title: Great disdyakis dodecahedron
text: # Great disdyakis dodecahedron
  
  Great disdyakis dodecahedron :
  
  | Type | Star polyhedron |
  | - | - |
  | Face | |
  | Elements | F = 48, E = 72 V = 26 (χ = 2) |
  | Symmetry group | $O_h,$ [4,3], *432 |
  | Index references | $DU_{20}$ |
  | dual polyhedron | Great truncated cuboctahedron |
  
  In geometry, the great disdyakis dodecahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform great truncated cuboctahedron. It has 48 triangular faces.
  
  ## Proportions
  
  The triangles have one angle of $\arccos({\frac {3}{4}}+{\frac {1}{8}}{\sqrt {2}})\approx 22.062\,191\,157\,54^{\circ }$, one of $\arccos(-{\frac {1}{6}}-{\frac {1}{12}}{\sqrt {2}})\approx 106.530\,027\,150\,22^{\circ }$ and one of $\arccos(-{\frac {1}{12}}+{\frac {1}{2}}{\sqrt {2}})\approx 51.407\,781\,692\,24^{\circ }$. The dihedral angle equals $\arccos({\frac {-71+12{\sqrt {2}}}{97}})\approx 123.848\,891\,579\,44^{\circ }$. Part of each triangle lies within the solid, hence is invisible in solid models.
  
  ## Related polyhedra
  
  The great disdyakis dodecahedron is topologically identical to the convex Catalan solid, disdyakis dodecahedron, which is dual to the truncated cuboctahedron.
  
  