title: Compound of six tetrahedra with rotational freedom
text: # Compound of six tetrahedra with rotational freedom
  
  Compound of six tetrahedra with rotational freedom :
  
  | Type | Uniform compound |
  | - | - |
  | Index | $UC_1$ |
  | Polyhedra | 6 tetrahedra |
  | Faces | 24 triangles |
  | Edges | 36 |
  | Vertices | 24 |
  | Symmetry group | tetrahedral $(T_d)$ |
  | Subgroup restricting to one constituent | 4-fold improper rotation $(S_4)$ |
  
  The compound of six tetrahedra with rotational freedom is a uniform polyhedron compound made of a symmetric arrangement of 6 tetrahedra, considered as antiprisms. It can be constructed by superimposing six tetrahedra within a cube, and then rotating them in pairs about the three axes that pass through the centres of two opposite cubic faces. Each tetrahedron is rotated by an equal (and opposite, within a pair) angle θ. Equivalently, a tetrahedron may be inscribed within each cube in the compound of six cubes with rotational freedom, in such a way as to preserve tetrahedral symmetry.
  
  When θ = 0, all six tetrahedra coincide. When θ is 45 degrees, the more symmetric compound of six tetrahedra (without rotational freedom) arises.
  
  ## Gallery
  
  * Compounds of six tetrahedra with rotational freedom
  * File:Tetrahedron.stlθ = 0°
  * File:Compound_of_six_tetrahedra_with_rotational_freedom_(5°).stlθ = 5°
  * File:Compound_of_six_tetrahedra_with_rotational_freedom_(10°).stlθ = 10°
  * File:Compound_of_six_tetrahedra_with_rotational_freedom_(15°).stlθ = 15°
  * File:Compound_of_six_tetrahedra_with_rotational_freedom_(20°).stlθ = 20°
  * File:Compound_of_six_tetrahedra_with_rotational_freedom_(25°).stlθ = 25°
  * File:Compound_of_six_tetrahedra_with_rotational_freedom_(30°).stlθ = 30°
  * File:Compound_of_six_tetrahedra_with_rotational_freedom_(35°).stlθ = 35°
  * File:Compound_of_six_tetrahedra_with_rotational_freedom_(40°).stlθ = 40°
  * File:Compound_of_six_tetrahedra.stlθ = 45°
  