Small-icosiicosahedron-in-icosidodecahedron

The

compound Compound may refer to: Architecture and built environments * Compound (enclosure), a cluster of buildings having a shared purpose, usually inside a fence or wall ** Compound (fortification), a version of the above fortified with defensive struct ...

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

. This compound was first described by

Edmund Hess Edmund Hess (17 February 1843 – 24 December 1903) was a German mathematician who discovered several regular polytopes. See also * Schläfli–Hess polychoron * Hess polytope References * ''Regular Polytopes In mathematics, a regu ...

in 1876. It is unique among the regular compounds for not having a regular convex hull.

As a stellation

It is the second

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 In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Cata ...

with rhombic-based

pyramid A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilat ...

s 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 In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram. The outer vertices of a compound can be connected ...

of five

octahedra In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...

arranged in

icosahedral symmetry In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the ...

(I_h). The

spherical A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...

and

stereographic 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.

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.

References

* Peter R. Cromwell, ''Polyhedra'', Cambridge, 1997. * * (1st Edn University of Toronto (1938)) * H.S.M. Coxeter, ''

Regular Polytopes In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...

'', (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 VRML (Virtual Reality Modeling Language, pronounced ''vermal'' or by its initials, originally—before 1995—known as the Virtual Reality Markup Language) is a standard file format for representing 3-dimensional (3D) interactive vector graphi ...

model

* Polyhedral stellation Polyhedral compounds {{polyhedron-stub