Compound of five octahedra


The compound of five octahedra is one of the five regular polyhedron compounds. This polyhedron can be seen as either a polyhedral stellation or a compound. This compound was first described by Edmund Hess in 1876. It is unique among the regular compounds for not having a regular convex hull.

As a stellation

It is the second stellation of the icosahedron, and given as Wenninger model index 23.
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.
It has a density of greater than 1.
Stellation diagramStellation coreConvex hull

Icosahedron
Icosidodecahedron

As a compound

It can also be seen as a polyhedral compound of five octahedra arranged in icosahedral symmetry.
It shares its edges and half of its triangular faces with the compound of five tetrahemihexahedra.

Compound of five tetrahemihexahedra
As a spherical tiling the octahedra edges match the disdyakis triacontahedron
Stereographic projection

As a faceting

It is also a faceting of an icosidodecahedron, shown at left.

Other 5-octahedra compounds

A second 5-octahedra compound, with octahedral symmetry, also exists. It can be generated by adding a fifth octahedra to the standard 4-octahedra compound.