# Great icosicosidodecahedron

Great icosicosidodecahedron
Rank3
TypeUniform
Notation
Bowers style acronymGiid
Coxeter diagramo3/2x3x5*a ()
Elements
Faces20 triangles, 12 pentagons, 20 hexagons
Edges60+60
Vertices60
Vertex figureCrossed isosceles trapezoid, edge lengths 1, 3, (1+5)/2, 3
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {17-3{\sqrt {5}}}{8}}}\approx 1.13423}$
Volume${\displaystyle {\frac {29{\sqrt {5}}-30}{3}}\approx 11.61533}$
Dihedral angles5–6: ${\displaystyle \arccos \left({\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 79.18768^{\circ }}$
3–6: ${\displaystyle \arccos \left({\frac {\sqrt {5}}{3}}\right)\approx 41.81032^{\circ }}$
Central density6
Number of external pieces1232
Level of complexity75
Related polytopes
ArmyTid, edge length ${\displaystyle {\frac {3-{\sqrt {5}}}{2}}}$
RegimentGidditdid
DualGreat icosacronic hexecontahedron
ConjugateSmall icosicosidodecahedron
Convex coreIcosahedron
Abstract & topological properties
Flag count480
Euler characteristic–8
OrientableYes
Genus5
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The great icosicosidodecahedron, or giid, is a uniform polyhedron. It consists of 20 triangles, 12 pentagons, and 20 hexagons. One triangle, one pentagon, and two hexagons join at each vertex.

It is a faceting of the great ditrigonal dodecicosidodecahedron, using its 12 pentagons and 20 triangles along with 20 additional hexagons.

A semi-uniform variant of this polyhedron can be constructed as a rectified great ditrigonary icosidodecahedron.

## Vertex coordinates

Its vertices are the same as those of its regiment colonel, the great ditrigonal dodecicosidodecahedron.