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Mathematics is the study of numbers, quantity, space, pattern, 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. It is used for calculation and considered as the most important subject. 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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Riemann Sphere.jpg
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.

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low-resolution ASCII-art depiction of the Mandelbrot set
Credit: Elphaba

This is a modern reproduction of the first published image of the Mandelbrot set, which appeared in 1978 in a technical paper on Kleinian groups by Robert W. Brooks and Peter Matelski. The Mandelbrot set consists of the points c in the complex plane that generate a bounded sequence of values when the recursive relation zn+1 = zn2 + c is repeatedly applied starting with z0 = 0. The boundary of the set is a highly complicated fractal, revealing ever finer detail at increasing magnifications. The boundary also incorporates smaller near-copies of the overall shape, a phenomenon known as quasi-self-similarity. The ASCII-art depiction seen in this image only hints at the complexity of the boundary of the set. Advances in computing power and computer graphics in the 1980s resulted in the publication of high-resolution color images of the set (in which the colors of points outside the set reflect how quickly the corresponding sequences of complex numbers diverge), and made the Mandelbrot set widely known by the general public. Named by mathematicians Adrien Douady and John H. Hubbard in honor of Benoit Mandelbrot, one of the first mathematicians to study the set in detail, the Mandelbrot set is closely related to the Julia set, which was studied by Gaston Julia beginning in the 1910s.

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