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} |
| 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 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
- (±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 |
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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
References
- Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 730208