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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.
Kepler's definitionIn 1619 Kepler defined stellation for polygons and polyhedra, as the process of extending edges or faces until they meet to form a new polygon or polyhedron. He stellated the dodecahedron to obtain two of the regular star polyhedra (two of the Kepler-Poinsot polyhedra).
Stellated polygons A stellation of a regular polygon is a star polygon or polygon compound.
It can be represented by the symbol , 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 . And so the (m-1)st stellation is .
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 is made of 2 triangles , and is made of 2 pentagrams .
A regular n-gon has (n-4)/2 stellations if n is even, and (n-3)/2 stellations if n is odd.
Like the heptagon, the octagon also has two octagrammic stellations, one, being a star polygon, and the other, , being the compound of two squares.
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:
Seventeen of the nonconvex uniform polyhedra are stellations of Archimedean solids.
Miller's rulesJ.C.P. Miller's rules for the regular icosahedron are :
- "(i) The faces must lie in 12 planes of faces of the regular icosahedron.
- "(ii) All parts composing the faces must be the same in each plane, although they may be quite disconnected.
- "(iii) The parts included in any one plane must have trigonal symmetry, without or with reflection.
- "(iv) The parts included in any plane must all be "accessible" in the completed solid (i.e. they must be on the "outside". In certain cases we should require models of enormous size in order to see all the outside. With a model of ordinary size, some parts of the "outside" could only be explored by a crawling insect).
- "(v) We exclude from consideration cases where the parts can be divided into two sets, each giving a solid with as much symmetry as the whole figure. But we allow the combination of an enantiomorphous pair having no common part (which actually occurs in just one case)."
Rules (i) to (iii) are just requirements for icosahedral symmetry. Rule (iv) excludes buried holes, so that no two stellations look outwardly identical. Rule (v) is meant to prevent disconnected compounds of other stellations: The 59 icosahedra understood that vertex-connected or edge-connected concentric compounds were not allowed either, though the rule appears to be ambiguous.
Under Miller's rules we find:
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 stellationMiller'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 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.
Naming stellationsThe first systematic naming of stellated polyhedra was Cayley's naming of the regular star polyhedra. This system was widely, but not always systematicall,y adopted for other polyhedra and higher polytopes.
John Conway devised a terminology for stellated polygons, polyhedra and polychora (Coxeter 1974). In this system the process of extending edges to create a new figure is called stellation, that of extending faces is called greatening and that of extending cells is called aggrandizement (this last does not apply to polyhedra). This allows a systematic use of words such as 'stellated', 'great, and 'grand' in devising names for the resulting figures. For example Conway proposed some minor variations to the names of the Kepler-Poinsot polyhedra.
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.
External links -
- - Software for exploring polyhedra and printing nets for their physical construction. Includes uniform polyhedra, stellations, compounds, Johnson solids, etc.
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