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The region between two loxodromes on a geometric sphere.

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 metrically 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.

Picture of the month

Missing square puzzle animation.

The missing square puzzle is an optical illusion used in mathematics classes to help students reason about geometrical figures. It depicts two arrangements of shapes, each of which apparently forms a 13×5 right-angled triangle, but one of which has a 1×1 hole in it.

The puzzle works as the blue and red triangles are not similar so their hypotenuses are not parallel. In one arrangement the line bends one way, in one arrangement it bends the other way, and the gap between these bent lines is exactly one unit.

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Did you know...

• ...that some functions can be written as an infinite sum of trigonometric polynomials? This sum is called the Fourier series of that function.
• ...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 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 (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?
• ...that you cannot knot strings in 4-dimensions? You can, however, knot 2-dimensional surfaces like spheres.

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