# Great stellated dodecahedron

Great stellated dodecahedron
Rank3
TypeRegular
Notation
Bowers style acronymGissid
Coxeter diagramx5/2o3o ()
Schläfli symbol${\displaystyle \{5/2,3\}}$
Elements
Faces12 pentagrams
Edges30
Vertices20
Vertex figureTriangle, edge length (5–1)/2
Petrie polygons6 skew decagrams
Measures (edge length 1)
Circumradius${\displaystyle {\frac {{\sqrt {15}}-{\sqrt {3}}}{4}}\approx 0.53523}$
Edge radius${\displaystyle {\frac {3-{\sqrt {5}}}{4}}\approx 0.19098}$
Inradius${\displaystyle {\sqrt {\frac {25-11{\sqrt {5}}}{40}}}\approx 0.10041}$
Volume${\displaystyle {\frac {7{\sqrt {5}}-15}{4}}\approx 0.16312}$
Dihedral angle${\displaystyle \arccos \left({\frac {\sqrt {5}}{5}}\right)\approx 63.43495^{\circ }}$
Central density7
Number of external pieces60
Level of complexity3
Related polytopes
ArmyDoe, edge length ${\displaystyle {\frac {3-{\sqrt {5}}}{4}}}$
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

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

• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {{\sqrt {5}}-1}{4}}\right)}$,

along with all even permutations and all sign changes of

• ${\displaystyle \left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,0\right)}$.

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

## Related polytopes

### Alternative realizations

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.

### In vertex figures

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} ${\displaystyle {\frac {1+{\sqrt {5}}}{2}}}$