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Mathematics is the study of representing and reasoning about abstract objects (such as numbers, points, spaces, sets, structures, and games). 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. (Full article...)

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three lines connecting corresponding vertices of a larger triangle on the left and a smaller one on the right converge at a point further to the right called the "center of perspectivity"
three lines connecting corresponding vertices of a larger triangle on the left and a smaller one on the right converge at a point further to the right called the "center of perspectivity"
Credit: User:Jujutacular, based on an original by User:DynaBlast
In projective geometry, Desargues' theorem states that two triangles are in perspective axially if and only if they are in perspective centrally. Lines through the triangle sides meet in pairs at collinear points along the axis of perspectivity. Lines through corresponding pairs of vertices on the triangles meet at a point called the center of perspectivity.

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  • ... that circle packings in the form of a Doyle spiral were used to model plant growth long before their mathematical investigation by Doyle?
  • ... that the music of math rock band Jyocho has been alternatively described as akin to "madness" or "contemplative and melancholy"?
  • ... that Ewa Ligocka cooked another mathematician's goose?
  • ... that despite published scholarship to the contrary, Andrew Planta neither received a doctorate nor taught mathematics at Erlangen?
  • ... that owner Matthew Benham influenced both Brentford FC in the UK and FC Midtjylland in Denmark to use mathematical modelling to recruit undervalued football players?
  • ... that the word algebra is derived from an Arabic term for the surgical treatment of bonesetting?
  • ... that despite a mathematical model deeming the ice cream bar flavour Goody Goody Gum Drops impossible, it was still created?
  • ... that the prologue to The Polymath was written by Martin Kemp, a leading expert on Leonardo da Vinci?

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The region between two loxodromes on a geometric sphere.
Image credit: Karthik Narayanaswami

The Riemann sphere is a way of extending the plane of complex numbers with one additional point at infinity, in a way that makes expressions such as

well-behaved and useful, at least in certain contexts. It is named after 19th century mathematician Bernhard Riemann. It is also called the complex projective line, denoted CP1.

On a purely algebraic level, the complex numbers with an extra infinity element constitute a number system known as the extended complex numbers. Arithmetic with infinity does not obey all of the usual rules of algebra, and so the extended complex numbers do not form a field. However, the Riemann sphere is geometrically and analytically well-behaved, even near infinity; it is a one-dimensional complex manifold, also called a Riemann surface.

In complex analysis, the Riemann sphere facilitates an elegant theory of meromorphic functions. The Riemann sphere is ubiquitous in projective geometry and algebraic geometry as a fundamental example of a complex manifold, projective space, and algebraic variety. It also finds utility in other disciplines that depend on analysis and geometry, such as quantum mechanics and other branches of physics. (Full article...)

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Topics in mathematics

General Foundations Number theory Discrete mathematics


Algebra Analysis Geometry and topology Applied mathematics
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