Portal:Mathematics
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Mathematics is the study of numbers, quantity, space, structure, and change. Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.
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There are approximately 31,444 mathematics articles in Wikipedia.
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3D illustration of a stereographic projection from the north pole onto a plane below the sphere. Image credit: Mark Howison 
In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it preserves angles. It is neither isometric nor areapreserving: that is, it preserves neither distances nor the areas of figures.
Intuitively, then, the stereographic projection is a way of picturing the sphere as the plane, with some inevitable compromises. Because the sphere and the plane appear in many areas of mathematics and its applications, so does the stereographic projection; it finds use in diverse fields including complex analysis, cartography, geology, and photography.
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The Lorenz attractor is an iconic example of a strange attractor in chaos theory. This threedimensional fractal structure, resembling a butterfly or figure eight, reflects the longterm behavior of solutions to the Lorenz system, a set of three differential equations used by mathematician and meteorologist Edward N. Lorenz as a simple description of fluid circulation in a shallow layer (of liquid or gas) uniformly heated from below and cooled from above. To be more specific, the figure is set in a threedimensional coordinate system whose axes measure the rate of convection in the layer (x), the horizontal temperature variation (y), and the vertical temperature variation (z). As these quantities change over time, a path is traced out within the coordinate system reflecting a particular solution to the differential equations. Lorentz's analysis revealed that while all solutions are completely deterministic, some choices of input parameters and initial conditions result in solutions showing complex, nonrepeating patterns that are highly dependent on the exact values chosen. As stated by Lorenz in his 1963 paper Deterministic Nonperiodic Flow: "Two states differing by imperceptible amounts may eventually evolve into two considerably different states". He later coined the term "butterfly effect" to describe the phenomenon. One implication is that computing such chaotic solutions to the Lorentz system (i.e., with a computer program) to arbitrary precision is not possible, as any realworld computer will have a limitation on the precision with which it can represent numerical values. The particular solution plotted in this animation is based on the parameter values used by Lorenz (σ = 10, ρ = 28, and β = 8/3, constants reflecting certain physical attributes of the fluid). Note that the animation repeatedly shows one solution plotted over a specific period of time; as previously mentioned, the true solution never exactly retraces itself. Not all solutions are chaotic, however. Some choices of parameter values result in solutions that tend toward equilibrium at a fixed point (as seen, for example, in this image). Initially developed to describe atmospheric convection, the Lorenz equations also arise in simplified models for lasers, electrical generators and motors, and chemical reactions.
Did you know...
 ...that a regular heptagon is the regular polygon with the fewest number of sides which is not constructible with a compass and straightedge?
 ...that the Gudermannian function relates the regular trigonometric functions and the hyperbolic trigonometric functions without the use of complex numbers?
 ...that the Catalan numbers solve a number of problems in combinatorics such as the number of ways to completely parenthesize an algebraic expression with n+1 factors?
 ...that a ball can be cut up and reassembled into two balls the same size as the original (BanachTarski paradox)?
 ...that it is impossible to devise a single formula involving only polynomials and radicals for solving an arbitrary quintic equation?
 ...that Euler found 59 more amicable numbers while for 2000 years, only 3 pairs had been found before him?
 ...that you cannot knot strings in 4dimensions? You can, however, knot 2dimensional surfaces like spheres.
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