Great inverted snub icosidodecahedron
| Great inverted 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 | |5/3 2 3 |
| Symmetry group | I, [5,3]+, 532 |
| Index references | U69, C73, W113 |
| Dual polyhedron | Great inverted pentagonal hexecontahedron |
| Vertex figure |  34.5/3 |
| Bowers acronym | Gisid |
In geometry, the great inverted snub icosidodecahedron is a uniform star polyhedron, indexed as U69. It is given a Schläfli symbol sr{5/3,3}.
Cartesian coordinates
Cartesian coordinates for the vertices of a great inverted 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 greater positive real solution to ξ3−2ξ=−1/τ, or approximately 1.2224727. 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 inverted pentagonal hexecontahedron
| Great inverted pentagonal hexecontahedron | |
|---|---|
 | |
| Type | Star polyhedron |
| Face |  |
| Elements | F = 60, E = 150 V = 92 (χ = 2) |
| Symmetry group | I, [5,3]+, 532 |
| Index references | DU69 |
| dual polyhedron | Great inverted snub icosidodecahedron |
The great inverted pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is composed of 60 self-intersecting pentagonal faces, 150 edges and 92 vertices.
It is the dual of the uniform great inverted snub icosidodecahedron.
See also
References
- Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 730208 p. 126