Truncated great icosahedron

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.^[1] It is given a Schläfli symbol t{3,5⁄2} or t_0,1{3,5⁄2} as a truncated great icosahedron.

Truncated great icosahedron
Type	Uniform star polyhedron
Elements	F = 32, E = 90 V = 60 (χ = 2)
Faces by sides	12{5/2}+20{6}
Coxeter diagram
Wythoff symbol	2 5/2 | 3 2 5/3 | 3
Symmetry group	Ih, [5,3], *532
Index references	U55, C71, W95
Dual polyhedron	Great stellapentakis dodecahedron
Vertex figure	6.6.5/2
Bowers acronym	Tiggy

3D model of 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

$${\displaystyle {\begin{array}{crccc}{\Bigl (}&\pm \,1,&0,&\pm \,{\frac {3}{\varphi }}&{\Bigr )}\\{\Bigl (}&\pm \,2,&\pm \,{\frac {1}{\varphi }},&\pm \,{\frac {1}{\varphi ^{3}}}&{\Bigr )}\\{\Bigl (}&\pm {\bigl [}1+{\frac {1}{\varphi ^{2}}}{\bigr ]},&\pm \,1,&\pm \,{\frac {2}{\varphi }}&{\Bigr )}\end{array}}}$$

where ${\displaystyle \varphi ={\tfrac {1+{\sqrt {5}}}{2}}}$ is the golden ratio. Using ${\displaystyle {\tfrac {1}{\varphi ^{2}}}=1-{\tfrac {1}{\varphi }}}$ one verifies that all vertices are on a sphere, centered at the origin, with the radius squared equal to ${\displaystyle 10-{\tfrac {9}{\varphi }}.}$ The edges have length 2.

Related polyhedra

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	DU55
dual polyhedron	Truncated great icosahedron

3D model of a great stellapentakis dodecahedron

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

See also

• List of uniform polyhedra