Great snub icosidodecahedron

Great snub icosidodecahedron
	(3D model); (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Gosid
Coxeter diagram	s5/2s3s ()
Elements
Faces	20+60 triangles, 12 pentagrams
Edges	30+60+60
Vertices	60
Vertex figure	Irregular pentagon, edge lengths 1, 1, 1, 1, (√5–1)/2 ;
Measures (edge length 1)
Circumradius	≈ 0.81608
Volume	≈ 7.67391
Dihedral angles	5/2–3: ≈ 138.82237°
	3–3: ≈ 126.82315°
Central density	7
Number of external pieces	300
Level of complexity	26
Related polytopes
Army	Non-uniform Snid
Regiment	Gosid
Dual	Great pentagonal hexecontahedron
Conjugates	Snub dodecahedron, Great inverted snub icosidodecahedron, great inverted retrosnub icosidodecahedron
Abstract & topological properties
Flag count	600
Euler characteristic	2
Orientable	Yes
Genus	0
Properties
Symmetry	H3+, order 60
Convex	No
Nature	Tame

The great snub icosidodecahedron or gosid is a uniform polyhedron. It consists of 60 snub triangles, 20 additional triangles, and 12 pentagrams. Four triangles and one pentagram meeting at each vertex.

Measures[edit | edit source]

The circumradius R ≈ 0.81608 of the great snub icosidodecahedron with unit edge length is the second to largest real root of:

4096x^{12}-27648x^{10}+47104x^8-35776x^6+13872x^4-2696x^2+209.

Its volume V ≈ 7.67391 is given by the second to largest real root of:

{\displaystyle \begin{align}&2176782336x^{12}-3195335070720x^{10}+162223191936000x^8+1030526618040000x^6\\ {} &+6152923794150000x^4-182124351550575000x^2+187445810737515625.\end{align}}

These same polynomials define the circumradii and volumes of the snub dodecahedron, the great inverted snub icosidodecahedron, and the great inverted retrosnub icosidodecahedron.

Related polyhedra[edit | edit source]

The great disnub icosidodecahedron is a uniform polyhedron compound composed of the 2 opposite chiral forms of the great snub icosidodecahedron.

o3o5/2o truncations
Name	OBSA	Schläfli symbol	CD diagram
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 ()