title: Small hexagrammic hexecontahedron
text: # 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.
  
  