title: Rhombicosacron
text: # Rhombicosacron
  
  Rhombicosacron :
  
  | Type | Star polyhedron |
  | - | - |
  | Face | |
  | Elements | F = 60, E = 120 V = 50 (χ = −10) |
  | Symmetry group | $I_h,$ [5,3], *532 |
  | Index references | $DU_{56}$ |
  | dual polyhedron | Rhombicosahedron |
  
  In geometry, the rhombicosacron (or midly dipteral ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform rhombicosahedron, U56. It has 50 vertices, 120 edges, and 60 crossed-quadrilateral faces.
  
  ## Proportions
  
  Each face has two angles of $\arccos({\frac {3}{4}})\approx 41.409\,622\,109\,27^{\circ }$ and two angles of $\arccos(-{\frac {1}{6}})\approx 99.594\,068\,226\,86^{\circ }$. The diagonals of each antiparallelogram intersect at an angle of $\arccos({\frac {1}{8}}+{\frac {7{\sqrt {5}}}{24}})\approx 38.996\,309\,663\,87^{\circ }$. The dihedral angle equals $\arccos(-{\frac {5}{7}})\approx 135.584\,691\,402\,81^{\circ }$. The ratio between the lengths of the long edges and the short ones equals ${\frac {3}{2}}+{\frac {1}{2}}{\sqrt {5}}$, which is the square of the golden ratio.
  
  