In geometry, a polyhedron is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
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 mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by .
In mathematics, an integral polytope has an associated Ehrhart polynomial that encodes the relationship between the volume of a polytope and the number of integer points the polytope contains. The theory of Ehrhart polynomials can be seen as a higher-dimensional generalization of Pick's theorem in the Euclidean plane.
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.
In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric. Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in three-dimensional space, or as the three-dimensional projection of a hypercube. Zonohedra were originally defined and studied by E. S. Fedorov, a Russian crystallographer. More generally, in any dimension, the Minkowski sum of line segments forms a polytope known as a zonotope.
In geometry and combinatorics, an arrangement of hyperplanes is an arrangement of a finite set A of hyperplanes in a linear, affine, or projective space S. Questions about a hyperplane arrangement A generally concern geometrical, topological, or other properties of the complement, M(A), which is the set that remains when the hyperplanes are removed from the whole space. One may ask how these properties are related to the arrangement and its intersection semilattice. The intersection semilattice of A, written L(A), is the set of all subspaces that are obtained by intersecting some of the hyperplanes; among these subspaces are S itself, all the individual hyperplanes, all intersections of pairs of hyperplanes, etc.. These intersection subspaces of A are also called the flats ofA. The intersection semilattice L(A) is partially ordered by reverse inclusion.
In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines.
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, in the branch of combinatorics, a graded poset is a partially-ordered set (poset) P equipped with a rank functionρ from P to the set N of all natural numbers. ρ must satisfy the following two properties:
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 mathematics, specifically in combinatorial commutative algebra, a convex lattice polytope P is called normal if it has the following property: given any positive integer n, every lattice point of the dilation nP, obtained from P by scaling its vertices by the factor n and taking the convex hull of the resulting points, can be written as the sum of exactly n lattice points in P. This property plays an important role in the theory of toric varieties, where it corresponds to projective normality of the toric variety determined by P. Normal polytopes have popularity in algebraic combinatorics. These polytopes also represent the homogeneous case of the Hilbert bases of finite positive rational cones and the connection to algebraic geometry is that they define projectively normal embeddings of toric varieties.
In algebraic combinatorics, the h-vector of a simplicial polytope is a fundamental invariant of the polytope which encodes the number of faces of different dimensions and allows one to express the Dehn–Sommerville equations in a particularly simple form. A characterization of the set of h-vectors of simplicial polytopes was conjectured by Peter McMullen and proved by Lou Billera and Carl W. Lee and Richard Stanley (g-theorem). The definition of h-vector applies to arbitrary abstract simplicial complexes. The g-conjecture stated that for simplicial spheres, all possible h-vectors occur already among the h-vectors of the boundaries of convex simplicial polytopes. It was proven in December 2018 by Karim Adiprasito.
Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra.
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 geometry and combinatorics, a simpliciald-sphere is a simplicial complex homeomorphic to the d-dimensional sphere. Some simplicial spheres arise as the boundaries of convex polytopes, however, in higher dimensions most simplicial spheres cannot be obtained in this way.
Polymake is software for the algorithmic treatment of convex polyhedra.
In geometry and polyhedral 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 hull of its integer points. Integral polytopes are also called lattice polytopes or Z-polytopes. The special cases of two- and three-dimensional integral polytopes may be called polygons or polyhedra instead of polytopes, respectively.
A purely combinatorial approach to mirror symmetry was suggested by Victor Batyrev using the polar duality for -dimensional convex polyhedra. The most famous examples of the polar duality provide Platonic solids: e.g., the cube is dual to octahedron, the dodecahedron is dual to icosahedron. There is a natural bijection between the -dimensional faces of a -dimensional convex polyhedron and -dimensional faces of the dual polyhedron and one has . In Batyrev's combinatorial approach to mirror symmetry the polar duality is applied to special -dimensional convex lattice polytopes which are called reflexive polytopes.
In mathematics, the order polytope of a finite partially ordered set is a convex polytope defined from the set. The points of the order polytope are the monotonic functions from the given set to the unit interval, its vertices correspond to the upper sets of the partial order, and its dimension is the number of elements in the partial order. The order polytope is a distributive polytope, meaning that coordinatewise minima and maxima of pairs of its points remain within the polytope.
