Portal:Mathematics
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Mathematics is used throughout the world in many fields, including science, engineering, medicine and economics. These fields both inspire and make use of new discoveries in mathematics. The application of mathematics to such fields is often called applied mathematics. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any practical application in mind; although applications for what began as pure mathematics are often discovered later.
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At present, there are approximately 16588 mathematical articles in Wikipedia.
The homotopy groups of spheres describe the different ways spheres of various dimensions can be wrapped around each other. They are studied as part of algebraic topology. The topic can be hard to understand because the most interesting and surprising results involve spheres in higher dimensions. These are defined as follows: an n-dimensional sphere, n-sphere, consists of all the points in a space of n+1 dimensions that are a fixed distance from a center point. This definition is a generalization of the familiar circle (1-sphere) and sphere (2-sphere).
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| A homotopy from a circle around a sphere down to a single point. |
The goal of algebraic topology is to categorize or classify topological spaces. Homotopy groups were invented in the late 19th century as a tool for such classification, in effect using the set of mappings from an n-sphere in to a space as a way to probe the structure of that space. An obvious question was how this new tool would work on n-spheres themselves. No general solution to this question has been found to date, but many homotopy groups of spheres have been computed and the results are surprisingly rich and complicated. The study of the homotopy groups of spheres has led to the development of many powerful tools used in algebraic topology.
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An escape-time fractal, similar to the famous Mandelbrot set, associated with the Collatz conjecture shown near the real axis.
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- ...that it is unknown whether π and e are algebraically independent?
- ...that outstanding mathematician Grigori Perelman was offered a Fields Medal in 2006, in part for his proof of the Poincaré conjecture, which he declined?
- ...that a regular heptagon is the regular polygon with the least 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 sphere can be cut up and reassembled into two spheres the same size as the original (Banach-Tarski 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?
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The Mathematics WikiProject is the center for mathematics related editing on Wikipedia. Join the discussion on the project's talk page.
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