In geometry, a compound of four tetrahedra can be constructed by four tetrahedra in a number of different symmetry positions.
A uniform compound of four tetrahedra can be constructed by rotating tetrahedra along an axis of symmetry C<sub>2</sub> (that is the middle of an edge) in multiples of . It has dihedral symmetry, D<sub>8h</sub>, 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<sub>2</sub> plane of symmetry, with one pair of tetrahedra shifted . 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:
Four tetrahedra that are not spread equally in angles over C<sub>2</sub> 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<sub>2</sub> that is rotated by equal and opposite angles. This compound is indexed as UC<sub>22</sub>, with parameters p/q = 2 and n = 4 as well.
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
.
This compound is self-dual, meaning its dual polyhedron is the same compound polyhedron.