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 n as an n-dimensional polytope or n-polytope.

What is a Polytope? Any standard text in polyhedral geometry will tell you one (or both) of the following de nitions, usually preceded or followed by a chunk of linear algebra explaining what these words actually mean. 1.A d-polytope P is the convex hull of nitely many points in Rd.

2025年1月20日 · The word polytope is used to mean a number of related, but slightly different mathematical objects. A convex polytope may be defined as the convex hull of a finite set of points (which are always bounded), or as a bounded intersection of a finite set of half-spaces.

polytope has a description by a nite system of inequalities, and every H-polytope can be obtained as the convex hull of a nite set of points (its vertices). Any V-polytope can be viewed as the image of an (n 1)-dimensional simplex

The standard cube is a V-polytope. The standard n-cross-polytope (or n-co-cube) is the set {(x1,...,x n) ∈ En | n i=1 |x i|≤1}. It is also a V-polytope. What happens if we take the dual of a V-polytope (resp. an H-polytope)? The following proposition, although very simple, is an important step in answering the above question: Proposition 4. ...

2024年10月15日 · Polytope is a partition of Euclidean space, resulting in a polytope filling a large portion. A regular polytope is generally of platonic form at arbitrary levels. The Swiss mathematician Ludwig Schläfli discovered the regular polytope Ludwig Schläfli as …

A regular 5-polytope {,,,} exists only if {,,} and {,,} are regular 4-polytopes. The space it fits in is based on the expression: cos 2 ⁡ ( π q ) sin 2 ⁡ ( π p ) + cos 2 ⁡ ( π r ) sin 2 ⁡ ( π s ) {\displaystyle {\frac {\cos ^{2}\left({\frac {\pi }{q}}\right)}{\sin ^{2}\left({\frac {\pi }{p}}\right)}}+{\frac {\cos ^{2}\left({\frac ...

A polytope is figure in geometry. It is a geometric object with flat sides, in any number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on. Sometimes the number of dimensions is shown as a number, as in a polyhedron being a 3-polytope or a tesseract a convex regular 4-polytope.

In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry.

This article is a broad overview of polytope concepts commonly found on this site, with references to important texts.