title: Compound of six decagonal prisms
text: # Compound of six decagonal prisms
  
  Compound of six decagonal prisms :
  
  | Type | Uniform compound |
  | - | - |
  | Index | $UC_{40}$ |
  | Polyhedra | 6 decagonal prisms |
  | Faces | 12 decagons, 60 squares |
  | Edges | 180 |
  | Vertices | 120 |
  | Symmetry group | icosahedral $(I_h)$ |
  | Subgroup restricting to one constituent | 5-fold antiprismatic $(D_{5d})$ |
  
  This uniform polyhedron compound is a symmetric arrangement of 6 decagonal prisms, aligned with the axes of fivefold rotational symmetry of a dodecahedron.
  
  ## Cartesian coordinates
  
  Cartesian coordinates for the vertices of this compound are all the cyclic permutations of
  
      $(±√(τ^{−1}/√5),$ ±2τ, ±√(τ/√5))
      $(±(√(τ^{−1}/√5)−τ^2),$ ±1, ±(√(τ/√5)+τ))
      $(±(√(τ^{−1}/√5)−τ),$ $±τ^2,$ ±(√(τ/√5)+1))
      $(±(√(τ^{−1}/√5)+τ),$ $±τ^2,$ ±(√(τ/√5)−1))
      $(±(√(τ^{−1}/√5)+τ^2),$ ±1, ±(√(τ/√5)−τ))
  
  where τ = (1+√5)/2 is the golden ratio (sometimes written φ).
  
  