title: Tridyakis icosahedron
text: # Tridyakis icosahedron
  
  Tridyakis icosahedron :
  
  | Type | Star polyhedron |
  | - | - |
  | Face | |
  | Elements | F = 120, E = 180 V = 44 (χ = −16) |
  | Symmetry group | $I_h,$ [5,3], *532 |
  | Index references | $DU_{45}$ |
  | dual polyhedron | Icositruncated dodecadodecahedron |
  
  In geometry, the tridyakis icosahedron is the dual polyhedron of the nonconvex uniform polyhedron, icositruncated dodecadodecahedron. It has 44 vertices, 180 edges, and 120 scalene triangular faces.
  
  ## Proportions
  
  The triangles have one angle of $\arccos({\frac {3}{5}})\approx 53.130\,102\,354\,16^{\circ }$, one of $\arccos({\frac {1}{3}}+{\frac {4}{15}}{\sqrt {5}})\approx 21.624\,633\,927\,143^{\circ }$ and one of $\arccos({\frac {1}{3}}-{\frac {4}{15}}{\sqrt {5}})\approx 105.245\,263\,718\,70^{\circ }$. The dihedral angle equals $\arccos(-{\frac {7}{8}})\approx 151.044\,975\,628\,14^{\circ }$. Part of each triangle lies within the solid, hence is invisible in solid models.
  
  