Snub dodecadodecahedron

Cartesian coordinates

Cartesian coordinates for the vertices of a snub dodecadodecahedron are all the even permutations of

(±2α, ±2, ±2β),(±(α+β/τ+τ), ±(-ατ+β+1/τ), ±(α/τ+βτ-1)),(±(-α/τ+βτ+1), ±(-α+β/τ-τ), ±(ατ+β-1/τ)),(±(-α/τ+βτ-1), ±(α-β/τ-τ), ±(ατ+β+1/τ)) and(±(α+β/τ-τ), ±(ατ-β+1/τ), ±(α/τ+βτ+1)),

with an even number of plus signs, where

β = (α²/τ+τ)/(ατ−1/τ),

where τ = (1+√5)/2 is the golden mean and α is the positive real root of τα⁴−α³+2α²−α−1/τ, or approximately 0.7964421. 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 pentagonal hexecontahedron

The medial pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the snub dodecadodecahedron. It has 60 intersecting irregular pentagonal faces.