Small hexagonal hexecontahedron

Treating it as a simple non-convex solid (without intersecting surfaces), it has 180 faces (all triangles), 270 edges, and 92 vertices (twelve with degree 10, twenty with degree 12, and sixty with degree 3), giving an Euler characteristic of 92 − 270 + 180 = +2.

The faces are irregular hexagons. Denoting the golden ratio by ${\displaystyle \phi }$ and putting ${\displaystyle \xi ={\frac {1}{4}}-{\frac {1}{4}}{\sqrt {1+4\phi }}\approx -0.433\,380\,199\,59}$, the hexagons have five equal angles of ${\displaystyle \arccos(\xi )\approx 115.682\,268\,170\,75^{\circ }}$ and one of ${\displaystyle \arccos(\phi ^{-2}\xi -\phi ^{-1})\approx 141.588\,659\,146\,23^{\circ }}$. Each face has four long and two short edges. The ratio between the edge lengths is

${\displaystyle 1/2+1/2\times {\sqrt {(1-\xi )/(\phi ^{3}-\xi )}}\approx 0.777\,024\,337\,46}$.

The dihedral angle equals ${\displaystyle \arccos(\xi /(1+\xi ))\approx 139.893\,813\,264\,51^{\circ }}$.