# Great icosidodecahedron

Great icosidodecahedron
Rank3
TypeUniform
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${\displaystyle {\frac {{\sqrt {5}}-1}{2}}\approx 0.61803}$
Volume${\displaystyle {\frac {45-17{\sqrt {5}}}{6}}\approx 1.16447}$
Dihedral angle${\displaystyle \arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 100.81232^{\circ }}$
Central density7
Number of external pieces132
Level of complexity10
Related polytopes
ArmyId, edge length ${\displaystyle {\frac {3-{\sqrt {5}}}{4}}}$
RegimentGid
DualGreat rhombic triacontahedron
ConjugateIcosidodecahedron
Convex coreIcosahedron
Abstract & topological properties
Flag count240
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryH3, order 120
Flag orbits2
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

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

and even permutations of

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

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

## Related polyhedra

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