Medial hexagonal hexecontahedron

The faces of the medial hexagonal hexecontahedron are irregular nonconvex hexagons. Denote the golden ratio by ${\displaystyle \phi }$, and let ${\displaystyle \xi \approx -0.377\,438\,833\,12}$ be the real zero of the polynomial ${\displaystyle 8x^{3}-4x^{2}+1}$. The number ${\displaystyle \xi }$ can be written as ${\displaystyle \xi =-1/(2\rho )}$, where ${\displaystyle \rho }$ is the plastic number. Then each face has four equal angles of ${\displaystyle \arccos(\xi )\approx 112.175\,128\,045\,27^{\circ }}$, one of ${\displaystyle \arccos(\phi ^{2}\xi +\phi )\approx 50.958\,265\,917\,31^{\circ }}$ and one of ${\displaystyle 360^{\circ }-\arccos(\phi ^{-2}\xi -\phi ^{-1})\approx 220.341\,221\,901\,59^{\circ }}$. Each face has two long edges, two of medium length and two short ones. If the medium edges have length ${\displaystyle 2}$, the long ones have length ${\displaystyle 1+{\sqrt {(1-\xi )/(-\phi ^{-3}-\xi )}}\approx 4.121\,448\,816\,41}$ and the short ones ${\displaystyle 1-{\sqrt {(1-\xi )/(\phi ^{3}-\xi )}}\approx 0.453\,587\,559\,98}$. The dihedral angle equals ${\displaystyle \arccos(\xi /(\xi +1))\approx 127.320\,132\,197\,62^{\circ }}$.

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

• Weisstein, Eric W. "Medial hexagonal hexecontahedron". MathWorld.