There are a number of attractive compounds of two regular tetrahedra. The most symmetrical is the arrangement whose outer hull is the stella octangula (left figure), while another attractive
arrangement arises by taking two opposite tetrahedra of the tetrahedron
5-compound (Cundy and Rollett 1989).

The vertices of the first tetrahedron 2-compound are among those of an equilateral augmented cube (i.e., a cube with faces replaced by outward-pointing square pyramidal
caps) and Escher's solid.

These tetrahedron 2-compounds are illustrated above together with their duals
and common midspheres.

The common solids and convex hulls of these compounds are illustrated above. The interior of the first compound is a regular
octahedron and the interior of the second has the connectivity of Dürer's
solid, while the convex hull of the first compound
is a cube.

The second compound consists of two tetrahedra with one flipped vertically about the plane through their common centroid. One of the two tetrahedra is then rotated by an angle

(1)

about the -axis,
bringing its base vertices into coincidence with the vertices of a dodecahedron
sharing vertices with the two tetrahedron. The compound can be built by beginning
with a base tetrahedron, placing a "cap" around one of the apexes, and
then affixing a triangular pyramid to the opposite face.

For a tetrahedron 2-compound inscribed in a dodecahedron
with unit edge lengths, the tetrahedron edges will have length

(2)

The position, size, and orientation of the cap are illustrated in the diagram above, where

(3)

(4)

(5)

(6)

(7)

(8)

(9)

The edge lengths and angles of the cap are given by

Cundy, H. and Rollett, A. "Five Tetrahedra in a Dodecahedron." §3.10.8 in Mathematical
Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 139-141, 1989.