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F-vector
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. Research in polyhedral combinatorics falls into two distinct areas. Mathematicians in this area study the combinatorics of polytopes; for instance, they seek inequalities that describe the relations between the numbers of vertices, edges, and faces of higher dimensions in arbitrary polytopes or in certain important subclasses of polytopes, and study other combinatorial properties of polytopes such as their connectivity and diameter (number of steps needed to reach any vertex from any other vertex). Additionally, many computer scientists use the phrase “polyhedral combinatorics” to describe research into precise descriptions of the faces of certain specific polytopes (especially 0-1 polytopes, whose vertices are subsets of a hypercube) arising from integer p ...
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H-vector
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. Stanley introduced a generalization of the ''h''-vector, the toric ''h''-vector, which is defined for an arbitrary ranked poset, and proved that for the class of Eulerian posets, the Dehn–Sommerville equations continue to hold. A differe ...
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Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and mathematical analysis, analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of mathematical object, abstract objects and the use of pure reason to proof (mathematics), prove them. These objects consist of either abstraction (mathematics), abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of inference rule, deductive rules to already established results. These results include previously proved theorems, axioms ...
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Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only regular hexahedron and is one of the five Platonic solids. It has 6 faces, 12 edges, and 8 vertices. The cube is also a square parallelepiped, an equilateral cuboid and a right rhombohedron a 3-zonohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations. The cube is dual to the octahedron. It has cubical or octahedral symmetry. The cube is the only convex polyhedron whose faces are all squares. Orthogonal projections The ''cube'' has four special orthogonal projections, centered, on a vertex, edges, face and normal to its vertex figure. The first and third correspond to the A2 and B2 Coxeter planes. Spherical tiling The cube can also be represented as a spherical tiling, and p ...
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Neighborly Polytope
In geometry and polyhedral combinatorics, a -neighborly polytope is a convex polytope in which every set of 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 -neighborly polytope (other than a simplex) requires a dimension of or more. A -simplex is -neighborly. A polytope is said to be neighborly, without specifying , if it is -neighborly for . If we exclude simplices, this is the maximum possible : in fact, every polytope that is -neighborly for some is a simplex. In a -neighborly polytope with , every 2-face must be a triangle, and in a -neighborly polytope with , every 3-face must be a tetrahedron. More generally, in any -neighborly polytope, all faces of dimension less than are simplices. The cyclic polytopes formed as the convex h ...
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Upper Bound Theorem
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. Originally known as the upper bound conjecture, this statement was formulated by Theodore Motzkin, proved in 1970 by Peter McMullen, and strengthened from polytopes to subdivisions of a sphere in 1975 by Richard P. Stanley. Cyclic polytopes The cyclic polytope \Delta(n,d) may be defined as the convex hull of n vertices on the moment curve, the set of d-dimensional points with coordinates (t,t^2,t^3,\dots). The precise choice of which n points on this curve are selected is irrelevant for the combinatorial structure of this polytope. The number of i-dimensional faces of \Delta(n,d) is given by the formula f_i(\Delta(n,d)) = \binom \quad \textrm \quad 0 \leq i < \left\lfloor\frac\right\rfloor and (f_0,\ldots,f_) comp ...
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Dehn–Sommerville Equations
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. Statement Let ''P'' be a ''d''-dimensional simplicial polytope. For ''i'' = 0, 1, ..., ''d'' − 1, let ''f''''i'' denote the number of ''i''-dimensional faces of ''P''. The sequence : f(P)=(f_0,f_1,\ldots,f_) is called the ''f''-vector of the polytope ''P''. Additionally, set : f_=1, f_d=1. Then for any ''k'' = −1, 0, ..., ''d'' − 2, the following Dehn–Sommerville equatio ...
