In elementary geometry, a polytope is a geometric object with flat sides (faces). It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. In this context, "flat sides" means that the sides of a (k + 1)-polytope consist of k-polytopes that may have (k – 1)-polytopes in common. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope.
In geometry, the tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells. The tesseract is one of the six convex regular 4-polytopes.
In solid geometry, a face is a flat surface that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by faces is a polyhedron.
In mathematics, an abstract polytope is an algebraic partially ordered set (poset) which captures the combinatorial properties of a traditional polytope without specifying purely geometric properties such as angles or edge lengths.
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the -dimensional Euclidean space . Most texts use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary.
In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In total it has 30 triangle faces, 30 edges, and 10 vertices. Each vertex is surrounded by 3 octahedra and 2 tetrahedra; the vertex figure is a triangular prism.
In geometry, the rectified 24-cell or rectified icositetrachoron is a uniform 4-dimensional polytope, which is bounded by 48 cells: 24 cubes, and 24 cuboctahedra. It can be obtained by rectification of the 24-cell, reducing its octahedral cells to cubes and cuboctahedra.
The Birkhoff polytopeBn is the convex polytope in RN whose points are the doubly stochastic matrices, i.e., the n × n matrices whose entries are non-negative real numbers and whose rows and columns each add up to 1. It is named after Garrett Birkhoff.
In mathematical programming and polyhedral combinatorics, the Hirsch conjecture is the statement that the edge-vertex graph of an n-facet polytope in d-dimensional Euclidean space has diameter no more than n − d. That is, any two vertices of the polytope must be connected to each other by a path of length at most n − d. The conjecture was first put forth in a letter by Warren M. Hirsch to George B. Dantzig in 1957 and was motivated by the analysis of the simplex method in linear programming, as the diameter of a polytope provides a lower bound on the number of steps needed by the simplex method. The conjecture is now known to be false in general.
In geometry, a vertex, often denoted by letters such as , , , , is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices.
In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a line segment on the boundary, and is often called a polygon side. In a polyhedron or more generally a polytope, an edge is a line segment where two faces meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal.
In mathematics, the permutohedron of order n is an (n − 1)-dimensional polytope embedded in an n-dimensional space. Its vertex coordinates (labels) are the permutations of the first n natural numbers. The edges identify the shortest possible paths that connect two vertices (permutations). Two permutations connected by an edge differ in only two places, and the numbers on these places are neighbors.
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, published in his 1900 paper. He called it an 8-ic semi-regular figure.
Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes.
In mathematics, an associahedronKn is an (n – 2)-dimensional convex polytope in which each vertex corresponds to a way of correctly inserting opening and closing parentheses in a string of n letters, and the edges correspond to single application of the associativity rule. Equivalently, the vertices of an associahedron correspond to the triangulations of a regular polygon with n + 1 sides and the edges correspond to edge flips in which a single diagonal is removed from a triangulation and replaced by a different diagonal. Associahedra are also called Stasheff polytopes after the work of Jim Stasheff, who rediscovered them in the early 1960s after earlier work on them by Dov Tamari.
In six-dimensional geometry, a rectified 6-orthoplex is a convex uniform 6-polytope, being a rectification of the regular 6-orthoplex.
In geometry and polyhedral combinatorics, a k-neighborly polytope is a convex polytope in which every set of k or fewer vertices forms a face. For instance, a 2-neighborly polytope is a polytope in which every pair of vertices is connected by an edge, forming a complete graph. 2-neighborly polytopes with more than four vertices may exist only in spaces of four or more dimensions, and in general a k-neighborly polytope requires a dimension of 2k or more. A d-simplex is d-neighborly. A polytope is said to be neighborly, without specifying k, if it is k-neighborly for k = ⌊d⁄2⌋. If we exclude simplices, this is the maximum possible k: in fact, every polytope that is k-neighborly for some k ≥ 1 + ⌊d⁄2⌋ is a simplex.
In the geometry of 4 dimensions, the 3-3 duoprism or triangular duoprism is a four-dimensional convex polytope. It can be constructed as the Cartesian product of two triangles and is the simplest of an infinite family of four-dimensional polytopes constructed as Cartesian products of two polygons, the duoprisms.
In polyhedral combinatorics, the Gale transform turns the vertices of any convex polytopes into a set of vectors or points in a space of a different dimension, the Gale diagram of the polytope. It can be used to describe high-dimensional polytopes with few vertices, by transforming them into sets of points in a space of a much lower dimension. The process can also be reversed, to construct polytopes with desired properties from their Gale diagrams. The Gale transform and Gale diagram are named after David Gale, who introduced these methods in a 1956 paper on neighborly polytopes.
