Inverted snub dodecadodecahedron

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.

Denote the golden ratio by ${\displaystyle \phi }$, and let ${\displaystyle \xi \approx -0.236\,993\,843\,45}$ be the largest (least negative) real zero of the polynomial ${\displaystyle P=8x^{4}-12x^{3}+5x+1}$. Then each face has three equal angles of ${\displaystyle \arccos(\xi )\approx 103.709\,182\,219\,53^{\circ }}$, one of ${\displaystyle \arccos(\phi ^{2}\xi +\phi )\approx 3.990\,130\,423\,41^{\circ }}$ and one of ${\displaystyle 360^{\circ }-\arccos(\phi ^{-2}\xi -\phi ^{-1})\approx 224.882\,322\,917\,99^{\circ }}$. Each face has one medium length edge, two short and two long ones. If the medium length is ${\displaystyle 2}$, then the short edges have length

${\displaystyle 1-{\sqrt {(1-\xi )/(\phi ^{3}-\xi )}}\approx 0.474\,126\,460\,54}$,

and the long edges have length

${\displaystyle 1+{\sqrt {(1-\xi )/(-\phi ^{-3}-\xi )}}\approx 37.551\,879\,448\,54}$.

The dihedral angle equals ${\displaystyle \arccos(\xi /(\xi +1))\approx 108.095\,719\,352\,34^{\circ }}$. The other real zero of the polynomial ${\displaystyle P}$ plays a similar role for the medial pentagonal hexecontahedron.

• List of uniform polyhedra
• Snub dodecadodecahedron

• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208 p. 124

• Weisstein, Eric W. "Medial inverted pentagonal hexecontahedron". MathWorld.
• Weisstein, Eric W. "Inverted snub dodecadodecahedron". MathWorld.