Snub icosidodecadodecahedron

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 = τ²ρ²+τ²ρ+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.

Medial hexagonal hexecontahedron

The medial hexagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform snub icosidodecadodecahedron.