title: Inverted snub dodecadodecahedron
text: 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 
  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.
  
  Related polyhedra
  
  Medial inverted pentagonal hexecontahedron
  
  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.
  
  Proportions
  
  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.
  
  See also 
   List of uniform polyhedra
   Snub dodecadodecahedron
  
  References
   p. 124
  
  External links 
   
   
  
  Uniform polyhedra