| Snub icosidodecadodecahedron | |
|---|---|
) | |
| Type | Uniform star polyhedron |
| Elements | F = 104, E = 180 V = 60 (χ = −16) |
| Faces by sides | (20+60){3}+12{5}+12{5/2} |
| Wythoff symbol | | 5/3 3 5 |
| Symmetry group | I, [5,3]+, 532 |
| Index references | U46, C58, W112 |
| Dual polyhedron | Medial hexagonal hexecontahedron |
| Vertex figure | ) 3.3.3.5.3.5/3 |
| Bowers acronym | Sided |
)
In geometry, the snub icosidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U46. It has 104 faces (80 triangles, 12 pentagons, and 12 pentagrams), 180 edges, and 60 vertices.[1]
As the name indicates, it belongs to the family of snub polyhedra.
Cartesian coordinates
Cartesian coordinates for the vertices of a snub icosidodecadodecahedron are all the even permutations of
- (±2α, ±2γ, ±2β),
- (±(α+β/τ+γτ), ±(-ατ+β+γ/τ), ±(α/τ+βτ-γ)),
- (±(-α/τ+βτ+γ), ±(-α+β/τ-γτ), ±(ατ+β-γ/τ)),
- (±(-α/τ+βτ-γ), ±(α-β/τ-γτ), ±(ατ+β+γ/τ)) and
- (±(α+β/τ-γτ), ±(ατ-β+γ/τ), ±(α/τ+βτ+γ)),
with an even number of plus signs, where
- α = ρ+1 = ρ3,
- β = τ2ρ2+τ2ρ+τ = τ2ρ4+τ,
- γ = ρ2+τρ,
and where τ = (1+√5)/2 is the golden mean and ρ is the real solution to ρ3=ρ+1, or approximately 1.3247180. ρ is called the plastic constant. 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
Medial hexagonal hexecontahedron
| Medial hexagonal hexecontahedron | |
|---|---|
) | |
| Type | Star polyhedron |
| Face | ) |
| Elements | F = 60, E = 180 V = 104 (χ = −16) |
| Symmetry group | I, [5,3]+, 532 |
| Index references | DU46 |
| dual polyhedron | Snub icosidodecadodecahedron |
)
The medial hexagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform snub icosidodecadodecahedron.
See also
References
- ^ Maeder, Roman. "46: snub icosidodecadodecahedron". MathConsult.
- Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208
External links
- Weisstein, Eric W. "Snub icosidodecadodecahedron". MathWorld.
- Weisstein, Eric W. "Medial hexagonal hexecontahedron". MathWorld.