Inverted snub dodecadodecahedron

In geometry, the inverted snub dodecadodecahedron (or vertisnub dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U60. It is given a Schläfli symbol sr{5/3,5}. 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+)/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. The medial inverted pentagonal hexecontahedron (or midly petaloid ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform inverted snub dodecadodecahedron. Its faces are irregular nonconvex pentagons, with one very acute angle. Denote the golden ratio by , and let be the largest (least negative) real zero of the polynomial . Then each face has three equal angles of , one of and one of . Each face has one medium length edge, two short and two long ones. If the medium length is , then the short edges have length and the long edges have length The dihedral angle equals . The other real zero of the polynomial plays a similar role for the medial pentagonal hexecontahedron.