title: Compound of ten hexagonal prisms
text: # Compound of ten hexagonal prisms
  
  Compound of ten hexagonal prisms :
  
  | Type | Uniform compound |
  | - | - |
  | Index | $UC_{39}$ |
  | Polyhedra | 10 hexagonal prisms |
  | Faces | 20 hexagons, 60 squares |
  | Edges | 180 |
  | Vertices | 120 |
  | Symmetry group | icosahedral $(I_h)$ |
  | Subgroup restricting to one constituent | 3-fold antiprismatic $(D_{3d})$ |
  
  This uniform polyhedron compound is a symmetric arrangement of 10 hexagonal prisms, aligned with the axes of three-fold rotational symmetry of an icosahedron.
  
  ## Cartesian coordinates
  
  Cartesian coordinates for the vertices of this compound are all the cyclic permutations of
  
      (±√3, $±(τ^{−1}−τ√3),$ $±(τ+τ^{−1}√3))$
      (±2√3, $±τ^{−1},$ ±τ)
      (±(1+√3), ±(1−τ√3), $±(1+τ^{−1}√3))$
      $(±(τ−τ^{−1}√3),$ ±√3, $±(τ^{−1}+τ√3))$
      $(±(1−τ^{−1}√3),$ ±(1−√3), ±(1+τ√3))
  
  where τ = (1+√5)/2 is the golden ratio (sometimes written φ).
  
  