title: Great dodecacronic hexecontahedron
text: # Great dodecacronic hexecontahedron
  
  Great dodecacronic hexecontahedron :
  
  | Type | Star polyhedron |
  | - | - |
  | Face | |
  | Elements | F = 60, E = 120 V = 44 (χ = −16) |
  | Symmetry group | $I_h,$ [5,3], *532 |
  | Index references | $DU_{61}$ |
  | dual polyhedron | Great dodecicosidodecahedron |
  
  In geometry, the great dodecacronic hexecontahedron (or great lanceal ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform great dodecicosidodecahedron. Its 60 intersecting quadrilateral faces are kites. Part of each kite lies inside the solid, hence is invisible in solid models.
  
  ## Proportions
  
  Each kite has two angles of $\arccos({\frac {5}{8}}-{\frac {1}{8}}{\sqrt {5}})\approx 69.788\,198\,194\,11^{\circ }$, one of $\arccos(-{\frac {1}{4}}+{\frac {1}{10}}{\sqrt {5}})\approx 91.512\,394\,720\,74^{\circ }$ and one of $\arccos(-{\frac {1}{8}}-{\frac {9}{40}}{\sqrt {5}})\approx 128.911\,208\,891\,04^{\circ }$. The dihedral angle equals $\arccos({\frac {-19+8{\sqrt {5}}}{41}})\approx 91.553\,403\,672\,16^{\circ }$. The ratio between the lengths of the long and short edges is ${\frac {21+3{\sqrt {5}}}{22}}\approx 1.259\,463\,815\,11$.
  
  