The compound of five octahedra is one of the five regular polyhedron compounds, and can also be seen as a stellation. It was first described by Edmund Hess in 1876. It is unique among the regular compounds for not having a regular convex hull. It can also be called a small icosicosahedron.
It can be constructed by a rhombic triacontahedron with rhombic-based pyramids added to all the faces, as shown by the five colored model image. (This construction does not generate the regular compound of five octahedra, but shares the same topology and can be smoothly deformed into the regular compound.)
The spherical and stereographic projections of this compound look the same as those of the disdyakis triacontahedron. But the convex solid's vertices on 3- and 5-fold symmetry axes (gray in the images below) correspond only to edge crossings in the compound.
Spherical polyhedron
Stereographic projections
2-fold
3-fold
5-fold
The area in the black circles below corresponds to the frontal hemisphere of the spherical polyhedron.
A second 5-octahedra compound, with octahedral symmetry, also exists. It can be generated by adding a fifth octahedron to the standard 4-octahedra compound.
E. Hess 1876 Zugleich Gleicheckigen und Gleichflächigen Polyeder, Schriften der Gesellschaft zur Berörderung der Gasammten Naturwissenschaften zu Marburg 11 (1876) pp 5–97.
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