A polyhedral compound is a polyhedron which is itself composed of several other polyhedra sharing a common centre, the three-dimensional analogs of polygonal compounds such as the hexagram. A polyhedron is a geometric shape which in mathematics is defined by three related meanings. ... A hexagram (also known as sexagram or a magicians/sorcerers star) is a six-pointed star, a type of complex star polygon. ...

The best known is the compound of two tetrahedra called the stella octangula, a name given to it by Kepler. The vertices of the two tetrahedra define a cube and the intersection of the two an octahedron, which shares the same face-planes as the compound. Thus it is a stellation of the octahedron, and in fact, the only stellation thereof. For academic journal, see Tetrahedron A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Stella octangula The stella octangula (eight-pointed star), also known as the stellated octahedron, is the polyhedral compound of two tetrahedra. ... Johannes Kepler Johannes Kepler (December 27, 1571 – November 15, 1630), a key figure in the scientific revolution, was a German astronomer, mathematician and astrologer. ... Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Stellation is a process of constructing new polygons (in two dimensions), new polyhedra in three dimensions, or in general new polytopes in n dimensions. ...

The stella octangula is one of only five compounds that are vertex-, edge-, and face-uniform, called regular compounds:

Regular compounds

Components	Picture	Vertices	Face-planes	Symmetry
2 tetrahedra		Cube	Octahedron	Oh
5 tetrahedra		Dodecahedron	Icosahedron	I
10 tetrahedra		Dodecahedron	Icosahedron	Ih
5 cubes		Dodecahedron	Rhombic triacontahedron	Ih
5 octahedra		Icosidodecahedron	Icosahedron	Ih

The compound of 5 tetrahedra actually comes in two enantiomorphic versions, which together make up the compound of 10 tetrahedra. Each of the tetrahedral compounds is self-dual, and the compound of 5 cubes is dual to the compound of 5 octahedra. The symmetry group of an object (e. ... Image File history File links Stella_octangula. ... Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Image File history File links Download high resolution version (868x802, 26 KB)Compound of five tetraahedra, Wenninger model 25 External link VRML model Licensing File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A dodecahedron is literally a polyhedron with 12 faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve pentagonal faces, with three meeting at each vertex. ... An icosahedron [ˌaıkəsəhiːdrən] noun (plural: -drons, -dra [-drə]) is a polyhedron having 20 faces, but usually a regular icosahedron is meant. ... Image File history File links Download high resolution version (892x826, 42 KB)Compound of ten tetraahedra, Wenninger model 25 External link VRML model Licensing File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A dodecahedron is literally a polyhedron with 12 faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve pentagonal faces, with three meeting at each vertex. ... An icosahedron [ˌaıkəsəhiːdrən] noun (plural: -drons, -dra [-drə]) is a polyhedron having 20 faces, but usually a regular icosahedron is meant. ... Image File history File links Download high resolution version (782x782, 35 KB)Compound of five cubes External link VRML model Licensing File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A dodecahedron is literally a polyhedron with 12 faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve pentagonal faces, with three meeting at each vertex. ... The Rhombic triacontahedron is a convex polyhedron with 30 rhombic faces. ... Image File history File links Download high resolution version (813x780, 32 KB)Compound of five octahedra, Wenninger model 23 External link VRML model Licensing This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ... An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. ... An icosahedron [ˌaıkəsəhiːdrən] noun (plural: -drons, -dra [-drə]) is a polyhedron having 20 faces, but usually a regular icosahedron is meant. ... In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone. ...

The stella octangula can also be regarded as a compound of a tetrahedron with its dual polyhedron, inscribed in a common sphere so that the vertices of one line up with the face centres of the other. The corresponding cube-octahedron and dodecahedron-icosahedron compounds are the first stellations of the cuboctahedron and icosidodecahedron, respectively. In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ... A sphere is a perfectly symmetrical geometrical object. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. ...

Dual-regular compounds

There are four compounds which are composed of a regular polyhedron and its dual.

Components	Picture	Symmetry
2 tetrahedra		Oh
Cube and octahedron		Oh
Dodecahedron and icosahedron		Ih
Great icosahedron and Great stellated dodecahedron		Ih

The symmetry group of an object (e. ... Image File history File links Stella_octangula. ... Image File history File links Download high resolution version (801x798, 21 KB)Compound of cube and octahedron External link VRML model Licensing This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ... Image File history File links Download high resolution version (800x800, 27 KB)Compound_of_dodecahedron_and_icosahedron. ... Image File history File links Download high resolution version (829x830, 28 KB)Great Icosahedron-Great Stellated Dodecahedron Compound, Wenninger model 61 External link VRML model Licensing This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ...

External link

• Compound polyhedra – from Virtual Reality Polyhedra
• Polyhedral Compounds
• http://users.skynet.be/polyhedra.fleurent/Compounds_2/Compounds_2.htm
• http://www.progonos.com/furuti/Origami/Modular/virtual.html