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The four charts each map part of the

circle to an open interval, and

together cover the whole circle.

A manifold is an abstract mathematical space in which every point has a neighborhood which

resembles Euclidean space, but in which the global structure may be more complicated. Indiscussing manifolds, the idea of dimension is important. For example, lines are one-

dimensional, and planes two-dimensional.

In a one dimensional manifold (or one-manifold), every point has a neighborhood that looks likea segment of a line. Examples of one-manifolds include a line, a circle, and two separate circles.

In a two-manifold, every point has a neighborhood that looks like a disk . Examples include a plane, the surface of a sphere, and the surface of a torus.

Manifolds are important objects in mathematics and physics because they allow morecomplicated structures to be expressed and understood in terms of the relatively well-understood

 properties of simpler spaces.

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Picture of the month 

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Illustration for the paradoxical decomposition of F2 used in the proof of the Banach-Tarski

paradox.

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editDid you know... 

y  ...properties of Pascal's triangle have application in many fields of mathematics including

combinatorics, algebra, calculus and geometry?

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y  ...work in artificial intelligence makes use of Swarm intelligence, which has foundationsin the behavorial examples found in nature of ants, birds, bees, and fish among others?

y  ...that statistical properties dictated by Benford's Law are used in auditing of financialaccounts as one means of detecting fraud?

y  ...that Modular arithmetic has application in at least ten different fields of study,

including the arts, computer science, and chemistry in addition to mathematics?y  ...that outstanding mathematician Grigori Perelman was offered a Fields Medal in 2006,

in part for his proof of the Poincaré conjecture, which he declined?y  ...that a regular heptagon is the regular polygon with the fewest number of sides which is

not constructible with a compass and straightedge?y  ...that the Gudermannian function relates the regular trigonometric functions and the

hyperbolic trigonometric functions without the use of complex numbers?y  ...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?y  ...that a ball can be cut up and reassembled into two balls the same size as the original

(Banach-Tarski paradox)?

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TheMathematics WikiProject is the center for mathematics-related editing on Wikipedia. Join

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