# Great icosihemidodecahedron

Great icosihemidodecahedron
Rank3
TypeUniform
Notation
Bowers style acronymGeihid
Coxeter diagram(o3/2x5/3x3*a)/2
( )/2
Elements
Faces20 triangles, 6 decagrams
Edges60
Vertices30
Vertex figureBowtie, edge lengths 1 and (5–5)/2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {{\sqrt {5}}-1}{2}}\approx 0.61803}$
Dihedral angle${\displaystyle \arccos \left({\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 37.37737^{\circ }}$
Number of external pieces540
Level of complexity28
Related polytopes
ArmyId, edge length ${\displaystyle {\frac {3-{\sqrt {5}}}{2}}}$
RegimentGid
DualGreat icosihemidodecacron
ConjugateSmall icosihemidodecahedron
Abstract & topological properties
Flag count240
Euler characteristic–4
OrientableNo
Genus6
Properties
SymmetryH3, order 120
Flag orbits2
ConvexNo
NatureTame

The great icosihemidodecahedron, or geihid, is a quasiregular polyhedron and one of 10 uniform hemipolyhedra. It consists of 20 equilateral triangles and 6 "hemi" decagrams, with two of each joining at a vertex.

It can be derived as a rectified petrial great stellated dodecahedron.

It is a faceting of the great icosidodecahedron, keeping the original's triangles while also using its equatorial decagrams.

## Name

Its face planes are the same as those of an icosahedron and a dodecahedron in the dual configuration, hence the name "icosihemidodecahedron". The "great" prefix, used for stellations in general, distinguishes it from the small icosihemidodecahedron, which also has this face arrangement.

## Vertex coordinates

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