Small hexagrammic hexecontahedron
# Small hexagrammic hexecontahedron

Small hexagrammic hexecontahedron :

| Type | Star polyhedron |
| - | - |
| Face | |
| Elements | F = 60, E = 180 V = 112 (χ = −8) |
| Symmetry group | $I_h,$ [5,3], *532 |
| Index references | $DU_{72}$ |
| dual polyhedron | Small retrosnub icosicosidodecahedron |

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.

## Geometry

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

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

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

