Compound of five octahedra

Compound of five octahedra
File:Compound of five octahedra.png (see here for a 3D model)
Type	Regular compound
Index	UC17, W23
Coxeter symbol	[5{3,4}]2{3,5}
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)

File:Small-icosiicosahedron-in-icosidodecahedron.png

It is also a faceting of the icosidodecahedron.

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.

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. (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.)

It has a density of greater than 1.

Stellation diagram	Stellation core	Convex hull
Stellation facets	File:Icosahedron.png Icosahedron	File:Icosidodecahedron.png Icosidodecahedron

As a compound

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

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
File:Spherical disdyakis triacontahedron as compound of five octahedra.png	File:Disdyakis triacontahedron stereographic d2 colored.svg	File:Disdyakis triacontahedron stereographic d3 colored.svg	File:Disdyakis triacontahedron stereographic d5 colored.svg
	File:Disdyakis triacontahedron stereographic d2 colored crop.svg	File:Disdyakis triacontahedron stereographic d3 colored crop.svg	File:Disdyakis triacontahedron stereographic d5 colored crop.svg
The area in the black circles below corresponds to the frontal hemisphere of the spherical polyhedron.

Replacing the octahedra by tetrahemihexahedra leads to the compound of five tetrahemihexahedra.

Other 5-octahedra compounds

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.

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.
• (1st Edn University of Toronto (1938))
• H.S.M. Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, , 3.6 The five regular compounds, pp.47-50, 6.2 Stellating the Platonic solids, pp.96-104
• 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
• Octahedron5-Compound as Gauss Pentagramma Mirificum

Notable stellations of the icosahedron
Regular	Regular star	Uniform duals
(Convex) icosahedron	Great icosahedron	Small triambic icosahedron	Medial triambic icosahedron	Great triambic icosahedron
File:Zeroth stellation of icosahedron.svg	File:Sixteenth stellation of icosahedron.png	File:First stellation of icosahedron.png	File:Ninth stellation of icosahedron.png
File:Stellation diagram of icosahedron.svg	File:Great icosahedron stellation facets.svg	File:Small triambic icosahedron stellation facets.svg	File:Great triambic icosahedron stellation facets.svg
Regular compounds			Others
Compound of five octahedra	Compound of five tetrahedra	Compound of ten tetrahedra	Excavated dodecahedron	Final stellation
File:First compound stellation of icosahedron.png	File:Second compound stellation of icosahedron.png	File:Third compound stellation of icosahedron.png	File:Third stellation of icosahedron.svg	File:Seventeenth stellation of icosahedron.png
File:Compound of five octahedra stellation facets.svg	File:Compound of five tetrahedra stellation facets.svg	File:Compound of ten tetrahedra stellation facets.svg	File:Excavated dodecahedron stellation facets.svg	File:Echidnahedron stellation facets.svg