0-polytope

In geometry, polytope is a generic term that can refer to a two-dimensional polygon, a three-dimensional polyhedron, or any of the various generalizations thereof, including generalizations to higher dimensions (such as a polychoron in four dimensions) and other abstractions (such as unbounded polytopes, tessellations, and abstract polytopes).

When referring to an n-dimensional generalization, the term n-polytope is used. For example, a polygon is a 2-polytope, a polyhedron is a 3-polytope, and a polychoron is a 4-polytope.

History

The Platonic solids, or regular polytopes in three dimensions, were a major focus of study of ancient Greek mathematicians (most notably Euclid's Elements), probably because of their intrinsic aesthetic qualities. In modern times, polytopes and related concepts have found important applications in computer graphics, optimization, search engines and numerous other fields.

Properties

A polytope may be convex. The convex polytopes are the simplest kind of polytopes, and form the basis for different generalizations of the concept of polytopes.

A polytope may be regular. The regular polytopes are a class of highly-symmetrical and aesthetically-pleasing polytopes, including the Platonic solids, which have been studied extensively since ancient times.

A non-convex polytope may be self-intersecting; this class of polytopes include the star polytopes.

Elements

An n-dimensional polytope is bounded by a number of (n−1)-dimensional facets. These facets are themselves polytopes, whose facets are (n−2)-dimensional ridges of the original polytope. Every ridge arises as the intersection of two facets (but the intersection of two facets need not be a ridge). Ridges are once again polytopes whose facets give rise to (n−3)-dimensional boundaries of the original polytope, and so on. These bounding sub-polytopes may be referred to as faces, or specifically k-dimensional faces or k-faces. A 0-dimensional face is called a vertex, and consists of a single point. A 1-dimensional face is called an edge, and consists of a line segment. A 2-dimensional face consists of a polygon, and a 3-dimensional face, sometimes called a cell, consists of a polyhedron.

Note that this terminology is not fully standardized. Existing literature uses terms that are not entirely consistent across different authors. Some authors use face to refer to a facet, while others use face to denote a 2-face specifically. Some sources use edge to refer to a ridge.

Different approaches to defining polytopes

The term polytope is a broad term that covers a wide class of objects, and different definitions are attested in mathematical literature. Many of these definitions are not equivalent, resulting in different sets of objects being called "polytopes". They represent different approaches of generalizing the convex polytopes to include other objects with similar properties and aesthetic beauty.

For example, one approach begins with the 0-dimensional vertex as a 0-polytope (vertices). A 1-dimensional 1-polytope (edge) is constructed by two 0-polytopes. Then 2-polytopes (polygons) are defined as objects whose facets (edges) are 1-polytopes, and 3-polytopes (polyhedra) are defined as objects whose facets (faces) are 2-polytopes, and so forth.

A more modern approach defines a polytope any set of points that admits a simplicial decomposition (see section below), under the observation that all convex polytopes can be decomposed into a union of simplices satisfying certain properties.

A polytope may also be regarded as a tessellation of some given manifold. Under this definition, plane tilings and space tilings are considered to be polytopes, and convex polytopes are regarded to be tilings of elliptic space. Tilings of hyperbolic space are also included under this definition.

The theory of abstract polytopes attempts to detach polytopes from the space containing them, considering their purely combinatorial properties. This allows the definition of the term to be extended to include objects for which it is difficult to define clearly a natural underlying space.

Simplicial decomposition

Given a convex r-dimensional polytope P, a subset of its vertices containing (r+1) linearly independent points defines an r-simplex. It is possible to form a collection of subsets such that the union of the corresponding simplices is equal to P, and the intersection of any two simplices is either empty or a lower-dimensional simplex.

For example, a square has four vertices (its corners). It can be decomposed into two triangles (2-simplices) by dividing it along one of its diagonals. The union of these two triangles is the original square, and their intersection is the diagonal of the square (a 1-simplex, which is the same as a line segment).

Any set of points that can be decomposed into r-simplices in this way is said to have a r-simplicial decomposition. So, the square in the above example has a 2-simplicial decomposition. All convex polytopes have simplicial decompositions; hence, one may define an r-polytope as a set that admits an r-simplicial decomposition. (This definition is attributed to Alexandrov. For a more abstract treatment, see simplicial complex.)

Note that although this definition includes the convex polytopes, it also admits a more general class of objects. For example, a line segment is a 1-polytope under this definition, and so is anything obtained by joining line segments end-to-end:

If every vertex in the 1-polytope has exactly two line segments meeting at it (all except the last example above), it is a topological curve, called a polygonal curve. One may categorize these as open or closed, depending on whether the ends are joined, and as simple or complex, depending on whether they intersect themselves. Closed polygonal curves are called polygons.

Simple polygons in the plane are Jordan curves: they have an interior that is a topological disk. So does a 2-polytope (as can be seen in the third example above), and these are often treated interchangeably with their boundary, the word polygon referring to either.

Now polygons, in turn, can be joined along edges (1-faces), giving a polyhedral surface, called a skew polygon when open and a polyhedron when closed. Simple (non-self-intersecting) polyhedra are interchangeable with their interiors, which are 3-polytopes that can be used to build 4-dimensional forms (sometimes called polychora), and the process can be repeated to obtain the higher polytopes.

Uses

References

Contents