title: Compound of four tetrahedra
text: # Compound of four tetrahedra
  
  Compound of 4 digonal antiprisms :
  
  | Type | Uniform compound |
  | - | - |
  | Index | $UC_{23}$ (n=4, p=2, q=1) |
  | Polyhedra | 4 digonal antiprisms (tetrahedra) |
  | Faces | 16 triangles |
  | Edges | 32 |
  | Vertices | 16 |
  | Symmetry group | $D_{8h},$ order 32 |
  | Subgroup restricting to one stella octangula | $O_h,$ order 48 $D_{4h},$ order 16 |
  
  In geometry, a compound of four tetrahedra can be constructed by four tetrahedra in a number of different symmetry positions.
  
  ## Uniform compounds
  
  A uniform compound of four tetrahedra can be constructed by rotating tetrahedra along an axis of symmetry $C_2$ (that is the middle of an edge) in multiples of $\pi /4$. It has dihedral symmetry, $D_{8h},$ and the same vertex arrangement as the convex octagonal prism.
  
  This compound can also be seen as two compounds of stella octangulae fit evenly on the same $C_2$ plane of symmetry, with one pair of tetrahedra shifted $\pi /4$. It is a special case of a p/q-gonal prismatic compound of antiprisms, where in this case the component p/q = 2 is a digonal antiprism, or tetrahedron.
  
  Below are two perspective viewpoints of the uniform compound of four tetrahedra, with each color representing one regular tetrahedron:
  
  Perspectives :
  
  | Top view | Side view |
  | - | - |
  
  Four tetrahedra that are not spread equally in $\pi /4$ angles over $C_2$ can still hold uniform symmetry when allowed rotational freedom. In this case, these tetrahedra share a symmetric arrangement over the common axis of symmetry $C_2$ that is rotated by equal and opposite angles. This compound is indexed as $UC_{22},$ with parameters p/q = 2 and n = 4 as well.
  
  ## Other compounds
  
  A nonuniform compound can be generated by rotating tetrahedra about lines extending from the center of each face and through the centroid (as altitudes), with varying degrees of rotation.
  
  A model for this compound polyhedron was first published by Robert Webb, using his program Stella, in 2004, following studies of polyhedron models:
  
  With edge-length as a unit, it has a surface area equal to
  
  $S={\frac {1291}{210\cdot {\sqrt {3}}}}\approx 3.5493$.
  
  This compound is self-dual, meaning its dual polyhedron is the same compound polyhedron.
  
  