Polytope


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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).

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

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