# Polytope

(redirected from Polytopes)

## polytope

[′päl·i‚tōp]
(mathematics)
A finite region in n-dimensional space (n = 2, 3, 4, …), enclosed by a finite number of hyperplanes; it is the n-dimensional analog of a polygon (n = 2) and a polyhedron (n = 3).
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Polytope

(1) A polyhedron.

(2) A geometric figure that is the union of a finite number of convex polyhedrons of an arbitrary number of dimensions arbitrarily arranged in n-dimensional space. This concept is often made use of in topology and can easily be extended to the case of n-dimensional space.

Let us consider a half space in the n-dimensional space Rn that is, the set of all points located on one side of some in — 1)-dimensional hyperplane of the space along with the points of the hyperplane itself. Analytically, the half space is the set of all points of Rn whose coordinates satisfy a linear inequality of the form a1x1 + a2x2 + … + anxn + b ≥0. The intersection of a finite number of half spaces—if it is bounded— is the most general convex polyhedron of arbitrary dimension ≤ n located in Rn. A poly tope in the general sense of the word is the union of a finite number of such polyhedrons. When n = 2, we obtain two-dimensional polytopes, or polygons, which are not necessarily convex. One-dimensional polytopes are broken lines that need not be connected and may be branched—at any vertex any number of segments may meet. A zero-dimensional polytope is a finite set of points. A three-dimensional polytope can always be partitioned into polyhedrons of the simplest type —that is, into simplexes. Simplexes of dimension 0, 1, 2, and 3 correspond, respectively, to a point, a line segment, a triangle, and a tetrahedron, which is in general irregular. This partitioning, moreover, can be performed in such a way that either two of the resulting simplexes have no points in common or they share a face. Such partitions of a polytope into simplexes are called triangulations and constitute a fundamental research technique in combinatorial topology.

The concept of polytope permits of various generalizations. For example, curved polytopes are the images of polytopes under topological mappings; thus an arbitrary curved surface may be regarded as the topological image of a polyhedral surface. Another example is infinite polytope, which consists of an infinite set of convex polyhedrons (simplexes).

### REFERENCES

Aleksandrov, P. S. Lektsii po analiticheskoi geometrii…. Moscow, 1968.
Aleksandrov, P. S. Kombinatornaia topologiia. Moscow-Leningrad, 1947.
Pontriagin, L. S. Osnovy kombinatornoi topologii. Moscow-Leningrad, 1947.
Aleksandrov, P. S., and B. A. Pasynkov. Vvedenie v teoriiu razmernosti. Moscow, 1973.

P. S. ALEKSANDROV

References in periodicals archive ?
For n = 3 and 0 [less than or equal to] [k.sub.1] [less than or equal to] [[??].sub.1] [less than or equal to] 1 the following formula can be obtained via triangulation method by decoupling polytopes of the two RV sets [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] into simplexes with origin as the common vertex:
[15.] Grunbaum, Branko (2003), Convex Polytopes, Graduate Texts in Mathematics (2nd ed.), Springer, ISBN 9780387004242
Several other convex approximations of the stability region such as boxes [3,4], ellipsoids [5,6], polytopes [7,8], or other convex sets [9,10] are widely used in robust control.
The cells are called Dirichlet regions, Thiessen polytopes, or Voronoi polygons.
KRIZEK, Gradient superconvergence on uniform simplicial partitions of polytopes, IMA J.
The volume is divided into four sections, dealing in turn with Xenakis's tutelage in the Le Corbusier studio, his writings on architecture, projects undertaken as an independent architect following his acrimonious split from Le Corbusier, and his various Polytopes, which perhaps represent Xenakis's most, comprehensive and coherent synthesis of music and architecture.
For example, there are six regular polytopes in four-dimensions that are the analogues of the Platonic Solids.
Other topics include metric graph theory and geometry, extremal problems for convex lattice polytopes, expansive motions, unfolding orthogonal polyhedra, the discharging method in combinatorial geometry, and line transversals to families of translated ovals.
Coxeter's work, especially his treatise entitled Regular Polytopes, went on to influence various people, including Buckminster Fuller, who credits Coxeter's vision in developing his famous geodesic domes.
POLYTOPES" IS THE COLLECTIVE NAME of a series of multimedia installations, including sound, light and architecture, conceived by IANNIS XENAKIS during the 1960s and 1970s.
The mathematics necessary for a sweep-plane algorithm to generate uniform random variates over simple polytopes in high dimensions was collected in Leydold and Hormann [1998].

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