title: Small icosacronic hexecontahedron
text: # Small icosacronic hexecontahedron
  
  Small icosacronic hexecontahedron :
  
  | Type | Star polyhedron |
  | - | - |
  | Face | |
  | Elements | F = 60, E = 120 V = 52 (χ = −8) |
  | Symmetry group | $I_h,$ [5,3], *532 |
  | Index references | $DU_{31}$ |
  | dual polyhedron | Small icosicosidodecahedron |
  
  In geometry, the small icosacronic hexecontahedron (or small lanceal trisicosahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform small icosicosidodecahedron. Its faces are kites. Part of each kite lies inside the solid, hence is invisible in solid models.
  
  ## Proportions
  
  The kites have two angles of $\arccos({\frac {3}{4}}-{\frac {1}{20}}{\sqrt {5}})\approx 50.342\,524\,343\,87^{\circ }$, one of $\arccos(-{\frac {1}{12}}-{\frac {19}{60}}{\sqrt {5}})\approx 142.318\,554\,460\,55^{\circ }$ and one of $\arccos(-{\frac {5}{12}}-{\frac {1}{60}}{\sqrt {5}})\approx 116.996\,396\,851\,70^{\circ }$. The dihedral angle equals $\arccos({\frac {-44-3{\sqrt {5}}}{61}})\approx 146.230\,659\,755\,53^{\circ }$. The ratio between the lengths of the long and short edges is ${\frac {31+5{\sqrt {5}}}{38}}\approx 1.110\,008\,944\,41$.
  
  