Great icosidodecahedron

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.

Great icosidodecahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Gid
Coxeter diagram	o5/2x3o ()
Elements
Faces	20 triangles, 12 pentagrams
Edges	60
Vertices	30
Vertex figure	Rectangle, edge lengths 1 and (√5–1)/2
Measures (edge length 1)
Circumradius	$\frac{\sqrt5-1}{2} ≈ 0.61803$
Volume	$\frac{45-17\sqrt5}{6} ≈ 1.16447$
Dihedral angle	$\arccos\left(-\sqrt{\frac{5-2\sqrt5}{15}}\right) ≈ 100.81232^\circ$
Central density	7
Number of external pieces	132
Level of complexity	10
Related polytopes
Army	Id, edge length $\frac{3-\sqrt5}{4}$
Regiment	Gid
Dual	Great rhombic triacontahedron
Conjugate	Icosidodecahedron
Convex core	Icosahedron
Abstract & topological properties
Flag count	240
Euler characteristic	2
Orientable	Yes
Genus	0
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

Vertex coordinates

A great icosidodecahedron of side length 1 has vertex coordinates given by all permutations of

• $\left(±\frac{\sqrt5-1}{2},\,0,\,0\right),$

and even permutations of

• $\left(±\frac{3-\sqrt5}{4},\,±\frac{\sqrt5-1}{4},\,±\frac12\right).$

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 ()

External links

• Bowers, Jonathan. "Polyhedron Category 3: Quasiregulars" (#30).

• Bowers, Jonathan. "Batch 3: Id, Did, and Gid Facetings" (#1 under gid).

• Klitzing, Richard. "gid".

• Wikipedia Contributors. "Great icosidodecahedron".
• McCooey, David. "Great Icosidodecahedron"