# Truncated great icosahedron

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Truncated great icosahedron Type Uniform star polyhedron
Elements F = 32, E = 90
V = 60 (χ = 2)
Faces by sides12{5/2}+20{6}
Wythoff symbol 2 5/2| 3
2 5/3| 3
Symmetry group Ih, [5,3], *532
Index references U 55, C 71, W 95
Dual polyhedron Great stellapentakis dodecahedron
Vertex figure 6.6.5/2
Bowers acronymTiggy

In geometry, the truncated great icosahedron (or great truncated icosahedron) is a nonconvex uniform polyhedron, indexed as U55. It has 32 faces (12 pentagrams and 20 hexagons), 90 edges, and 60 vertices.  It is given a Schläfli symbol t{3,52} or t0,1{3,52} as a truncated great icosahedron.

## Cartesian coordinates

Cartesian coordinates for the vertices of a truncated great icosahedron centered at the origin are all the even permutations of

(±1, 0, ±3/τ)
(±2, ±1/τ, ±1/τ3)
(±(1+1/τ2), ±1, ±2/τ)

where τ = (1+√5)/2 is the golden ratio (sometimes written φ). Using 1/τ2 = 1 − 1/τ one verifies that all vertices are on a sphere, centered at the origin, with the radius squared equal to 10−9/τ. The edges have length 2.

This polyhedron is the truncation of the great icosahedron:

The truncated great stellated dodecahedron is a degenerate polyhedron, with 20 triangular faces from the truncated vertices, and 12 (hidden) pentagonal faces as truncations of the original pentagram faces, the latter forming a great dodecahedron inscribed within and sharing the edges of the icosahedron.

Name Great
stellated
dodecahedron
Truncated great stellated dodecahedron Great
icosidodecahedron
Truncated
great
icosahedron
Great
icosahedron
Coxeter-Dynkin
diagram                                   Picture     ### Great stellapentakis dodecahedron

Great stellapentakis dodecahedron Type Star polyhedron
Face Elements F = 60, E = 90
V = 32 (χ = 2)
Symmetry group Ih, [5,3], *532
Index references DU 55
dual polyhedron Truncated great icosahedron

The great stellapentakis dodecahedron is a nonconvex isohedral polyhedron. It is the dual of the truncated great icosahedron. It has 60 intersecting triangular faces.

## Related Research Articles In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra. In geometry, the rhombicosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces. In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges. In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol {5,5/2} and Coxeter–Dynkin diagram of . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces, with five pentagons meeting at each vertex, intersecting each other making a pentagrammic path. In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol {​52,5}. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex. In geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedron, with Schläfli symbol {​52,3}. It is one of four nonconvex regular polyhedra. In geometry, the great icosahedron is one of four Kepler-Poinsot polyhedra, with Schläfli symbol {3,​52} and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence. In geometry, the small ditrigonal icosidodecahedron (or small ditrigonary icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U30. It has 32 faces (20 triangles and 12 pentagrams), 60 edges, and 20 vertices. It has extended Schläfli symbol a{5,3}, as an altered dodecahedron, and Coxeter diagram or . In geometry, the dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U36. It is the rectification of the great dodecahedron (and that of its dual, the small stellated dodecahedron). It was discovered independently by Hess (1878), Badoureau (1881) and Pitsch (1882). In geometry, the great icosidodecahedron is a nonconvex uniform polyhedron, indexed as U54. It has 32 faces (20 triangles and 12 pentagrams), 60 edges, and 30 vertices. It is given a Schläfli symbol r{3,​52}. It is the rectification of the great stellated dodecahedron and the great icosahedron. It was discovered independently by Hess (1878), Badoureau (1881) and Pitsch (1882). In geometry, the truncated great dodecahedron is a nonconvex uniform polyhedron, indexed as U37. It has 24 faces (12 pentagrams and 12 decagons), 90 edges, and 60 vertices. It is given a Schläfli symbol t{5,​52}. In geometry, the great stellated truncated dodecahedron (or quasitruncated great stellated dodecahedron or great stellatruncated dodecahedron) is a nonconvex uniform polyhedron, indexed as U66. It has 32 faces (20 triangles and 12 pentagrams), 90 edges, and 60 vertices. It is given a Schläfli symbol t0,1{​53,3}. In geometry, the rhombidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U38. It has 54 faces (30 squares, 12 pentagons and 12 pentagrams), 120 edges and 60 vertices. It is given a Schläfli symbol t0,2{​52,5}, and by the Wythoff construction this polyhedron can also be named a cantellated great dodecahedron. In geometry, the truncated dodecadodecahedron (or stellatruncated dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U59. It is given a Schläfli symbol t0,1,2{​53,5}. It has 54 faces (30 squares, 12 decagons, and 12 decagrams), 180 edges, and 120 vertices. The central region of the polyhedron is connected to the exterior via 20 small triangular holes. In geometry, the great truncated icosidodecahedron (or great quasitruncated icosidodecahedron or stellatruncated icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U68. It has 62 faces (30 squares, 20 hexagons, and 12 decagrams), 180 edges, and 120 vertices. It is given a Schläfli symbol t0,1,2{​53,3}, and Coxeter-Dynkin diagram, . In geometry, the great snub icosidodecahedron is a nonconvex uniform polyhedron, indexed as U57. It has 92 faces (80 triangles and 12 pentagrams), 150 edges, and 60 vertices. It can be represented by a Schläfli symbol sr{​52,3}, and Coxeter-Dynkin diagram . In geometry, the snub dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U40. It has 84 faces (60 triangles, 12 pentagons, and 12 pentagrams), 150 edges, and 60 vertices. It is given a Schläfli symbol sr{​52,5}, as a snub great dodecahedron. In geometry, the ditrigonal dodecadodecahedron (or ditrigonary dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U41. It has 24 faces (12 pentagons and 12 pentagrams), 60 edges, and 20 vertices. It has extended Schläfli symbol b{5,​52}, as a blended great dodecahedron, and Coxeter diagram . It has 4 Schwarz triangle equivalent constructions, for example Wythoff symbol 3 | ​53 5, and Coxeter diagram . In geometry, the nonconvex great rhombicosidodecahedron is a nonconvex uniform polyhedron, indexed as U67. It has 62 faces (20 triangles, 30 squares and 12 pentagrams), 120 edges, and 60 vertices. It is also called the quasirhombicosidodecahedron. It is given a Schläfli symbol t0,2{​53,3}. Its vertex figure is a crossed quadrilateral. In geometry, the great stellapentakis dodecahedron is a nonconvex isohedral polyhedron. It is the dual of the truncated great icosahedron. It has 60 intersecting triangular faces.

1. Maeder, Roman. "55: great truncated icosahedron". MathConsult.