Small hexagrammic hexecontahedron

In geometry, the small hexagrammic hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the small retrosnub icosicosidodecahedron. It is partially degenerate, having coincident vertices, as its dual has coplanar triangular faces.

Its faces are hexagonal stars with two short and four long edges. Denoting the golden ratio by ${\displaystyle \phi }$and putting ${\displaystyle \xi ={\frac {1}{4}}+{\frac {1}{4}}{\sqrt {1+4\phi }}\approx 0.933\,380\,199\,59}$, the stars have five equal angles of ${\displaystyle \arccos(\xi )\approx 21.031\,988\,967\,51^{\circ }}$and one of ${\displaystyle 360^{\circ }-\arccos(\phi ^{-2}\xi -\phi ^{-1})\approx 254.840\,055\,162\,43^{\circ }}$. Each face has four long and two short edges. The ratio between the edge lengths is

${\displaystyle 1/2-1/2\times {\sqrt {(1-\xi )/(\phi ^{3}-\xi )}}\approx 0.428\,986\,992\,12}$.

The dihedral angle equals ${\displaystyle \arccos(\xi /(1+\xi ))\approx 61.133\,452\,273\,64^{\circ }}$. Part of each face is inside the solid, hence is not visible in solid models.

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

• Weisstein, Eric W. "Small hexagrammic hexecontahedron". MathWorld.