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 |      
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| 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{3,5⁄2} or t0,1{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
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8ee2d99dd874675246b0ecc2631988e590c21a0)
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.
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 |
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| Coxeter-Dynkin diagram |
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Great stellapentakis dodecahedron
| Great stellapentakis dodecahedron | |
|---|---|
| |
| Type | Star polyhedron |
| Face | 
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| 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
References
- ^ Maeder, Roman. "55: great truncated icosahedron". MathConsult.
- Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, 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
