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

In geometry, the truncated great icosahedron is a nonconvex uniform polyhedron, indexed as U₅₅. It is given a Schläfli symbol t{3,5/2} or t_0,1{3,5/2} 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.

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

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

References

• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208

External links

• Weisstein, Eric W. "Truncated great icosahedron". MathWorld.
• Weisstein, Eric W. "Great stellapentakis dodecahedron". MathWorld.
• Uniform polyhedra and duals