Great snub icosidodecahedron
| Great snub icosidodecahedron | |
|---|---|
 | |
| Type | Uniform star polyhedron |
| Elements | F = 92, E = 150 V = 60 (χ = 2) |
| Faces by sides | (20+60){3}+12{5/2} |
| Wythoff symbol | |2 5/2 3 |
| Symmetry group | I, [5,3]+, 532 |
| Index references | U57, C88, W116 |
| Dual polyhedron | Great pentagonal hexecontahedron |
| Vertex figure |  34.5/2 |
| Bowers acronym | Gosid |
In geometry, the great snub icosidodecahedron is a nonconvex uniform polyhedron, indexed as U57. It can be represented by a Schläfli symbol sr{5/2,3}, and Coxeter-Dynkin diagram 
.
This polyhedron is the snub member of a family that includes the great icosahedron, the great stellated dodecahedron and the great icosidodecahedron.
Cartesian coordinates
Cartesian coordinates for the vertices of a great snub icosidodecahedron are all the even permutations of
- (±2α, ±2, ±2β),
- (±(α−βτ−1/τ), ±(α/τ+β−τ), ±(−ατ−β/τ−1)),
- (±(ατ−β/τ+1), ±(−α−βτ+1/τ), ±(−α/τ+β+τ)),
- (±(ατ−β/τ−1), ±(α+βτ+1/τ), ±(−α/τ+β−τ)) and
- (±(α−βτ+1/τ), ±(−α/τ−β−τ), ±(−ατ−β/τ+1)),
with an even number of plus signs, where
- α = ξ−1/ξ
and
- β = −ξ/τ+1/τ2−1/(ξτ),
where τ = (1+√5)/2 is the golden mean and ξ is the negative real root of ξ3−2ξ=−1/τ, or approximately −1.5488772. Taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph of the other one.
Related polyhedra
Great pentagonal hexecontahedron
| Great pentagonal hexecontahedron | |
|---|---|
.png) | |
| Type | Star polyhedron |
| Face |  |
| Elements | F = 60, E = 150 V = 92 (χ = 2) |
| Symmetry group | I, [5,3]+, 532 |
| Index references | DU57 |
| dual polyhedron | Great snub icosidodecahedron |
The great pentagonal hexecontahedron is a nonconvex isohedral polyhedron and dual to the uniform great snub icosidodecahedron. It has 60 intersecting irregular pentagonal faces, 120 edges, and 92 vertices.
See also
References
- Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 730208