title: Great triakis icosahedron
text: # Great triakis icosahedron
  
  Great triakis icosahedron :
  
  | Type | Star polyhedron |
  | - | - |
  | Face | |
  | Elements | F = 60, E = 90 V = 32 (χ = 2) |
  | Symmetry group | $I_h,$ [5,3], *532 |
  | Index references | $DU_{66}$ |
  | dual polyhedron | Great stellated truncated dodecahedron |
  
  In geometry, the great triakis icosahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform great stellated truncated dodecahedron. Its faces are isosceles triangles. Part of each triangle lies within the solid, hence is invisible in solid models.
  
  ## Proportions
  
  The triangles have one angle of $\arccos(-{\frac {3}{20}}+{\frac {3}{20}}{\sqrt {5}})\approx 79.314\,951\,312\,25^{\circ }$ and two of $\arccos({\frac {3}{4}}-{\frac {1}{20}}{\sqrt {5}})\approx 50.342\,524\,343\,87^{\circ }$. The dihedral angle equals $\arccos({\frac {-24+15{\sqrt {5}}}{61}})\approx 81.001\,410\,024\,84^{\circ }$.
  
  