title: Compound of six decagrammic prisms
text: # Compound of six decagrammic prisms
  
  Compound of six decagrammic prisms :
  
  | Type | Uniform compound |
  | - | - |
  | Index | $UC_{41}$ |
  | Polyhedra | 6 decagrammic prisms |
  | Faces | 12 decagrams, 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 decagrammic 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/√5), $±2τ^{−1},$ $±√(τ^{−1}/√5))$
      $(±(√(τ/√5)+τ^{−2}),$ ±1, $±(√(τ^{−1}/√5)−τ^{−1}))$
      $(±(√(τ/√5)−τ^{−1}),$ $±τ^{−2},$ $±(√(τ^{−1}/√5)+1))$
      $(±(√(τ/√5)+τ^{−1}),$ $±τ^{−2},$ $±(√(τ^{−1}/√5)−1))$
      $(±(√(τ/√5)−τ^{−2}),$ ±1, $±(√(τ^{−1}/√5)+τ^{−1}))$
  
  where τ = (1+√5)/2 is the golden ratio (sometimes written φ).
  
  