Compound of five octahedra

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Compound of five octahedra
Compound of five octahedra.png
Type Regular compound
Index UC17, W23
Coxeter symbol [5{3,4}]2{3,5}[1]
(As a compound)
5 octahedra:
F = 40, E = 60, V = 30
Dual compound Compound of five cubes
Symmetry group icosahedral (Ih)
Subgroup restricting to one constituent pyritohedral (Th)

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.

As a stellation[edit]

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. (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.)

Stellation diagram Stellation core Convex hull
Stellation facets Icosahedron.png

As a compound[edit]

It can also be seen as a polyhedral compound of five octahedra arranged in icosahedral symmetry (Ih).

It shares its edges and half of its triangular faces with the compound of five tetrahemihexahedra.

UC18-5 tetrahemihexahedron.png
Compound of five tetrahemihexahedra
Spherical compound of five octahedra.png
As a spherical tiling the octahedra edges match the disdyakis triacontahedron

As a facetting[edit]

Five octahedra in an icosidodecahedron

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

See also[edit]


  1. ^ Regular polytopes, pp.49-50, p.98

External links[edit]

Notable stellations of the icosahedron
Regular Uniform duals Regular compounds Regular star Others
(Convex) icosahedron Small triambic icosahedron Medial triambic icosahedron Great triambic icosahedron Compound of five octahedra Compound of five tetrahedra Compound of ten tetrahedra Great icosahedron Excavated dodecahedron Final stellation
Zeroth stellation of icosahedron.png First stellation of icosahedron.png Ninth stellation of icosahedron.png First compound stellation of icosahedron.png Second compound stellation of icosahedron.png Third compound stellation of icosahedron.png Sixteenth stellation of icosahedron.png Third stellation of icosahedron.png Seventeenth stellation of icosahedron.png
Zeroth stellation of icosahedron facets.png First stellation of icosahedron facets.png Ninth stellation of icosahedron facets.png First compound stellation of icosahedron facets.png Second compound stellation of icosahedron facets.png Third compound stellation of icosahedron facets.png Sixteenth stellation of icosahedron facets.png Third stellation of icosahedron facets.png Seventeenth stellation of icosahedron facets.png
The stellation process on the icosahedron creates a number of related polyhedra and compounds with icosahedral symmetry.