title: Compound of five truncated cubes
text: # Compound of five truncated cubes
  
  Compound of five truncated cubes :
  
  | Type | Uniform compound |
  | - | - |
  | Index | $UC_{57}$ |
  | Polyhedra | 5 truncated cubes |
  | Faces | 40 triangles, 30 octagons |
  | Edges | 180 |
  | Vertices | 120 |
  | Symmetry group | icosahedral $(I_h)$ |
  | Subgroup restricting to one constituent | pyritohedral $(T_h)$ |
  
  This uniform polyhedron compound is a composition of 5 truncated cubes, formed by truncating each of the cubes in the compound of 5 cubes.
  
  ## Cartesian coordinates
  
  Cartesian coordinates for the vertices of this compound are all the cyclic permutations of
  
      (±(2+√2), ±√2, ±(2+√2))
      (±τ, $±(τ^{−1}+τ^{−1}√2),$ ±(2τ−1+τ√2))
      (±1, $±(τ^{−2}−τ^{−1}√2),$ $±(τ^2+τ√2))$
      (±(1+√2), $±(−τ^{−2}−√2),$ $±(τ^2+√2))$
      (±(τ+τ√2), $±(−τ^{−1}),$ $±(2τ−1+τ^{−1}√2))$
  
  where τ = (1+√5)/2 is the golden ratio (sometimes written φ).
  
  