title: Compound of five truncated tetrahedra
text: # Compound of five truncated tetrahedra
  
  Compound of five truncated tetrahedra :
  
  | Type | Uniform compound |
  | - | - |
  | Index | $UC_{55}$ |
  | Polyhedra | 5 truncated tetrahedra |
  | Faces | 20 triangles, 20 hexagons |
  | Edges | 90 |
  | Vertices | 60 |
  | Dual | Compound of five triakis tetrahedra |
  | Symmetry group | chiral icosahedral (I) |
  | Subgroup restricting to one constituent | chiral tetrahedral (T) |
  
  The compound of five truncated tetrahedra is a uniform polyhedron compound. It's composed of 5 truncated tetrahedra rotated around a common axis. It may be formed by truncating each of the tetrahedra in the compound of five tetrahedra. A far-enough truncation creates the compound of five octahedra. Its convex hull is a nonuniform snub dodecahedron.
  
  ## Cartesian coordinates
  
  Cartesian coordinates for the vertices of this compound are all the cyclic permutations of
  
      (±1, ±1, ±3)
      $(±τ^{−1},$ $±(−τ^{−2}),$ ±2τ)
      (±τ, $±(−2τ^{−1}),$ $±τ^2)$
      $(±τ^2,$ $±(−τ^{−2}),$ ±2)
      (±(2τ−1), ±1, ±(2τ − 1))
  
  with an even number of minuses in the choices for '±', where τ = (1+√5)/2 is the golden ratio (sometimes written φ).
  
  