properties
Flag count120
Euler characteristic2
Schläfli type{5,3}
OrientableYes
Genus0
Properties
SymmetryH3, order 120
Flag orbits1
ConvexNo
NatureTame
History
Discovered byJohannes Kepler[note 1]
First discovered1613

The great stellated dodecahedron, or gissid, is one of the four Kepler–Poinsot solids. It has 12 pentagrams as faces, joining 3 to a vertex.

It is the last stellation of the dodecahedron, from which its name is derived. It is also the only Kepler-Poinsot solid to share its vertices with the dodecahedron as opposed to the icosahedron. It has the smallest circumradius of any uniform polyhedron.

Great stellated dodecahedra appear as cells in two star regular polychora, namely the great stellated hecatonicosachoron and great grand stellated hecatonicosachoron.

Vertex coordinates[edit | edit source]

The vertices of a great stellated dodecahedron of edge length 1, centered at the origin, are all sign changes of

  • ,

along with all even permutations and all sign changes of

  • .

The first set of vertices corresponds to a scaled cube which can be inscribed into the great stellated dodecahedron's vertices.

Related polytopes[edit | edit source]

Alternative realizations[edit | edit source]

PointDodecahedronGreat stellated dodecahedronSkew pure dodecahedronHemidodecahedron (4-dimensional)Hemidodecahedron (5-dimensional)Hemidodecahedron (9-dimensional)Dodecahedron (6-dimensional)Dodecahedron (cross-polytope realization)
Symmetric realizations of {5,3}. Click on a node to be taken to the page for that realization.

The dodecahedron and the great stellated dodecahedron are conjugates. Thus they are realizations of the same underlying abstract regular polytope {5,3}. These are the only faithful symmetric realizations of this polytope in 3-dimensional Euclidean space, however there are many more skew faithful symmetric realizations. In total there are 28 faithful symmetric realizations, of which 3 are pure.

Faithful symmetric realizations of {5,3}
Dimension Components Name
3 Dodecahedron Dodecahedron
3 Great stellated dodecahedron Great stellated dodecahedron
4 Skew pure dodecahedron Skew pure dodecahedron
6 Dodecahedron
Great stellated dodecahedron
Dodecahedron (6-dimensional)
7 Dodecahedron
Skew pure dodecahedron
7 Great stellated dodecahedron
Skew pure dodecahedron
7 Dodecahedron
Hemidodecahedron (4-dimensional)
7 Great stellated dodecahedron
Hemidodecahedron (4-dimensional)
8 Dodecahedron
Hemidodecahedron (5-dimensional)
8 Great stellated dodecahedron
Hemidodecahedron (5-dimensional)
8 Skew pure dodecahedron
Hemidodecahedron (4-dimensional)
10 Dodecahedron
Great stellated dodecahedron
Skew pure dodecahedron
Dodecahedron (cross-polytope realization)
10 Dodecahedron
Great stellated dodecahedron
Hemidodecahedron (4-dimensional)
11 Dodecahedron
Skew pure dodecahedron
Hemidodecahedron (4-dimensional)
11 Great stellated dodecahedron
Skew pure dodecahedron
Hemidodecahedron (4-dimensional)
11 Dodecahedron
Great stellated dodecahedron
Hemidodecahedron (5-dimensional)
12 Dodecahedron
Skew pure dodecahedron
Hemidodecahedron (5-dimensional)
12 Great stellated dodecahedron
Skew pure dodecahedron
Hemidodecahedron (5-dimensional)
12 Dodecahedron
Hemidodecahedron (4-dimensional)
Hemidodecahedron (5-dimensional)
12 Great stellated dodecahedron
Hemidodecahedron (4-dimensional)
Hemidodecahedron (5-dimensional)
13 Skew pure dodecahedron
Hemidodecahedron (4-dimensional)
Hemidodecahedron (5-dimensional)
14 Dodecahedron
Great stellated dodecahedron
Skew pure dodecahedron
Hemidodecahedron (4-dimensional)
15 Dodecahedron
Great stellated dodecahedron
Hemidodecahedron (4-dimensional)
Hemidodecahedron (5-dimensional)
15 Dodecahedron
Great stellated dodecahedron
Skew pure dodecahedron
Hemidodecahedron (5-dimensional)
16 Dodecahedron
Skew pure dodecahedron
Hemidodecahedron (4-dimensional)
Hemidodecahedron (5-dimensional)
16 Great stellated dodecahedron
Skew pure dodecahedron
Hemidodecahedron (4-dimensional)
Hemidodecahedron (5-dimensional)
19 Dodecahedron
Great stellated dodecahedron
Skew pure dodecahedron
Hemidodecahedron (4-dimensional)
Hemidodecahedron (5-dimensional)

In vertex figures[edit | edit source]

The great stellated dodecahedron appears as a vertex figure of one Schläfli–Hess polychoron.

Name Picture Schläfli symbol Edge length
Great grand hecatonicosachoron
{5,5/2,3}

External links[edit | edit source]

Notes[edit | edit source]

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