Great inverted snub icosidodecahedron

Cartesian coordinates for the vertices of a great inverted snub icosidodecahedron are all the even permutations of

(±2α, ±2, ±2β),

(±(α−βτ−1/τ), ±(α/τ+β−τ), ±(−ατ−β/τ−1)),

(±(ατ−β/τ+1), ±(−α−βτ+1/τ), ±(−α/τ+β+τ)),

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

(±(α−βτ+1/τ), ±(−α/τ−β−τ), ±(−ατ−β/τ+1)),

with an even number of plus signs, where

α = ξ−1/ξ

and

β = −ξ/τ+1/τ²−1/(ξτ),

where τ = (1+√5)/2 is the golden mean and ξ is the greater positive real solution to ξ³−2ξ=−1/τ, or approximately 1.2224727. 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 circumradius for unit edge length is

${\displaystyle R={\frac {1}{2}}{\sqrt {\frac {2-x}{1-x}}}=0.816081\dots }$

where ${\displaystyle x}$ is the appropriate root of ${\displaystyle x^{3}+2x^{2}={\Big (}{\tfrac {1\pm {\sqrt {5}}}{2}}{\Big )}^{2}}$. The four positive real roots of the sextic in ${\displaystyle R^{2},}$

${\displaystyle 4096R^{12}-27648R^{10}+47104R^{8}-35776R^{6}+13872R^{4}-2696R^{2}+209=0}$

are the circumradii of the snub dodecahedron (U₂₉), great snub icosidodecahedron (U₅₇), great inverted snub icosidodecahedron (U₆₉), and great retrosnub icosidodecahedron (U₇₄).

Denote the golden ratio by ${\displaystyle \phi }$. Let ${\displaystyle \xi \approx 0.252\,780\,289\,27}$ be the smallest positive zero of the polynomial ${\displaystyle P=8x^{3}-8x^{2}+\phi ^{-2}}$. Then each pentagonal face has four equal angles of ${\displaystyle \arccos(\xi )\approx 75.357\,903\,417\,42^{\circ }}$ and one angle of ${\displaystyle 360^{\circ }-\arccos(-\phi ^{-1}+\phi ^{-2}\xi )\approx 238.568\,386\,330\,33^{\circ }}$. Each face has three long and two short edges. The ratio ${\displaystyle l}$ between the lengths of the long and the short edges is given by

${\displaystyle l={\frac {2-4\xi ^{2}}{1-2\xi }}\approx 3.528\,053\,034\,81}$.

The dihedral angle equals ${\displaystyle \arccos(\xi /(\xi +1))\approx 78.359\,199\,060\,62^{\circ }}$. Part of each face lies inside the solid, hence is invisible in solid models. The other two zeroes of the polynomial ${\displaystyle P}$ play a similar role in the description of the great pentagonal hexecontahedron and the great pentagrammic hexecontahedron.

• List of uniform polyhedra
• Great snub icosidodecahedron
• Great retrosnub icosidodecahedron

• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208 p. 126

• Weisstein, Eric W. "Great inverted pentagonal hexecontahedron". MathWorld.
• Weisstein, Eric W. "Great inverted snub icosidodecahedron". MathWorld.