In elementary geometry, a polytope is a geometric object with flat sides (faces). 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. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-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.
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, a simplicial complex is a set composed of points, line segments, triangles, and their n-dimensional counterparts. Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. To distinguish a simplicial complex from an abstract simplicial complex, the former is often called a geometric simplicial complex.
In mathematics, Sperner's lemma is a combinatorial result on colorings of triangulations, analogous to the Brouwer fixed point theorem, which is equivalent to it. It states that every Sperner coloring of a triangulation of an -dimensional simplex contains a cell whose vertices all have different colors.
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 mathematics, the "happy ending problem" is the following statement:
In geometry, a Schlegel diagram is a projection of a polytope from into through a point just outside one of its facets. The resulting entity is a polytopal subdivision of the facet in that, together with the original facet, is combinatorially equivalent to the original polytope. The diagram is named for Victor Schlegel, who in 1886 introduced this tool for studying combinatorial and topological properties of polytopes. In dimension 3, a Schlegel diagram is a projection of a polyhedron into a plane figure; in dimension 4, it is a projection of a 4-polytope to 3-space. As such, Schlegel diagrams are commonly used as a means of visualizing four-dimensional polytopes.
In mathematics, the Dehn–Sommerville equations are a complete set of linear relations between the numbers of faces of different dimension of a simplicial polytope. For polytopes of dimension 4 and 5, they were found by Max Dehn in 1905. Their general form was established by Duncan Sommerville in 1927. The Dehn–Sommerville equations can be restated as a symmetry condition for the h-vector of the simplicial polytope and this has become the standard formulation in recent combinatorics literature. By duality, analogous equations hold for simple polytopes.
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.
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 polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the 3-vertex-connected planar graphs. That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as polyhedral graphs.
In mathematics, a cyclic polytope, denoted C(n,d), is a convex polytope formed as a convex hull of n distinct points on a rational normal curve in Rd, where n is greater than d. These polytopes were studied by Constantin Carathéodory, David Gale, Theodore Motzkin, Victor Klee, and others. They play an important role in polyhedral combinatorics: according to the upper bound theorem, proved by Peter McMullen and Richard Stanley, the boundary Δ(n,d) of the cyclic polytope C(n,d) maximizes the number fi of i-dimensional faces among all simplicial spheres of dimension d − 1 with n vertices.
In geometry, the moment curve is an algebraic curve in d-dimensional Euclidean space given by the set of points with Cartesian coordinates of the form
In geometry and polyhedral combinatorics, the Kleetope of a polyhedron or higher-dimensional convex polytope P is another polyhedron or polytope PK formed by replacing each facet of P with a shallow pyramid. Kleetopes are named after Victor Klee.
In mathematics, a shelling of a simplicial complex is a way of gluing it together from its maximal simplices in a well-behaved way. A complex admitting a shelling is called shellable.
In geometry, the Perles configuration is a system of nine points and nine lines in the Euclidean plane for which every combinatorially equivalent realization has at least one irrational number as one of its coordinates. It can be constructed from the diagonals and symmetry lines of a regular pentagon, omitting one of the symmetry lines. In turn, it can be used to construct higher-dimensional convex polytopes that cannot be given rational coordinates, having the fewest vertices of any known example. All of the realizations of the Perles configuration in the projective plane are equivalent to each other under projective transformations.
In mathematics, the upper bound theorem states that cyclic polytopes have the largest possible number of faces among all convex polytopes with a given dimension and number of vertices. It is one of the central results of polyhedral combinatorics.
In polyhedral combinatorics, a stacked polytope is a polytope formed from a simplex by repeatedly gluing another simplex onto one of its facets.
In the mathematical discipline of polyhedral combinatorics, the Gale transform turns the vertices of any convex polytope 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 with the same number of points, but 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.
Convex Polytopes is a graduate-level mathematics textbook about convex polytopes, higher-dimensional generalizations of three-dimensional convex polyhedra. It was written by Branko Grünbaum, with contributions from Victor Klee, Micha Perles, and G. C. Shephard, and published in 1967 by John Wiley & Sons. It went out of print in 1970. A second edition, prepared with the assistance of Volker Kaibel, Victor Klee, and Günter M. Ziegler, was published by Springer-Verlag in 2003, as volume 221 of their book series Graduate Texts in Mathematics.
