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Pick's Theorem
In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary. The result was first described by Georg Alexander Pick in 1899. It was popularized in English by Hugo Steinhaus in the 1950 edition of his book ''Mathematical Snapshots''. It has multiple proofs, and can be generalized to formulas for certain kinds of non-simple polygons. Formula Suppose that a polygon has integer coordinates for all of its vertices. Let i be the number of integer points interior to the polygon, and let b be the number of integer points on its boundary (including both vertices and points along the sides). Then the area A of this polygon is: A = i + \frac - 1. The example shown has i=7 interior points and b=8 boundary points, so its area is A=7+\tfrac-1=10 square units. Proofs Via Euler's formula One proof of this theorem involves subdividing the polygon into triangles with three ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pick Theorem Simple
Pick may refer to: Places * Pick City, North Dakota, a town in the United States * Pick Lake (Cochrane District, Ontario), a lake in Canada * Pick Lake (Thunder Bay District), a lake in Canada * Pick Mere, a lake in Pickmere, England People with the name * Pick (surname), a list of people with this name * nickname of Percy Charles Pickard (1915–1944), British Royal Air Force pilot * Pick Temple (1911–1991), American folk singer and children's television star * Pick Withers (born 1948), drummer for the English rock band Dire Straits Arts, entertainment, and media * Plectrum or pick, a device for strumming a stringed instrument :*Guitar pick, specific to guitars and similar instruments * The Picks, a vocal quartet which backed Buddy Holly and the Crickets in 1957 * Pick (TV channel), a British television channel * "The Pick", an episode of the television show ''Seinfeld'' * Odds and evens or pick, a hand game * Pick (film), short drama film, directed by Alicia K. Harris Sci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyomino
A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling. Polyominoes have been used in popular puzzles since at least 1907, and the enumeration of pentominoes is dated to antiquity. Many results with the pieces of 1 to 6 squares were first published in ''Fairy Chess Review'' between the years 1937 to 1957, under the name of "dissection problems." The name ''polyomino'' was invented by Solomon W. Golomb in 1953, and it was popularized by Martin Gardner in a November 1960 "Mathematical Games" column in ''Scientific American''. Related to polyominoes are polyiamonds, formed from equilateral triangles; polyhexes, formed from regular hexagons; and other plane polyforms. Polyominoes have been generalized to higher dimensions by joining cubes to form polycubes, or hypercubes to form polyhypercubes. In statistical physics, the study ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reeve Tetrahedron
In geometry, the Reeve tetrahedra are a family of polyhedra in three-dimensional space with vertices at , , and where is a positive integer. They are named after John Reeve, who in 1957 used them to show that higher-dimensional generalizations of Pick's theorem do not exist. Counterexample to generalizations of Pick's theorem All vertices of a Reeve tetrahedron are lattice points (points whose coordinates are all integers). No other lattice points lie on the surface or in the interior of the tetrahedron. The volume of the Reeve tetrahedron with vertex is . In 1957 Reeve used this tetrahedron to show that there exist tetrahedra with four lattice points as vertices, and containing no other lattice points, but with arbitrarily large volume. In two dimensions, the area of every polyhedron with lattice vertices is determined as a formula of the number of lattice points at its vertices, on its boundary, and in its interior, according to Pick's theorem. The Reeve tetrahedra imply th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reeve Tetrahedrons
Reeve may refer to: Titles *Reeve (Canada), an elected chief executive of some counties, townships, and equivalents *Reeve (England), an official elected annually by the serfs to supervise lands for a lord *High-reeve, a title taken by some English magistrates during the 10th and 11th centuries *Shire reeve, an official position that originated the term Sheriff *Vogt, an official in many European countries, often translated ''reeve'' Other uses *Reeve (surname), list of notable people with the surname *Reeve, Wisconsin, an unincorporated community *Reeve knot, a stopper knot *Reeve, a female ruff (bird), a wading bird * Reeve (''Final Fantasy''), a character from the video game ''Final Fantasy VII'' *Reeve Electric Association Plant, listed on the National Register of Historic Places in Franklin County, Iowa *"The Reeve's Prologue and Tale", from ''The Canterbury Tales'' by Chaucer * Leander Reeve House, listed on the National Register of Historic Places in Franklin County, Iowa * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Winding Number
In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of turns. The winding number depends on the orientation of the curve, and it is negative if the curve travels around the point clockwise. Winding numbers are fundamental objects of study in algebraic topology, and they play an important role in vector calculus, complex analysis, geometric topology, differential geometry, and physics (such as in string theory). Intuitive description Suppose we are given a closed, oriented curve in the ''xy'' plane. We can imagine the curve as the path of motion of some object, with the orientation indicating the direction in which the object moves. Then the winding number of the curve is equal to the total number of counterclockwise turns that the object makes around the origin. When counting the total nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, acc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar Straight-line Graph
