title: Compound of five icosahedra
text: # Compound of five icosahedra
  
  Compound of five icosahedra :
  
  | Type | Uniform compound |
  | - | - |
  | Index | $UC_{47}$ |
  | Polyhedra | 5 icosahedra |
  | Faces | 40+60 Triangles |
  | Edges | 150 |
  | Vertices | 60 |
  | Symmetry group | icosahedral $(I_h)$ |
  | Subgroup restricting to one constituent | pyritohedral $(T_h)$ |
  
  The compound of five icosahedra is uniform polyhedron compound. It's composed of 5 icosahedra, rotated around a common axis. It has icosahedral symmetry $I_h.$
  
  The triangles in this compound decompose into two orbits under action of the symmetry group: 40 of the triangles lie in coplanar pairs in icosahedral planes, while the other 60 lie in unique planes.
  
  ## Cartesian coordinates
  
  Cartesian coordinates for the vertices of this compound are all the cyclic permutations of
  
      (0, ±2, ±2τ)
      $(±τ^{−1},$ ±1, $±(1+τ^2))$
      (±τ, $±τ^2,$ ±(2τ−1))
  
  where τ = (1+√5)/2 is the golden ratio (sometimes written φ).
  
  