Editing Stellation (section)

== Stellating polyhedra ==

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A polyhedron is stellated by extending the edges or face planes of a polyhedron until they meet again to form a new polyhedron or compound. The interior of the new polyhedron is divided by the faces into a number of cells. The face planes of a polyhedron may divide space into many such cells, and as the stellation process continues then more of these  cells will be enclosed. 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" [[H.S.M. Coxeter|Coxeter]], Du Val, Flather and Petrie record five rules suggested by [[J. C. P. Miller|Miller]].  Although these rules refer specifically to the icosahedron's geometry, they have been 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 usually considered a stellation of the [[cuboctahedron]].

Generalising Miller's rules 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 [[List of Wenninger polyhedron models#Stellations: models W19 to W66|Wenninger]]'s ''Polyhedron Models'')
* An unknown number of stellations of the [[icosidodecahedron]]; there are 7,071,671 non-[[chirality (mathematics)|chiral]] stellations, but the number of chiral stellations is unknown. (20 are shown in [[List of Wenninger polyhedron models#Stellations of icosidodecahedron|Wenninger]]'s ''Polyhedron Models''<!--W47 to W66; although W60 is called the 14th and W62 the 15th, W61, a compound of the great stellated dodecahedron and the great icosahedron, is also a stellation.-->)

Seventeen of the nonconvex uniform polyhedra are stellations of Archimedean solids.

===Miller's rules===
In the book ''[[The fifty nine icosahedra|The Fifty-Nine Icosahedra]]'', J.C.P. Miller proposed a [[The fifty nine icosahedra#Miller's rules|set of rules]] for defining which stellation forms should be considered "properly significant and distinct".

These rules have been adapted for use with stellations of many other polyhedra. Under Miller's rules we find:
* There are no stellations of the [[tetrahedron]], because all faces are adjacent
* There are no stellations of the [[cube]], because non-adjacent faces are parallel and thus cannot be extended to meet in new edges
* There is 1 stellation of the [[octahedron]], the [[stella octangula]]
* There are 3 stellations of the [[dodecahedron]]: the [[small stellated dodecahedron]], the [[great dodecahedron]] and the [[great stellated dodecahedron]], all of which are Kepler–Poinsot polyhedra.
* There are 58 stellations of the [[icosahedron]], including the [[great icosahedron]] (one of the Kepler–Poinsot polyhedra), and the [[Second stellation of icosahedron|second]] and [[Final stellation of the icosahedron|final]] stellations of the icosahedron.  The 59th model in ''The fifty nine icosahedra'' is the original icosahedron itself.

Many "Miller stellations" cannot be obtained directly by using Kepler's method. For example many have hollow centres where the original faces and edges of the core polyhedron are entirely missing: there is nothing left to be stellated. On the other hand, Kepler's method also yields stellations which are forbidden by Miller's rules since their cells are edge- or vertex-connected, even though their faces are single polygons. This discrepancy received no real attention until Inchbald (2002).

===Other rules for stellation=== 
Miller's rules by no means represent the "correct" way to enumerate stellations.  They are based on combining parts within the [[stellation diagram]] in certain ways, and don't take into account the topology of the resulting faces.  As such there are some quite reasonable stellations of the icosahedron that are not part of their list – one was identified by James Bridge in 1974, while some "Miller stellations" are questionable as to whether they should be regarded as stellations at all – one of the icosahedral set comprises several quite disconnected cells floating symmetrically in space.

As yet an alternative set of rules that takes this into account has not been fully developed. Most progress has been made based on the notion that stellation is the reciprocal or dual process to [[facetting]], whereby parts are removed from a polyhedron without creating any new vertices. For every stellation of some polyhedron, there is a [[Duality (mathematics)|dual]] facetting of the [[dual polyhedron]], and vice versa. By studying facettings of the dual, we gain insights into the stellations of the original. Bridge found his new stellation of the icosahedron by studying the facettings of its dual, the dodecahedron.

Some polyhedronists take the view that stellation is a two-way process, such that any two polyhedra sharing the same face planes are stellations of each other. This is understandable if one is devising a general algorithm suitable for use in a computer program, but is otherwise not particularly helpful.

Many examples of stellations can be found in the [[List of Wenninger polyhedron models#Stellations: models W19 to W66|list of Wenninger's stellation models]].