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 t0,1{3,​5⁄2} as a truncated great icosahedron.

Truncated great icosahedron
TypeUniform star polyhedron
ElementsF = 32, E = 90
V = 60 (χ = 2)
Faces by sides12{5/2}+20{6}
Wythoff symbol3
2 5/3 | 3
Symmetry groupIh, [5,3], *532
Index referencesU55, C71, W95
Dual polyhedronGreat stellapentakis dodecahedron
Vertex figure
6.6.5/2
Bowers acronymTiggy
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

(±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
TypeStar polyhedron
Face
ElementsF = 60, E = 90
V = 32 (χ = 2)
Symmetry groupIh, [5,3], *532
Index referencesDU55
dual polyhedronTruncated 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

References
  1. Maeder, Roman. "55: great truncated icosahedron". MathConsult.
Animated truncation sequence from {​5⁄2, 3} to {3, ​5⁄2}


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