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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]
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)

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.

Contents

As a stellationEdit

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
   
Icosahedron
 
Icosidodecahedron

As a compoundEdit

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.

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

As a facetingEdit

 
Five octahedra in an icosidodecahedron

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

See alsoEdit

ReferencesEdit

  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.

External linksEdit