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Steinitz's Theorem
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. This result provides a classification theorem for the three-dimensional convex polyhedra, something that is not known in higher dimensions. It provides a complete and purely combinatorial description of the graphs of these polyhedra, allowing other results on them, such as Eberhard's theorem on the realization of polyhedra with given types of faces, to be proven more easily, without reference to the geometry of these shapes. Additionally, it has been applied in graph drawing, as a way to construc ...
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Double Counting (proof Technique)
In combinatorics, double counting, also called counting in two ways, is a combinatorial proof technique for showing that two expressions are equal by demonstrating that they are two ways of counting the size of one set. In this technique, which call "one of the most important tools in combinatorics", one describes a finite set from two perspectives leading to two distinct expressions for the size of the set. Since both expressions equal the size of the same set, they equal each other. Examples Multiplication (of natural numbers) commutes This is a simple example of double counting, often used when teaching multiplication to young children. In this context, multiplication of natural numbers is introduced as repeated addition, and is then shown to be commutative by counting, in two different ways, a number of items arranged in a rectangular grid. Suppose the grid has n rows and m columns. We first count the items by summing n rows of m items each, then a second time by summing ...
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Euler Characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré 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 \chi ( Greek lower-case letter chi). The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids. It was stated for Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico. Leonhard Euler, for whom the concept is named, introduced it for convex polyhedra more generally but failed to rigorously prove that it is an invariant. In modern mathematics, the Euler characteristic arises from homology and, more abstractly, homological algebra. Polyhedra The Euler characteristic \chi was classically defined for the surfaces of polyhedra, ...
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Simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example, * a 0-dimensional simplex is a point, * a 1-dimensional simplex is a line segment, * a 2-dimensional simplex is a triangle, * a 3-dimensional simplex is a tetrahedron, and * a 4-dimensional simplex is a 5-cell. Specifically, a ''k''-simplex is a ''k''-dimensional polytope which is the convex hull of its ''k'' + 1 vertices. More formally, suppose the ''k'' + 1 points u_0, \dots, u_k \in \mathbb^ are affinely independent, which means u_1 - u_0,\dots, u_k-u_0 are linearly independent. Then, the simplex determined by them is the set of points : C = \left\ This representation in terms of weighted vertices is known as the barycentric coordinate system. A regular simplex is a simplex that is also a regular poly ...
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Simplicial Polytope
In geometry, a simplicial polytope is a polytope whose facets are all simplices. For example, a ''simplicial polyhedron'' in three dimensions contains only triangular facesPolyhedra, Peter R. Cromwell, 1997. (p.341) and corresponds via Steinitz's theorem to a maximal planar graph. They are topologically dual to simple polytopes. Polytopes which are both simple and simplicial are either simplices or two-dimensional polygons. Examples Simplicial polyhedra include: * Bipyramids * Gyroelongated dipyramids * Deltahedra (equilateral triangles) ** Platonic *** tetrahedron, octahedron, icosahedron ** Johnson solids: ***triangular bipyramid, pentagonal bipyramid, snub disphenoid, triaugmented triangular prism, gyroelongated square dipyramid * Catalan solids: ** triakis tetrahedron, triakis octahedron, tetrakis hexahedron, disdyakis dodecahedron, triakis icosahedron, pentakis dodecahedron, disdyakis triacontahedron Simplicial tilings: * Regular: ** triangular tiling *Laves ti ...
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Unimodal
In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object. Unimodal probability distribution In statistics, a unimodal probability distribution or unimodal distribution is a probability distribution which has a single peak. The term "mode" in this context refers to any peak of the distribution, not just to the strict definition of mode which is usual in statistics. If there is a single mode, the distribution function is called "unimodal". If it has more modes it is "bimodal" (2), "trimodal" (3), etc., or in general, "multimodal". Figure 1 illustrates normal distributions, which are unimodal. Other examples of unimodal distributions include Cauchy distribution, Student's ''t''-distribution, chi-squared distribution and exponential distribution. Among discrete distributions, the binomial distribution and Poisson distribution can be seen as unimodal, tho ...
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