Polytope

A polytope is an object that generalizes the intuitive notions of "flat" shapes like polygons and polyhedra to any amount of dimensions. An n-dimensional polytope, sometimes just called an n-polytope, consists of various (n–1)-dimensional facets. Each of these facets has itself various (n–2)-dimensional facets called ridges, so that at each of the polytope's ridges, two facets meet. This generalizes the condition that two edges must meet at a polygon's vertex, and that two faces must meet at a polyhedron's edge.

The term "polytope" can have many different and often contradictory meanings, depending on the context. These meanings often differ only on the implicit assumptions made about them, such as whether they must be embedded in a space of a certain dimension, or whether they can have an infinite number of facets. However, some of the objects the word is used to refer to, such as convex polytopes (in convex geometry) and abstract polytopes, are entirely different mathematically. This article presents these various different notions.

Basic concepts
All definitions of the word "polytope" satisfy certain common characteristics. Polytopes all have a basic notion of membership, whereby a polytope can be an element of another. This notion is transitive, meaning that a polytope's elements' elements must also be elements of the original polytope. If one of two polytopes is an element of the other, the two polytopes are said to be incident.

A derived concept is that of a flag. A flag of a polytope is a a maximal chain of elements under the incidence relation. In other words, a flag is a set of elements such that every two are incident to one another, not a subset of any larger such set. All polytopes are subject to the following condition regarding flags:


 * Every two flags of a polytope must be of equal length.

A related concept all polytopes share is that of dimension, or rank. The dimensionality of a polytope is defined recursively as the least integer greater than the dimensionality of all of its elements. Sometimes, a single polytope of dimension –1 called the nullitope is considered as being an element of any other polytope. However, in other contexts, 0-dimensional points take on the role of least elements. The (n–1)-dimensional elements of an n-dimensional polytope are known as its facets.

Polytopes of certain dimensions have special names, as do elements of polytopes of certain dimensions. These are summarized in the following table.

A last condition, common to virtually all definitions of a polytope, is the following:


 * For any two elements A and B of a polytope such that B is an element of A and such that their dimensionalities differ by 2, there are exactly two elements of A that contain B as an element.

In other words, at each of a polytope's ridges, exactly two facets meet. This guarantees that two edges meet at each of a polygon's vertices, two faces meet at each of a polyhedron's edges, two cells meet at each of a polychoron's faces, and so on.

Definitions
There are at least three standard approaches to defining the word "polytope", which despite their similar origins are often studied separately. The geometric approach first interprets the 0-dimensional elements of a polytope as points embedded in some space (most often Euclidean space), and each higher-dimensional element as either a set or a manifold defined by its facets. The convex approach interprets a polytope as a convex point set, and its elements as specific subsets on its surface. Lastly, the abstract approach simply encodes and refines the conditions from the previous section into a partially ordered set.

Geometric approach
Under the geometric approach, a polytope's 0-dimensional elements are interpreted as points in some space, most often Euclidean space. There are two branching approaches to defining higher-dimensional elements. These can either be defined as sets in the geometric-combinatorial approach, or as manifolds in the geometric-topological approach.

Geometric-combinatorial approach
Under the geometric-combinatorial approach, every one of a polytope's elements, save for its vertices, is defined as the set of its facets, or as any other equivalent structure thereof. As such, only the vertices are truly embedded in space, while any lines drawn or faces filled to represent such a polytope are only a graphical representation. This makes some of its properties, like density or volume quite hard if not impossible to define in the general case.

Additional restrictions
The disadvantage of the geometric approach to defining polytopes is that many extra restrictions which are often taken for granted have to be explicitly stated.

Types of polytopes
The most common way to classify polytopes is by their dimension. 2-dimensional polytopes being called polygons, followed by 3D polyhedra and 4D polychora. The general name used for a polytope of n-dimension is n-polytope. Specific names for polytopes of dimensions lower than 2D or greater than 4D exist, but are much rarer or unagreed upon.

If a polytope is isogonal and (geometrically) has one size of edges only, and all of its elements are realizable as such, it is called a uniform polytope. If the requirement of uniform elements is removed, allowing for Johnson solids as cells, then it is called a scaliform polytope, and if the requirement of vertex-transitivity is removed (or rather negated) and convexity would be added instead, then it is called a convex regular-faced polytope.