Stellated Polyhedra
The face planes of a polyhedron divide space into many discrete cells. For a symmetrical polyhedron, these cells will fall into groups, or sets, of congruent cells - we say that the cells in such a congruent set are of the same type. A common method of finding stellations involves selecting one or more cell types.
This can lead to a huge number of possible forms, so further criteria are often imposed to reduce the set to those stellations that are significant and unique in some way.
A set of cells forming a closed layer around its core is called a shell. For a symmetrical polyhedron, a shell may be made up of one or more cell types.
Based on such ideas, several restrictive categories of interest have been identified.
- Main-line stellations. Adding successive shells to the core polyhedron leads to the set of main-line stellations.
- Fully supported stellations. The underside faces of a cell can appear externally as an "overhang." In a fully supported stellation there are no such overhangs, and all visible parts of a face are seen from the same side.
- Monoacral stellations. Literally "single-peaked." Where there is only one kind of peak, or vertex, in a stellation (i.e. all vertices are congruent within a single symmetry orbit), the stellation is monoacral. All such stellations are fully supported.
- Primary stellations. Where a polyhedron has planes of mirror symmetry, edges falling in these planes are said to lie in primary lines. If all edges lie in primary lines, the stellation is primary. All primary stellations are fully supported.
- Miller stellations. In "The Fifty-Nine Icosahedra" Coxeter, Du Val, Flather and Petrie record five rules suggested by Miller. Although these rules refer specifically to the icosahedron's geometry, they can easily be adapted to work for arbitrary polyhedra. They ensure, among other things, that the rotational symmetry of the original polyhedron is preserved, and that each stellation is different in outward appearance. The four kinds of stellation just defined are all subsets of the Miller stellations.
We can also identify some other categories:
- A partial stellation is one where not all elements of a given dimensionality are extended.
- A sub-symmetric stellation is one where not all elements are extended symmetrically.
The Archimedean solids and their duals can also be stellated. Here we usually add the rule that all of the original face planes must be present in the stellation, i.e. we do not consider partial stellations. For example the cube is not considered a stellation of the cuboctahedron. There are:
- 4 stellations of the rhombic dodecahedron
- 187 stellations of the triakis tetrahedron
- 358,833,097 stellations of the rhombic triacontahedron
- 17 stellations of the cuboctahedron (4 are shown in Wenninger's "Polyhedron Models")
- Unknown stellations of the icosidodecahedron, but many more than above! (19 are shown in Wenninger's "Polyhedron Models")
Seventeen of the nonconvex uniform polyhedra are stellations of Archimedean solids.
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