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.

## 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,
β = τ2ρ22ρ+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