Truncated great icosahedron
Truncated great icosahedron | |
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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 |
![](http://upload.wikimedia.org/wikipedia/commons/thumb/3/32/Truncated_great_icosahedron.stl/220px-Truncated_great_icosahedron.stl.png)
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
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 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Picture | ![]() | ![]() | ![]() | ![]() | ![]() |
Great stellapentakis dodecahedron
Great stellapentakis dodecahedron | |
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![]() | |
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 |
![](http://upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Great_stellapentakis_dodecahedron.stl/220px-Great_stellapentakis_dodecahedron.stl.png)
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
![](http://upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Great_stellated_dodecahedron_truncations.gif/240px-Great_stellated_dodecahedron_truncations.gif)
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polyhedra (nonconvex
regular polyhedra)
of Kepler-Poinsot
polyhedra
hemipolyhedra
uniform polyhedra
- medial rhombic triacontahedron
- small stellapentakis dodecahedron
- medial deltoidal hexecontahedron
- small rhombidodecacron
- medial pentagonal hexecontahedron
- medial disdyakis triacontahedron
- great rhombic triacontahedron
- great stellapentakis dodecahedron
- great deltoidal hexecontahedron
- great disdyakis triacontahedron
- great pentagonal hexecontahedron
uniform polyhedra with
infinite stellations
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