Compound of five octahedra

Compound of five octahedra
Type	Regular compound
Index	UC17, W23
Elements (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)
Stellation diagram Stellation core Convex hull Icosahedron Icosidodecahedron

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

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.

As a compound

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

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

Compound of five octahedra

Compound of five tetrahemihexahedra

As a facetting

It is a facetting of the icosidodecahedron, as shown below:

At left only one octahedron is shown; all five are shown at right.

See also

• Compound of three octahedra
• Compound of four octahedra
• Compound of ten octahedra
• Compound of twenty octahedra

References

• Peter R. Cromwell, Polyhedra, Cambridge, 1997.
• Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9. 
• Coxeter, Harold Scott MacDonald; Du Val, P.; Flather, H. T.; Petrie, J. F. (1999). The fifty-nine icosahedra (3rd ed.). Tarquin. ISBN 978-1-899618-32-3. MR676126  (1st Edn University of Toronto (1938))
• E. Hess 1876 Zugleich Gleicheckigen und Gleichflächigen Polyeder, Schriften der Gesellschaft zur Berörderung der Gasammten Naturwissenschaften zu Marburg 11 (1876) pp 5–97.

External links

• MathWorld: Octahedron5-Compound
• Paper Model Compound of Five Octahedra
• VRML model: [1]

Notable stellations of the icosahedron
Regular	Uniform duals		Regular compounds			Regular star	Others
Icosahedron	Small triambic icosahedron	Great triambic icosahedron	Compound of five octahedra	Compound of five tetrahedra	Compound of ten tetrahedra	Great icosahedron	Excavated dodecahedron	Final stellation
The stellation process on the icosahedron creates a number of related polyhedra and compounds with icosahedral symmetry.