Политоптарды кеңейту арқылы жаңа фигуралар жасау

Extending the elements of a polytope to form a new figure

In geometry, stellation is the process of extending a polygon in two dimensions, 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 Latin stella, "star". Stellation is the reciprocal or dual process to faceting.

In 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 regular dodecahedron to obtain two regular star polyhedra, the small stellated dodecahedron and great stellated dodecahedron. He also stellated the regular octahedron to obtain the stella octangula, a regular compound of two tetrahedra.

Stellating a regular polygon symmetrically creates a regular star polygon or polygonal compound. These polygons are characterised by the number of times m that the polygonal boundary winds around the centre of the figure. Like all regular polygons, their vertices lie on a circle. m also corresponds to the number of vertices around the circle to get from one end of a given edge to the other, starting at 1. 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 factor). 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 digons are not considered), and stellations if n is odd. The pentagram, {5/2}, is the only stellation of a pentagonThe 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 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.

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 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.

The stellation process can be applied to higher dimensional polytopes as well. A stellation diagram of an n polytope exists in an (n − 1) dimensional hyperplane of a given facet. For example, in 4 space, the great grand stellated 120 cell is the final stellation of the regular 4 polytope 120 cell.

The first systematic naming of stellated polyhedra was Cayley's naming of the regular star polyhedra (nowadays known as the Kepler–Poinsot polyhedra). This system was widely, but not always systematically, 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.

Wenninger noticed that some polyhedra, such as the cube, do not have any finite stellations. However stellation cells can be constructed as prisms which extend to infinity. The figure comprising these prisms may be called a stellation to infinity. By most definitions of a polyhedron, however, these stellations are not strictly 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".

Alongside from his contributions to mathematics, Magnus Wenninger is described in the context of the relationship of mathematics and art as making "especially beautiful" models of complex stellated polyhedra. 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 lithographs by M. C. Escher: Contrast (Order and Chaos), 1950, and Gravitation, 1952.