Great icosihemidodecahedron

Great icosihemidodecahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Notation
Bowers style acronym	Geihid
Coxeter diagram	(o3/2x5/3x3*a)/2 ( )/2
Elements
Faces	20 triangles, 6 decagrams
Edges	60
Vertices	30
Vertex figure	Bowtie, edge lengths 1 and √(5–√5)/2
Measures (edge length 1)
Circumradius	${\frac {{\sqrt {5}}-1}{2}}\approx 0.61803$
Dihedral angle	$\arccos \left({\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 37.37737^{\circ }$
Number of external pieces	540
Level of complexity	28
Related polytopes
Army	Id, edge length ${\frac {3-{\sqrt {5}}}{2}}$
Regiment	Gid
Dual	Great icosihemidodecacron
Conjugate	Small icosihemidodecahedron
Abstract & topological properties
Flag count	240
Euler characteristic	–4
Orientable	No
Genus	6
Properties
Symmetry	H3, order 120
Flag orbits	2
Convex	No
Nature	Tame

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.

External links

• Bowers, Jonathan. "Polyhedron Category 3: Quasiregulars" (#32).

• Bowers, Jonathan. "Batch 3: Id, Did, and Gid Facetings" (#2 under gid).

• Klitzing, Richard. "geihid".
• Wikipedia contributors. "Great icosihemidodecahedron".
• McCooey, David. "Great Icosihemidodecahedron"