Compound_of_five_octahedra

Compound of five octahedra

Compound of five octahedra

Polyhedral compound

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.

Compound of five octahedra

(see here for a 3D model)
TypeRegular compound
IndexUC17, W23
Coxeter symbol[5{3,4}]2{3,5}[1]
Elements
(As a compound)		5 octahedra:
F = 40, E = 60, V = 30
Dual compoundCompound of five cubes
Symmetry groupicosahedral (Ih)
Subgroup restricting to one constituentpyritohedral (Th)
It is also a faceting of the icosidodecahedron.

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.

More information Stellation diagram, Stellation core ...

As a compound

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

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.

More information Spherical polyhedron, Stereographic projections ...

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 octahedra to the standard 4-octahedra compound.

See also

References

  1. [1]
    Regular polytopes, pp.49-50, p.98
  • 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. MR 0676126. (1st Edn University of Toronto (1938))
  • H.S.M. Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, 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.
More information (Convex) icosahedron, Small triambic icosahedron ...


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