The stella octangula is a polyhedron compound composed of a tetrahedron and its dual
(a second tetrahedron rotated with respect to the first). The stella octangula
is also (incorrectly) called the stellated tetrahedron, and is the only stellation
of the octahedron. A wireframe version of the stella
octangula is sometimes known as the merkaba and imbued with mystic properties.
The name "stella octangula" is due to Kepler (1611), but the solid was known earlier to many others, including Pacioli (1509), who called it the "octaedron elevatum," and Jamnitzer (1568); see Cromwell (1997, pp. 124 and 152).
It is implemented in the Wolfram Language as PolyhedronData["StellaOctangula"].
A stella octangula can be inscribed in a cube, deltoidal icositetrahedron, pentagonal icositetrahedron, rhombic dodecahedron, small triakis octahedron, and tetrakis hexahedron, (E. Weisstein, Dec. 24-25, 2009).
Two stella octangula, one solid compound in the lower left and one beveled wireframe in the upper right, appear as polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43). Escher also built his own model of the solid (Bool et al. 1982, p. 146).
The stella octangula can be constructed using the following net by cutting along the solid lines, folding back along the plain lines, and folding forward along the dotted lines.
Another construction builds a single tetrahedron, then attaches four tetrahedral caps, one to each face. This augmentation
of a unit edge-length octahedron uses pyramids with
height .
Combining two tetrahedra with unit edge lengths produces a stella octangula with edge lengths 1/2. This solid has surface area and volume
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(1)
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(2)
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The convex hull of the stella octangula is a cube.
The above diagrams show two projections of the stella octangula. The edges lying on tetrahedral faces are represented using dashed lines, while the edges of the two large tetrahedron are showing using solid lines.
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The solid common to both tetrahedra is an octahedron (left figure; Ball and Coxeter 1987), which is another way of saying that the stella octangula is a stellation of the octahedron (in fact, the only stellation). The edges of the two tetrahedra in the stella octangula form the 12 polyhedron diagonals of a cube (middle figure). Finally, the stella octangula can be constructed using eight of the 20 vertices of the dodecahedron (right figure).