Great pentagrammic hexecontahedron

Denote the golden ratio by ${\displaystyle \phi }$. Let ${\displaystyle \xi \approx 0.946\,730\,033\,56}$ be the largest positive zero of the polynomial ${\displaystyle P=8x^{3}-8x^{2}+\phi ^{-2}}$. Then each pentagrammic face has four equal angles of ${\displaystyle \arccos(\xi )\approx 18.785\,633\,958\,24^{\circ }}$ and one angle of ${\displaystyle \arccos(-\phi ^{-1}+\phi ^{-2}\xi )\approx 104.857\,464\,167\,03^{\circ }}$. 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

${\displaystyle l={\frac {2-4\xi ^{2}}{1-2\xi }}\approx 1.774\,215\,864\,94}$.

The dihedral angle equals ${\displaystyle \arccos(\xi /(\xi +1))\approx 60.901\,133\,713\,21^{\circ }}$. 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 inverted pentagonal hexecontahedron.

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

• Weisstein, Eric W. "Great pentagrammic hexecontahedron". MathWorld.