# Great inverted retrosnub icosidodecahedron

Great inverted retrosnub icosidodecahedron Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymGirsid
Coxeter diagrams5/3s3/2s (       )
Elements
Faces20+60 triangles, 12 pentagrams
Edges30+60+60
Vertices60
Vertex figureIrregular pentagram, edge lengths 1, 1, 1, 1, (5–1)/2 Measures (edge length 1)
Volume≈ 1.03760
Dihedral angles5/2–3: ≈ 67.31029°
3–3: ≈ 21.72466°
Central density37
Number of external pieces1800
Level of complexity139
Related polytopes
ArmyNon-uniform Snid
RegimentGirsid
DualGreat pentagrammic hexecontahedron
ConjugatesSnub dodecahedron, great snub icosidodecahedron, great inverted snub icosidodecahedron
Abstract & topological properties
Flag count600
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryH3+, order 60
ConvexNo
NatureTame

The great inverted retrosnub icosidodecahedron or girsid, also called the great retrosnub icosidodecahedron, is a uniform polyhedron. It consists of 60 snub triangles, 20 additional triangles, and 12 pentagrams. Four triangles and one pentagram meet at each vertex.

## Measures

The circumradius R ≈ 0.58000 of the great inverted retrosnub icosidodecahedron with unit edge length is the smallest positive real root of:

$4096x^{12}-27648x^{10}+47104x^8-35776x^6+13872x^4-2696x^2+209.$ Its volume V ≈ 1.03760 is given by the smallest positive real root of:

\begin{align}&2176782336x^{12}-3195335070720x^{10}+162223191936000x^8+1030526618040000x^6\\ {} &+6152923794150000x^4-182124351550575000x^2+187445810737515625.\end{align} These same polynomials define the circumradii and volumes of the snub dodecahedron, the great snub icosidodecahedron, and the great inverted snub icosidodecahedron.

## Related polyhedra

The great diretrosnub icosidodecahedron is a uniform polyhedron compound composed of the 2 opposite chiral forms of the great inverted retrosnub icosidodecahedron.