# Great stellated dodecahedron

Great stellated dodecahedron Rank3
TypeRegular
SpaceSpherical
Notation
Bowers style acronymGissid
Coxeter diagramx5/2o3o (       )
Schläfli symbol$\{5/2,3\}$ Elements
Faces12 pentagrams
Edges30
Vertices20
Vertex figureTriangle, edge length (5–1)/2 Measures (edge length 1)
Circumradius$\frac{\sqrt{15}-\sqrt3}{4} ≈ 0.53523$ Edge radius$\frac{3-\sqrt5}{4} ≈ 0.19098$ Inradius$\sqrt{\frac{25-11\sqrt5}{40}} ≈ 0.10041$ Volume$\frac{7\sqrt5-15}{4} ≈ 0.16312$ Dihedral angle$\arccos\left(\frac{\sqrt5}{5}\right) ≈ 63.43495^\circ$ Central density7
Number of pieces60
Level of complexity3
Related polytopes
ArmyDoe
RegimentGissid
DualGreat icosahedron
Petrie dualPetrial great stellated dodecahedron
ConjugateDodecahedron
Convex coreDodecahedron
Abstract properties
Flag count120
Euler characteristic2
Schläfli type{5,3}
Topological properties
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

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.

## Vertex coordinates

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

• $\left(±\frac{\sqrt5-1}{4},\,±\frac{\sqrt5-1}{4},\,±\frac{\sqrt5-1}{4}\right),$ along with all even permutations and all sign changes of

• $\left(±\frac{3-\sqrt5}{4},\,±\frac12,\,0\right).$ The first set of vertices corresponds to a scaled cube which can be inscribed into the great stellated dodecahedron's vertices.

## 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} $\frac{1+\sqrt{5}}{2}$ ## Related polyhedra

o3o5/2o truncations
Name OBSA Schläfli symbol CD diagram Picture
Great icosahedron gike {3,5/2} x3o5/2o (     )
Truncated great icosahedron tiggy t{3,5/2} x3x5/2o (     )
Great icosidodecahedron gid r{3,5/2} o3x5/2o (     )
Truncated great stellated dodecahedron (degenerate, ike+2gad) t{5/2,3} o3x5/2x (     )
Great stellated dodecahedron gissid {5/2,3} o3o5/2x (     )
Small complex rhombicosidodecahedron (degenerate, sidtid+rhom) sicdatrid rr{3,5/2} x3o5/2x (     )
Truncated great icosidodecahedron (degenerate, ri+12(10/2)) tr{3,5/2} x3x5/2x (     )
Great snub icosidodecahedron gosid sr{3,5/2} s3s5/2s (     )