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The compound of five octahedra is one of the five regular polyhedron compounds. This polyhedron can be seen as either a polyhedral
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. Replacing the octahedra by tetrahemihexahedra leads to the
MathWorld: Octahedron5-Compound
Paper Model Compound of Five Octahedra
* VRML model
* Polyhedral stellation Polyhedral compounds {{polyhedron-stub
The compound of five octahedra is one of the five regular polyhedron compounds. This polyhedron can be seen as either a polyhedral stellation
In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specific el ...
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.
As a stellation
It is the secondstellation
In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specific el ...
of the icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
, 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.
As a compound
It can also be seen as a polyhedral compound of five octahedra arranged in icosahedral symmetry (Ih). The spherical andstereographic
Stereoscopy (also called stereoscopics, or stereo imaging) is a technique for creating or enhancing the illusion of depth in an image by means of stereopsis for binocular vision. The word ''stereoscopy'' derives . Any stereoscopic image is ...
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. Replacing the octahedra by tetrahemihexahedra leads to the
compound of five tetrahemihexahedra
A compound of five tetrahemihexahedra is a uniform polyhedron compound and a symmetric arrangement of five tetrahemihexahedron, tetrahemihexahedra. It is Chirality (mathematics), chiral with icosahedral symmetry (I).
Related polyhedra
Its conv ...
.
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
* Compound of three octahedra * Compound of four octahedra *Compound of ten octahedra
The compounds of ten octahedra UC15 and UC16 are two uniform polyhedron compounds. They are composed of a symmetric arrangement of 10 octahedron, octahedra, considered as triangular antiprisms, aligned with the axes of three-fold rotational symm ...
* Compound of twenty octahedra
References
*Peter R. Cromwell
Peter may refer to:
People
* List of people named Peter, a list of people and fictional characters with the given name
* Peter (given name)
** Saint Peter (died 60s), apostle of Jesus, leader of the early Christian Church
* Peter (surname), a sur ...
, ''Polyhedra'', Cambridge, 1997.
*
* (1st Edn University of Toronto (1938))
* H.S.M. Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
, '' 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
* VRML model
* Polyhedral stellation Polyhedral compounds {{polyhedron-stub