Polytope


In elementary geometry, a polytope is a geometric object with flat sides. Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -dimensional polytope or -polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a -polytope consist of -polytopes that may have -polytopes in common.
Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations, decompositions or tilings of curved manifolds including spherical polyhedra, and set-theoretic abstract polytopes.
Polytopes of more than three dimensions were first discovered by Ludwig Schläfli before 1853, who called such a figure a polyschem. The German term Polytop was coined by the mathematician Reinhold Hoppe, and was introduced to English mathematicians as polytope by Alicia Boole Stott.

Definitions and terminology

Nowadays, the term polytope is a broad term that covers a wide class of objects, and various definitions appear in the mathematical literature. Many of these definitions are not equivalent to each other, resulting in different overlapping sets of objects being called polytopes. They represent different approaches to generalizing the convex polytopes to include other objects with similar properties.
The original approach broadly followed by Ludwig Schläfli, Thorold Gosset and others begins with the extension by analogy into four or more dimensions, of the idea of a polygon and polyhedron respectively in two and three dimensions.
Attempts to generalise the Euler characteristic of polyhedra to higher-dimensional polytopes led to the development of topology and the treatment of a decomposition or CW-complex as analogous to a polytope. In this approach, a polytope may be regarded as a tessellation or decomposition of some given manifold. An example of this approach defines a polytope as a set of points that admits a simplicial decomposition. In this definition, a polytope is the union of finitely many simplices, with the additional property that, for any two simplices that have a nonempty intersection, their intersection is a vertex, edge, or higher dimensional face of the two. However this definition does not allow star polytopes with interior structures, and so is restricted to certain areas of mathematics.
The discovery of star polyhedra and other unusual constructions led to the idea of a polyhedron as a bounding surface, ignoring its interior. In this light convex polytopes in p-space are equivalent to tilings of the -sphere, while others may be tilings of other elliptic, flat or toroidal -surfaces – see elliptic tiling and toroidal polyhedron. A polyhedron is understood as a surface whose faces are polygons, a 4-polytope as a hypersurface whose facets are polyhedra, and so forth.
The idea of constructing a higher polytope from those of lower dimension is also sometimes extended downwards in dimension, with an seen as a 1-polytope bounded by a point pair, and a point or vertex as a 0-polytope. This approach is used for example in the theory of abstract polytopes.
In certain fields of mathematics, the terms "polytope" and "polyhedron" are used in a different sense: a polyhedron is the generic object in any dimension and polytope means a bounded polyhedron. This terminology is typically confined to polytopes and polyhedra that are convex. With this terminology, a convex polyhedron is the intersection of a finite number of halfspaces and is defined by its sides while a convex polytope is the convex hull of a finite number of points and is defined by its vertices.
Polytopes in lower numbers of dimensions have standard names:
Dimension
of polytope		Description
−1Nullitope
0Monogon
1Digon
2Polygon
3Polyhedron
4Polychoron

Elements

A polytope comprises elements of different dimensionality such as vertices, edges, faces, cells and so on. Terminology for these is not fully consistent across different authors. For example, some authors use face to refer to an -dimensional element while others use face to denote a 2-face specifically. Authors may use j-face or j-facet to indicate an element of j dimensions. Some use edge to refer to a ridge, while H. S. M. Coxeter uses cell to denote an -dimensional element.
The terms adopted in this article are given in the table below:
Dimension
of element		Term
−1Nullity
0Vertex
1Edge
2Face
3Cell
jj-face – element of rank j = −1, 0, 1, 2, 3,..., n
n − 3Peak – -face
n − 2Ridge or subfacet – -face
n − 1Facet – -face
nThe polytope itself

An n-dimensional polytope is bounded by a number of -dimensional facets. These facets are themselves polytopes, whose facets are -dimensional ridges of the original polytope. Every ridge arises as the intersection of two facets. Ridges are once again polytopes whose facets give rise to -dimensional boundaries of the original polytope, and so on. These bounding sub-polytopes may be referred to as faces, or specifically j-dimensional faces or j-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.

Important classes of polytopes

Convex polytopes

A polytope may be convex. The convex polytopes are the simplest kind of polytopes, and form the basis for several different generalizations of the concept of polytopes. A convex polytope is sometimes defined as the intersection of a set of half-spaces. This definition allows a polytope to be neither bounded nor finite. Polytopes are defined in this way, e.g., in linear programming. A polytope is bounded if there is a ball of finite radius that contains it. A polytope is said to be pointed if it contains at least one vertex. Every bounded nonempty polytope is pointed. An example of a non-pointed polytope is the set. A polytope is finite if it is defined in terms of a finite number of objects, e.g., as an intersection of a finite number of half-planes.
It is an integral polytope if all of its vertices have integer coordinates.
A certain class of convex polytopes are reflexive polytopes. An integral is reflexive if for some integral matrix,, where denotes a vector of all ones, and the inequality is component-wise. It follows from this definition that is reflexive if and only if for all. In other words, a of differs, in terms of integer lattice points, from a of only by lattice points gained on the boundary. Equivalently, is reflexive if and only if its dual polytope is an integral polytope.

Regular polytopes

s have the highest degree of symmetry of all polytopes. The symmetry group of a regular polytope acts transitively on its flags; hence, the dual polytope of a regular polytope is also regular.
There are three main classes of regular polytope which occur in any number of dimensions:
Dimensions two, three and four include regular figures which have fivefold symmetries and some of which are non-convex stars, and in two dimensions there are infinitely many regular polygons of n-fold symmetry, both convex and star. But in higher dimensions there are no other regular polytopes.
In three dimensions the convex Platonic solids include the fivefold-symmetric dodecahedron and icosahedron, and there are also four star Kepler-Poinsot polyhedra with fivefold symmetry, bringing the total to nine regular polyhedra.
In four dimensions the regular 4-polytopes include one additional convex solid with fourfold symmetry and two with fivefold symmetry. There are ten star Schläfli-Hess 4-polytopes, all with fivefold symmetry, giving in all sixteen regular 4-polytopes.

Star polytopes

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

Properties

Euler characteristic

Since a convex polytope P in dimensions is contractible to a point, the Euler characteristic of its boundary ∂P is given by the alternating sum:
This generalizes Euler's formula for polyhedra.

Internal angles

The Gram–Euler theorem similarly generalizes the alternating sum of internal angles for convex polyhedra to higher-dimensional polytopes:

Generalisations of a polytope

Infinite polytopes

Not all manifolds are finite. Where a polytope is understood as a tiling or decomposition of a manifold, this idea may be extended to infinite manifolds. plane tilings, space-filling and hyperbolic tilings are in this sense polytopes, and are sometimes called apeirotopes because they have infinitely many cells.
Among these, there are regular forms including the regular skew polyhedra and the infinite series of tilings represented by the regular apeirogon, square tiling, cubic honeycomb, and so on.