# Great icosidodecahedron

Great icosidodecahedron
Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymGid
Coxeter diagramo5/2x3o ()
Elements
Faces20 triangles, 12 pentagrams
Edges60
Vertices30
Vertex figureRectangle, edge lengths 1 and (5–1)/2
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt5-1}{2} ≈ 0.61803}$
Volume${\displaystyle \frac{45-17\sqrt5}{6} ≈ 1.16447}$
Dihedral angle${\displaystyle \arccos\left(-\sqrt{\frac{5-2\sqrt5}{15}}\right) ≈ 100.81232°}$
Central density7
Number of pieces132
Level of complexity10
Related polytopes
ArmyId
RegimentGid
DualGreat rhombic triacontahedron
ConjugateIcosidodecahedron
Convex coreIcosahedron
Abstract properties
Euler characteristic2
Topological properties
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The great icosidodecahedron or gid is a quasiregular uniform polyhedron. It consists of 20 equilateral triangles and 12 pentagrams, with two of each joining at a vertex. It can be derived as a rectified great stellated dodecahedron or great icosahedron.

## Vertex coordinates

A great icosidodecahedron of side length 1 has vertex coordinates given by all permutations of

• ${\displaystyle \left(±\frac{\sqrt5-1}{2},\,0,\,0\right),}$

and even permutations of

• ${\displaystyle \left(±\frac{3-\sqrt5}{4},\,±\frac{\sqrt5-1}{4},\,±\frac12\right).}$

The first set of vertices corresponds to a scaled octahedron which can be inscribed into the icosidodecahedron.

## Related polyhedra

The great icosidodecahedron is the colonel of a three-member regiment that also includes the great icosihemidodecahedron and great dodecahemidodecahedron.

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 ()