In elementary geometry, a polytope is a geometric object with flat sides (faces). Polytopes are the generalization of three-dimensional polyhedra to any...

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n {\displaystyle n} -dimensional...

In mathematics, a regular 4-polytope or regular polychoron is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular...

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

generally, a vertex of a polyhedron or polytope is convex, if the intersection of the polyhedron or polytope with a sufficiently small sphere centered...

mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely...

regular 4-polytopes. The tesseract is also called an 8-cell, C8, (regular) octachoron, or cubic prism. It is the four-dimensional measure polytope, taken...

In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset...

A polytope is a geometric object with flat sides, which exists in any general number of dimensions. The following list of polygons, polyhedra and polytopes...

polytope is a higher-dimensional generalization of a prism. An n-dimensional prismatic polytope is constructed from two (n − 1)-dimensional polytopes...

In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in n-dimensional Euclidean space. A 2-dimensional...

octahedron is the three-dimensional case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan (ℓ1) metric. If the...

polytopes. (In other conventions, the words polyhedron and polytope are used in any dimension, with the distinction between the two that a polytope is...

examples of convex polyhedra. A polyhedron is a 3-dimensional example of a polytope, a more general concept in any number of dimensions. Convex polyhedra are...

In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure...

polyhedra and higher-dimensional polytopes, the term is also used to mean an element of any dimension of a more general polytope (in any number of dimensions)...

polyhedron, like this cube. Every edge is shared by three or more faces in a 4-polytope, as seen in this projection of a tesseract. In geometry, an edge is a particular...

6-dimensional geometry, there are 39 uniform polytopes with E6 symmetry. The two simplest forms are the 221 and 122 polytopes, composed of 27 and 72 vertices respectively...

regular polytopes in Euclidean, spherical and hyperbolic spaces. This table shows a summary of regular polytope counts by rank. Only counting polytopes of...

triangle faces. For a regular k-polytope, the Coxeter-Dynkin diagram of the characteristic k-orthoscheme is the k-polytope's diagram without the generating...

The polyhedral model (also called the polytope method) is a mathematical framework for programs that perform large numbers of operations -- too large to...

The Birkhoff polytope Bn (also called the assignment polytope, the polytope of doubly stochastic matrices, or the perfect matching polytope of the complete...

In geometry, a five-dimensional polytope (or 5-polytope) is a polytope in five-dimensional space, bounded by (4-polytope) facets, pairs of which share a...

In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol {3,3,3}. It is a 5-vertex four-dimensional object bounded by five tetrahedral cells...

mathematics, a matroid polytope, also called a matroid basis polytope (or basis matroid polytope) to distinguish it from other polytopes derived from a matroid...

, q , r , . . . } {\displaystyle \{p,q,r,...\}} that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century...

convex polytopes, a distributive polytope is a convex polytope for which coordinatewise minima and maxima of pairs of points remain within the polytope. For...

measure polytope (originally from Elte, 1912) is also used, notably in the work of H. S. M. Coxeter who also labels the hypercubes the γn polytopes. The...

geometry, there are 255 uniform polytopes with E8 symmetry. The three simplest forms are the 421, 241, and 142 polytopes, composed of 240, 2160 and 17280...

combinatorics, an integral polytope is a convex polytope whose vertices all have integer Cartesian coordinates. That is, it is a polytope that equals the convex...