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 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 (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{3,52} or t0,1{3,52} as a truncated great icosahedron.

3D model of a truncated great icosahedron

Cartesian coordinates

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

 

where   is the golden ratio. Using   one verifies that all vertices are on a sphere, centered at the origin, with the radius squared equal to   The edges have length 2.

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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

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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

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References

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  1. ^ Maeder, Roman. "55: great truncated icosahedron". MathConsult.
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Animated truncation sequence from {52, 3} to {3, 52}