Tetrahedron 2-Compound

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 compounds are implemented in the Wolfram Language as PolyhedronData["TetrahedronTwoCompound", n] for , 2.

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

			(10)
			(11)
			(12)
			(13)
			(14)
			(15)