Polyhedral compound

﻿
Polyhedral compound

A polyhedral compound is a polyhedron that is itself composed of several other polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram.

Neighbouring vertices of a compound can be connected to form a convex polyhedron called the convex hull. The compound is a facetting of the convex hull.

Another convex polyhedron is formed by the small central space common to all members of the compound. This polyhedron can be considered the core for a set of stellations including this compound. (See List of Wenninger polyhedron models for these compounds and more stellations.)

Regular compounds

A regular polyhedron compound can be defined as a compound which, like a regular polyhedron, is vertex-transitive, edge-transitive, and face-transitive. With this definition there are 5 regular compounds.

Components Picture Convex hull Core Symmetry Dual
Compound of two tetrahedra, or stella octangula Cube Octahedron Oh Self-dual
Compound of five tetrahedra, or chiro-icosahedron Dodecahedron Icosahedron I enantiomorph, or chiral twin
Compound of ten tetrahedra, compound of two chiro-icoshedra, or icosiicosahedron Dodecahedron Icosahedron Ih Self-dual
Compound of five cubes, or rhombihedron Dodecahedron Rhombic triacontahedron Ih Compound of five octahedra
Compound of five octahedra, or small icosiicosahedron Icosidodecahedron Icosahedron Ih Compound of five cubes

Best known is the compound of two tetrahedra, often 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 finite stellation thereof.

The stella octangula can also be regarded as a dual-regular compound.

The compound of five tetrahedra 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.

Dual-regular compounds

A dual-regular compound is composed of a regular polyhedron (one of the Platonic solids or Kepler-Poinsot polyhedra) and its regular dual, arranged reciprocally about a common intersphere or midsphere, such that the edge of one polyhedron intersects the dual edge of the dual polyhedron. There are five such compounds.

The dual-regular compound of a tetrahedron with its dual polyhedron is also the regular Stella octangula, since the tetrahedron is self-dual.

The cube-octahedron and dodecahedron-icosahedron dual-regular compounds are the first stellations of the cuboctahedron and icosidodecahedron, respectively.

The compound of the small stellated dodecahedron and great dodecahedron looks outwardly the same as the small stellated dodecahedron, because the great dodecahedron is completely contained inside. For this reason, the image shown above shows the small stellated dodecahedron in wireframe.

Uniform compounds

In 1976 John Skilling published Uniform Compounds of Uniform Polyhedra which enumerated 75 compounds (including 6 as infinite prismatic sets of compounds, #20-#25) made from uniform polyhedra with rotational symmetry. (Every vertex is vertex-transitive and every vertex is transitive with every other vertex.) This list includes the five regular compounds above. 

Here is a pictorial table of the 75 uniform compounds as listed by Skilling. Most are singularly colored by each polyhedron element. Some as chiral pairs are colored by symmetry of the faces within each polyhedron.

• 1-19: Miscellaneous (4,5,6,9,17 are the 5 regular compounds)
• 46-67: Tetrahedral symmetry embedded in octahedral or icosahedral symmetry,
• 68-75: enantiomorph pairs

Two polyhedra that are compounds but have their elements rigidly locked into place are the small complex icosidodecahedron (compound of icosahedron and great dodecahedron) and the great complex icosidodecahedron (compound of small stellated dodecahedron and great icosahedron). If the definition of a uniform polyhedron is generalised they are uniform.

Group theory

In terms of group theory, if G is the symmetry group of a polyhedral compound, and the group acts transitively on the polyhedra (so that each polyhedra can be sent to any of the others, as in uniform compounds), then if H is the stabilizer of a single chosen polyhedron, the polyhedra can be identified with the orbit space G/H – the coset gH corresponds to which polyhedron g sends the chosen polyhedron to.

Wikimedia Foundation. 2010.

Look at other dictionaries:

• polyhedral compound — poliedrinis junginys statusas T sritis chemija apibrėžtis Policiklinis junginys, kurio kiekvienas žiedas dviem ar trimis bendrais anglies atomais sujungtas su kitu žiedu. atitikmenys: angl. polyhedral compound rus. каркасное соединение;… …   Chemijos terminų aiškinamasis žodynas

• Compound of five octahedra — Type Regular compound Index UC17, W23 Elements (As a compound) 5 octahedra: F = 40, E = 60, V = 30 …   Wikipedia

• Compound of five cubes — Type Regular compound Stellation core rhombic triacontahedron Convex hull Dodecahedron Index …   Wikipedia

• Compound — may refer to: Chemical compounds, combinations of two or more elements 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… …   Wikipedia

• Compound of three octahedra — The compound of three octahedra In mathematics, the compound of three octahedra or octahedron 3 compound is a polyhedral compound formed from three regular octahedra, all sharing a common center but rotated with respect to each other. Although… …   Wikipedia

• Compound of ten tetrahedra — Type regular compound Index UC6, W25 Elements (As a compound) 10 tetrahedra: F = 40, E = 60, V = 20 …   Wikipedia

• Compound of cube and octahedron — Type Compound Stellation core cuboctahedron Convex hull Rhombic dodecahedron …   Wikipedia

• Compound of five tetrahedra — Type Regular compound Index UC5, W24 Elements (As a compound) 5 tetrahedra: F = 20 …   Wikipedia

• Compound of twenty tetrahemihexahedra — Type Uniform compound Index UC19 Polyhedra 20 tetrahemihexahedra Faces …   Wikipedia

• Compound of five tetrahemihexahedra — Type Uniform compound Index UC18 Polyhedra 5 tetrahemihexahedra Faces …   Wikipedia