title: Compound of ten octahedra
text: # Compound of ten octahedra
  
  Compounds of ten octahedra :
  
  | Type | Uniform compound |
  | - | - |
  | Index | $UC_{15}$ and $UC_{16}$ |
  | Polyhedra | 10 octahedra |
  | Faces | 20+60 triangles |
  | Edges | 120 |
  | Vertices | 60 |
  | Symmetry group | icosahedral $(I_h)$ |
  | Subgroup restricting to one constituent | 3-fold antiprismatic $(D_{3d})$ |
  
  The compounds of ten octahedra $UC_{15}$ and $UC_{16}$ are two uniform polyhedron compounds. They are composed of a symmetric arrangement of 10 octahedra, considered as triangular antiprisms, aligned with the axes of three-fold rotational symmetry of an icosahedron. The two compounds differ in the orientation of their octahedra: each compound may be transformed into the other by rotating each octahedron by 60 degrees.
  
  ## Cartesian coordinates
  
  Cartesian coordinates for the vertices of this compound are all the cyclic permutations of
  
      (0, $±(τ^{−1}√2$ + 2sτ), ±(τ√2 − $2sτ^{−1}))$
      (±(√2 − $sτ^2),$ ±(√2 + s(2τ − 1)), ±(√2 + $sτ^{−2}))$
      $(±(τ^{−1}√2$ − sτ), ±(τ√2 + $sτ^{−1}),$ ±3s)
  
  where τ = (1 + √5)/2 is the golden ratio (sometimes written φ) and s is either +1 or −1. Setting s = −1 gives $UC_{15},$ while s = +1 gives $UC_{16}.$
  
  