Polyhedral Compound - Regular Compounds

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 Subgroup
restricting
to one
constituent
Dual
Compound of two tetrahedra, or stella octangula Cube Octahedron *432

Oh
*332

Td
Self-dual
Compound of five tetrahedra, or chiro-icosahedron Dodecahedron Icosahedron 532
+
I
332
+
T
enantiomorph, or chiral twin
Compound of ten tetrahedra, compound of two chiro-icoshedra, or icosiicosahedron Dodecahedron Icosahedron *532

Ih
332

T
Self-dual
Compound of five cubes, or rhombihedron Dodecahedron Rhombic triacontahedron *532

Ih
3*2

Th
Compound of five octahedra
Compound of five octahedra, or small icosiicosahedron Icosidodecahedron Icosahedron *532

Ih
3*2

Th
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.

Read more about this topic:  Polyhedral Compound

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