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
Famous quotes containing the words regular and/or compounds:
“It was inspiriting to hear the regular dip of the paddles, as if they were our fins or flippers, and to realize that we were at length fairly embarked. We who had felt strangely as stage-passengers and tavern-lodgers were suddenly naturalized there and presented with the freedom of the lakes and woods.”
—Henry David Thoreau (1817–1862)
“We can come up with a working definition of life, which is what we did for the Viking mission to Mars. We said we could think in terms of a large molecule made up of carbon compounds that can replicate, or make copies of itself, and metabolize food and energy. So that’s the thought: macrocolecule, metabolism, replication.”
—Cyril Ponnamperuma (b. 1923)