Great inverted snub icosidodecahedron

Great inverted snub icosidodecahedron
	(3D model); (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Gisid
Coxeter diagram	s5/3s3s ()
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	≈ 0.64502
Volume	≈ 2.71387
Dihedral angles	3–3: ≈ 89.78760°
	5/2–3: ≈ 21.61047°
Central density	13
Number of external pieces	780
Level of complexity	65
Related polytopes
Army	Non-uniform Snid
Regiment	Gisid
Dual	Great inverted pentagonal hexecontahedron
Conjugates	Snub dodecahedron, Great snub icosidodecahedron, Great inverted retrosnub icosidodecahedron
Convex core	Chiral order-6 truncated pentakis dodecahedron
Abstract & topological properties
Flag count	600
Euler characteristic	2
Orientable	Yes
Genus	0
Properties
Symmetry	H3+, order 60
Convex	No
Nature	Tame

The great inverted snub icosidodecahedron or gisid, is a uniform polyhedron. It consists of 60 snub triangles, 20 additional triangles, and 12 pentagrams. Four triangles and one pentagram meet at each vertex. It can be constructed by alternation of the great quasitruncated icosidodecahedron after setting all edge lengths to be equal.

Measures[edit | edit source]

The circumradius R ≈ 0.64502 of the great inverted snub icosidodecahedron with unit edge length is the second to smallest positive real root of:

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

Its volume V ≈ 2.71387 is given by the second to smallest positive 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 snub icosidodecahedron, and the great inverted retrosnub icosidodecahedron.

Related polyhedra[edit | edit source]

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

o5/3o3o truncations
Name	OBSA	Schläfli symbol	CD diagram
Great stellated dodecahedron	gissid	{5/3,3}	x5/3o3o ()
Quasitruncated great stellated dodecahedron	quit gissid	t{5/3,3}	x5/3x3o ()
Great icosidodecahedron	gid	r{3,5/3}	o5/3x3o ()
Truncated great icosahedron	tiggy	t{3,5/3}	o5/3x3x ()
Great icosahedron	gike	{3,5/3}	o5/3o3x ()
Quasirhombicosidodecahedron	qrid	rr{3,5/3}	x5/3o3x ()
Great quasitruncated icosidodecahedron	gaquatid	tr{3,5/3}	x5/3x3x ()
Great inverted snub icosidodecahedron	gisid	sr{3,5/3}	s5/3s3s ()