Stellation

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Stellation is a process of constructing new polygons (in two dimensions), new polyhedra in three dimensions, or, in general, new polytopes in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again. The new figure is a stellation of the original. In 1619 Kepler defined stellation for polygons and polyhedra, and stellated the dodecahedron to obtain two of the regular star polyhedra (two of the Kepler-Poinsot solids).

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.

1 Stellated polygons
2 Stellated polyhedra
3 See also
4 References
5 External links

[edit] Stellated polygons

A stellation of a regular polygon is a star polygon or polygon compound.

It can be represented by the symbol {n/m}, where n is the number of vertices, and m is the step used in sequencing the edges around it. If m is one, it is the zeroth stellation, and a regular polygon {n}. And so the (m-1)^st stellation is {n/m}.

A polygon compound appears if n and m have a common divisor, and the full stellation require multiple cyclic paths to complete it. For example a hexagram {6/3} is made of 2 triangles {3}, and {10/4} is made of 2 pentagrams {5/2}.

A regular n-gon has (n-4)/2 stellations if n is even, and (n-3)/2 stellations if n is odd.

The pentagram, {5/2}, is the only stellation of a pentagon	The hexagram, {6/2}, the stellation of a hexagon and a compound of two triangles.	The enneagon has 3 enneagrammic forms: {9/2}, {9/3}, {9/4}, with {9/3} being 3 triangles.
The heptagon has two heptagrammic forms: {7/2}, {7/3}

Like the heptagon, the octagon also has two octagrammic stellations, one, {8/3} being a star polygon, and the other, {8/2}, being the compound of two squares.

[edit] 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 extended 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 appearance. The four kinds of stellation just defined are all subsets of the Miller stellations.

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 solids.
There are 58 stellations of the icosahedron, including the great icosahedron (one of the Kepler-Poinsot solids), and the 2nd and final stellations of the icosahedron. The 59th model in "The 59 Icosahedra" is the original icosahedron itself.

The Archimedean solids and their duals can also be stellated. Here we usually add the rule that all of the original faces must "contribute" to the stellation, so 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.

Other rules for stellation. Miller's rules by no means represent the "correct" way to enumerate stellations however. 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 process to facetting, whereby parts are removed from a polyhedron without creating any new vertices. For every stellation of some polyhedron, there is a 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's stellation models.

[edit] See also

List of Wenninger polyhedron models Includes 44 stellated forms of the octahedron, dodecahedron, icosahedron, and icosidodecahedron, enumerated the 1974 book "Polyhedron Models" by Magnus Wenninger
Polyhedral compound Includes 5 regular compounds and 4 dual regular compounds.

[edit] References

Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9.
Coxeter Du Val, P.; Flather, H. T.; and Petrie, J. F. The Fifty-Nine Icosahedra. Stradbroke, England: Tarquin Publications, 1999.
Messer, P.; Stellations of the rhombic triacontahedron and beyond, Symmetry: culture and science, 11 (2000), pp 201-230.
Bridge, N. J.; Facetting the dodecahedron, Acta Crystallographica A30 (1974), pp. 548-552.