Small hexagrammic hexecontahedron

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.

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