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In geometry, stellation is the process of extending a polygon in two , a polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure. Starting with an original figure, the process extends specific elements such as its edges or face planes, usually in a symmetrical way, until they meet each other again to form the closed boundary of a new figure. The new figure is a stellation of the original. The word stellation comes from the Latin stellātus, "starred", which in turn comes from the Latin stella, "star". Stellation is the reciprocal or dual process to faceting.
For polyhedra he distinguished two types of stellation: an echinus (Latin for hedgehog or sea urchin), now called edge-stellated, made by extending non-adjacent edges until they intersect; and ostrea (Latin for oyster), now called face-stellated, made by extending faces until they intersect.
He (edge-) stellated the regular dodecahedron to obtain the small stellated dodecahedron and the regular icosahedron to obtain the great stellated dodecahedron. He also (face-) stellated the regular octahedron to obtain the stella octangula, a regular compound of two tetrahedra.
A regular star polygon is represented by its Schläfli symbol { n/ m}, where n is the number of vertices, m is the step used in sequencing the edges around it, and m and n are coprime (have no common divisor). The case m = 1 gives the convex polygon { n}. m also must be less than half of n; otherwise the lines will either be parallel or diverge, preventing the figure from ever closing.
If n and m do have a common factor, then the figure is a regular compound. For example {6/2} is the regular compound of two triangles {3} or hexagram, while {10/4} is a compound of two pentagrams {5/2}.
Some authors use the Schläfli symbol for such regular compounds. Others regard the symbol as indicating a single path which is wound m times around vertex points, such that one edge is superimposed upon another and each vertex point is visited m times. In this case a modified symbol may be used for the compound, for example 2{3} for the hexagram and 2{5/2} for the regular compound of two pentagrams.
A regular n-gon has stellations if n is even (assuming compounds of multiple degenerate are not considered), and 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 (nonagon) {9} has 3 enneagrammic forms: {9/2}, {9/3}, {9/4}, with {9/3} being a compound of 3 triangles. |
The heptagon has two forms: {7/2}, {7/3} |
Like the heptagon, the octagon also has two stellations, one, {8/3} being a star polygon, and the other, {8/2}, being the compound of two squares.
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 method by extending the edges of a polyhedron rarely leads to a new polyhedron because at least 3 extended edges need to intersect at the same point. Examples of edge-stellated polyhedra are the small stellated dodecahedron (edge-stellation of the dodecahedron) and the great stellated dodecahedron (edge-stellation of the icosahedron).
Face-stellated polyhedra are much more common and the following is only about face-stellated polyhedra.
A polyhedron is face-stellated by extending the face planes of a polyhedron until they meet again to form a new polyhedron or compound. When the face planes of the polyhedron are extended indefinitely the space around the polyhedron is divided into unbounded sub spaces and often a number of bounded polyhedrons or cells. Different sets of cells yield different stellations.
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.
A set of cells forming a closed layer around its core is called a shell. A shell may be made up of one or more cell types. A stellation where only complete shells are considered are called main-line stellations.
The stellation where all cells are included is called the complete or final stellation of the polyhedron, for example the final stellation of the icosahedron, the great stellated dodecahedron (the final stellation of the dodecahedron) and the fourth stellation of the rhombic dodecahedron.
Face-stellation can lead to a huge number of different possible polyhedra, so further criteria are often imposed to reduce the set to those stellations that are properly significant and distinct in some way.
Based on such ideas, several restrictive categories of interest have been identified:
We can also identify some other categories:
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:
Seventeen of the nonconvex uniform polyhedra are stellations of Archimedean solids.
These rules have been adapted for use with stellations of many other polyhedra. 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).
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 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 .
For example, in 4-space, the great grand stellated 120-cell is the final stellation of the regular 4-polytope 120-cell.
John Conway devised a terminology for stellated , polyhedron and polychoron (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.
Wenninger's figures occurred as duals of the uniform hemipolyhedra, where the faces that pass through the center are sent to vertices "at infinity".
The Italian Renaissance artist Paolo Uccello created a floor mosaic showing a small stellated dodecahedron in the Basilica of St Mark, Venice, c. 1430. Uccello's depiction was used as the symbol for the Venice Biennale in 1986 on the topic of "Art and Science". The same stellation is central to two by M. C. Escher: Contrast (Order and Chaos), 1950, and Gravitation, 1952.
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