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.

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 type of edge, 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), then it is called a convex regular-faced polytope.

Common definitions for polytopes

 * Must be strictly convex
 * Must be a regular polytope
 * Must be a uniform polytope
 * Must not have any non-linear elements
 * Must not have an infinite number of element s
 * Must not extend infinitely
 * Must not self-intersect
 * Must exist completely within its rank (native dimension)
 * Must exist completely within Euclidean space
 * Must have a genus of 0