“Polytope”的概念、定义、习题解题方法及翻译-数学辞典

单词

Polytope

释义

Polytope

A convex polytope may be defined as the Convex Hull of a finite set of points (which are always bounded), or as theintersection of a finite set of half-spaces. Explicitly, a -dimensional polytope may be specified as the set of solutionsto a system of linear inequalities

where

is a real

Matrix and

is a real

-Vector. The positions of the verticesgiven by the above equations may be found using a process called Vertex Enumeration.

A regular polytope is a generalization of the Platonic Solids to an arbitrary Dimension. TheNecessary condition for the figure with Schläfli Symbol to be a finitepolytope is

Sufficiency can be established by consideration of the six figures satisfying this condition. The tablebelow enumerates the six regular polytopes in 4-D (Coxeter 1969, p. 414).

Name	Schläfli Symbol
Regular Simplex		5	10	10	5
Hypercube		16	32	24	8
16-Cell		8	24	32	16
24-Cell		24	96	96	24
120-Cell		600	1200	720	120
600-Cell		120	720	1200	600

Here, is the number of Vertices, the number of Edges, the number of Faces, and the number of cells. These quantities satisfy the identity

which is a version of the Polyhedral Formula.

For -D with , there are only three regular polytopes, the Measure Polytope, Cross Polytope, andregular Simplex (which are analogs of the Cube, Octahedron, and Tetrahedron).

See also 16-Cell, 24-Cell, 120-Cell, 600-Cell, Cross Polytope, Edge (Polytope),Face, Facet, Hypercube, Incidence Matrix, Measure Polytope, Ridge,Simplex, Tesseract, Vertex (Polyhedron)

References

Solid Geometry

Coxeter, H. S. M. ``Regular and Semi-Regular Polytopes I.'' Math. Z. 46, 380-407, 1940.

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969.

Eppstein, D. ``Polyhedra and Polytopes.''http://www.ics.uci.edu/~eppstein/junkyard/polytope.html.

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