Polytope

For an introductory account, see Introduction to polytopes.

A polytope is an object that generalizes the intuitive notions of "flat" shapes like polygons and polyhedra to more dimensions.

The term "polytope" can have many different and often contradictory meanings, depending on the context. These meanings often differ only in 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

Elements

Main article: Element

A square pyramid, with a diagram representing membership within its elements.

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.

Flags

Main article: Flag

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:

All flags of a polytope must be of equal length.

Rank

Main article: Rank

A related concept all polytopes share is that of rank. The rank of a polytope is the number of elements in its flags.

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

Polytopes by rank

Rank	Element	Polytope
0	Vertex	Point
1	Edge	Polytelon
2	Face	Polygon
3	Cell	Polyhedron
4	Teron	Polychoron

Gernally a polytope of rank n  can be called an n -polytope. Specific names for polytopes of ranks lower than 2 or greater than 4 exist, but are much rarer or nonstandard.

Rank is related to the concept of dimensionality, and in many definitions the two are the same, so they may be used interchangeably. Dimensionality is roughly the number of dimensions of the space a polytope occupies.

If a definition allows for skew polytopes then the rank and dimensionality can vary from each other. For example the square duocomb is a skew polytope which is rank 3, but 4-dimensional.

Diamond property

Main article: Dyadicity

A last condition, common to virtually all definitions of a polytope, is the diamond property. This can be stated as follows:

For n -polytope 𝓟 every rank n -2 element of 𝓟 (ridge) is an element of exactly two rank n -1 elements of 𝓟 (facets).

Since the elements of 𝓟 must themselves be polytopes this definition is applied recursively.

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 rank 0 elements of a polytope as points embedded in some space (most often Euclidean space), and each higher rank 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 rank 0 elements are interpreted as points in some space, most often Euclidean space. There are two branching approaches to defining higher rank 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 rank. Rank 2 polytopes being called polygons, followed by polyhedra (rank 3) and polychora (rank 5).

There are several specific named categories of polytopes based on their properties. If the symmetry of a polytope is transitive on the polytope's flags, then it is regular. If a polytope is isogonal and all its edges are the same length, and all its elements have the same property, it is called a uniform polytope. If the requirement of uniform elements is removed, allowing for things such as Johnson solids as cells, then it is called a scaliform polytope, and if the requirement of vertex-transitivity is removed and convexity is added instead, then it is called a convex regular-faced polytope.

Related concepts

The following concepts fail to meet the "classical" constraints of finite planar polytopes, but still have a valid abstract polytope structure:

• Tilings
• Infinite polytopes, including apeirotopes and dense polytopes
• Skew polytopes
• Spherical polytopes
• Projective polytopes
• Hyperbolic polytopes
• Polytwisters

The following concepts generally do not meet the definition of an abstract polytope, but are associated with the study of polytopes:

• Compounds
• Complex polytopes
• Maps and maniplexes
• Incidence geometries, including:
  • Configurations
  • Hypertopes
  • Hypermaps
  • Exotic polytopoids
  • Complex polytopes