Compound of three octahedra

In mathematics, the compound of three octahedra or octahedron 3-compound is a polyhedral compound formed from three regular octahedra, all sharing a common center but rotated with respect to each other. Although appearing earlier in the mathematical literature, it was rediscovered and popularized by M. C. Escher, who used it in the central image of his 1948 woodcut Stars.

A regular octahedron can be circumscribed around a cube in such a way that the eight edges of two opposite squares of the cube lie on the eight faces of the octahedron. The three octahedra formed in this way from the three pairs of opposite cube squares form the compound of three octahedra. The eight cube vertices are the same as the eight points in the compound where three edges cross each other. Each of the octahedron edges that participates in these triple crossings is divided by the crossing point in the ratio 1:√2. The remaining octahedron edges cross each other in pairs, within the interior of the compound; their crossings are at their midpoints and form right angles.

The compound of three octahedra can also be formed from three copies of a single octahedron by rotating each copy by an angle of π/4 around one of the three symmetry axes that pass through two opposite vertices of the starting octahedron. A third construction for the same compound of three octahedra is as the dual polyhedron of the compound of three cubes, one of the uniform polyhedron compounds.

The six vertices of one of the three octahedra may be given by the coordinates (0, 0, ±2) and (±√2, ±√2, 0). The other two octahedra have coordinates that may be obtained from these coordinates by exchanging the z coordinate for the x or y coordinate.

The compound of three octahedra has the same symmetry group as a single octahedron. It is an isohedral deltahedron, meaning that its faces are equilateral triangles and that it has a symmetry taking every face to every other face. There is one known infinite family of isohedral deltahedra, and 36 more that do not fall into this family; the compound of three octahedra is one of the 36 sporadic examples. However, its symmetry group does not take every vertex to every other vertex, so it is not itself a uniform polyhedron compound.

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