In computational geometry and geometric graph theory, a planar straight-line graph, in short ''PSLG'', (or ''straight-line plane graph'', or ''plane straight-line graph'') is a term used for an embedding of a planar graph in the plane such that its edges are mapped into straight-line segments. Fáry's theorem (1948) states that every planar graph has this kind of embedding. In computational geometry, PSLGs have often been called planar subdivisions, with an assumption or assertion that subdivisions are polygonal rather than having curved boundaries. PSLGs may serve as representations of various maps, e.g., geographical maps in geographical information systems. Special cases of PSLGs are triangulations ( polygon triangulation, point-set triangulation). Point-set triangulations are maximal PSLGs in the sense that it is impossible to add straight edges to them while keeping the graph planar. Triangulations have numerous applications in various areas. PSLGs may be seen as a speci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pick Theorem Hole
Pick may refer to: Places * Pick City, North Dakota, a town in the United States * Pick Lake (Cochrane District, Ontario), a lake in Canada * Pick Lake (Thunder Bay District), a lake in Canada * Pick Mere, a lake in Pickmere, England People with the name * Pick (surname), a list of people with this name * nickname of Percy Charles Pickard (1915–1944), British Royal Air Force pilot * Pick Temple (1911–1991), American folk singer and children's television star * Pick Withers (born 1948), drummer for the English rock band Dire Straits Arts, entertainment, and media * Plectrum or pick, a device for strumming a stringed instrument :*Guitar pick, specific to guitars and similar instruments * The Picks, a vocal quartet which backed Buddy Holly and the Crickets in 1957 * Pick (TV channel), a British television channel * "The Pick", an episode of the television show ''Seinfeld'' * Odds and evens or pick, a hand game * Pick (film), short drama film, directed by Alicia K. Harris Sci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof Assistant
In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human-machine collaboration. This involves some sort of interactive proof editor, or other interface, with which a human can guide the search for proofs, the details of which are stored in, and some steps provided by, a computer. System comparison * ACL2 – a programming language, a first-order logical theory, and a theorem prover (with both interactive and automatic modes) in the Boyer–Moore tradition. * Coq – Allows the expression of mathematical assertions, mechanically checks proofs of these assertions, helps to find formal proofs, and extracts a certified program from the constructive proof of its formal specification. * HOL theorem provers – A family of tools ultimately derived from the LCF theorem prover. In these systems the logical core is a library of their programming language. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Benchmark (computing)
In computing, a benchmark is the act of running a computer program, a set of programs, or other operations, in order to assess the relative Computer performance, performance of an object, normally by running a number of standard Software performance testing, tests and trials against it. The term ''benchmark'' is also commonly utilized for the purposes of elaborately designed benchmarking programs themselves. Benchmarking is usually associated with assessing performance characteristics of computer hardware, for example, the floating point operation performance of a Central processing unit, CPU, but there are circumstances when the technique is also applicable to software. Software benchmarks are, for example, run against compilers or database management systems (DBMS). Benchmarks provide a method of comparing the performance of various subsystems across different chip/system Computer architecture, architectures. Purpose As computer architecture advanced, it became more diffi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic Function
In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the function ::\mathbf_A\colon X \to \, :which for a given subset ''A'' of ''X'', has value 1 at points of ''A'' and 0 at points of ''X'' − ''A''. * There is an indicator function for affine varieties over a finite field: given a finite set of functions f_\alpha \in \mathbb_q _1,\ldots,x_n/math> let V = \left\ be their vanishing locus. Then, the function P(x) = \prod\left(1 - f_\alpha(x)^\right) acts as an indicator function for V. If x \in V then P(x) = 1, otherwise, for some f_\alpha, we have f_\alpha(x) \neq 0, which implies that f_\alpha(x)^ = 1, hence P(x) = 0. * The characteristic function in convex analysis, closely related to the indicator function of a set: *:\chi_A (x) := \begin 0, & x \in A; \\ + \infty, & x \not \in A. \end * In probability theory, the characteristic function of any probability distribution on the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Summation Formula
In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples of the original function's Fourier transform. And conversely, the periodic summation of a function's Fourier transform is completely defined by discrete samples of the original function. The Poisson summation formula was discovered by Siméon Denis Poisson and is sometimes called Poisson resummation. Forms of the equation Consider an aperiodic function s(x) with Fourier transform S(f) \triangleq \int_^ s(x)\ e^\, dx, alternatively designated by \hat s(f) and \mathcal\(f). The basic Poisson summation formula is: Also consider periodic functions, where parameters T>0 and P>0 are in the same units as x: :s_(x) \triangleq \sum_^ s(x + nP) \quad \text \quad S_(f) \triangleq \sum_^ S(f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
