# 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${\displaystyle \arccos\left(-\sqrt{\frac{5-2\sqrt5}{15}}\right) ≈ 100.81232^\circ}$
Central density7
Number of external pieces132
Level of complexity10
Related polytopes
ArmyId, edge length ${\displaystyle \frac{3-\sqrt5}{4}}$
RegimentGid
DualGreat rhombic triacontahedron
ConjugateIcosidodecahedron
Convex coreIcosahedron
Abstract & topological properties
Flag count240
Euler characteristic2
OrientableYes
Genus0
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 ()