Polytope in a sentence

For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope.

If a polytope has the same number of vertices as facets, of edges as ridges, and so forth, and the same connectivities, then the dual figure will be similar to the original and the polytope is self-dual.

In general, the facets of a polytope's dual will be the topological duals of the polytope's vertex figures.

Like any polytope, the elements of a 4-polytope cannot be subdivided into two or more sets which are also 4-polytopes, i.e. it is not a compound.

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.

A polytope is said to be pointed if it contains at least one vertex.

Approaches to definition The term polytope is nowadays a broad term that covers a wide class of objects, and different definitions are attested in mathematical literature.

A root system is often identical to the set of vertices of a regular polytope.

A similar result holds for the number of vertices (of the Gaussian polytope), the number of edges, and in fact, faces of all dimensions.

At the close of the 20th century these latter ideas merged with other work on incidence complexes to create the modern idea of an abstract polyhedron (as an abstract 3-polytope), notably presented by McMullen and Schulte.

Coxeter (1973) 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.

Depending on circumstance, the dual figure may or may not be another geometric polytope.

Elements A polytope comprises elements of different dimensionality such as vertices, edges, faces, cells and so on.

Essentially, these methods attempt to find the shortest pivot path on the arrangement polytope under the linear programming problem.

Every bounded nonempty polytope is pointed.

For an abstract polytope, this simply reverses the ordering of the set.

For example, a polygon has a two-dimensional body and no faces, while a 4-polytope has a four-dimensional body and an additional set of three-dimensional "cells".

For instance, the volume of a convex polytope (described using a membership oracle) can be approximated to high accuracy by a randomized polynomial time algorithm, but not by a deterministic one: see citation.

Generalisations of a polytope Infinite polytopes Not all manifolds are finite.

In rigorous treatments, a triangle is therefore called a 2- simplex (see also Polytope ).