Inverted snub dodecadodecahedron

Cartesian coordinates

Cartesian coordinates for the vertices of an inverted 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 negative real root of τα⁴−α³+2α²−α−1/τ, or approximately −0.3352090. 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 inverted pentagonal hexecontahedron

The medial inverted pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform inverted snub dodecadodecahedron.