Portal:Mathematics
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Mathematics is the study of numbers, quantity, space, 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. 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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There are approximately 31,444 mathematics articles in Wikipedia.
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Example of a Persian design with wallpaper group type "p6m" Image credit: Owen Jones 
A wallpaper group is a mathematical concept used to classify repetitive designs on twodimensional surfaces, such as floors and walls, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art. The mathematical study of such patterns reveals that exactly 17 different types of pattern can occur.
Wallpaper groups are examples of an abstract algebraic structure known as a group. Groups are frequently used in mathematics to study the notion of symmetry. Wallpaper groups are related to the simpler frieze groups, and to the more complex threedimensional crystallographic groups.
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A Klein bottle is an example of a closed surface (a twodimensional manifold) that is nonorientable (no distinction between the "inside" and "outside"). This image is a representation of the object in everyday threedimensional space, but a true Klein bottle is an object in fourdimensional space. When it is constructed in threedimensions, the "inner neck" of the bottle curves outward and intersects the side; in four dimensions, there is no such selfintersection (the effect is similar to a twodimensional representation of a cube, in which the edges seem to intersect each other between the corners, whereas no such intersection occurs in a true threedimensional cube). Also, while any real, physical object would have a thickness to it, the surface of a true Klein bottle has no thickness. Thus in three dimensions there is an inside and outside in a colloquial sense: liquid forced through the opening on the right side of the object would collect at the bottom and be contained on the inside of the object. However, on the fourdimensional object there is no inside and outside in the way that a sphere has an inside and outside: an unbroken curve can be drawn from a point on the "outer" surface (say, the object's lowest point) to the right, past the "lip" to the "inside" of the narrow "neck", around to the "inner" surface of the "body" of the bottle, then around on the "outer" surface of the narrow "neck", up past the "seam" separating the inside and outside (which, as mentioned before, does not exist on the true 4D object), then around on the "outer" surface of the body back to the starting point (see the light gray curve on this simplified diagram). In this regard, the Klein bottle is a higherdimensional analog of the Möbius strip, a twodimensional manifold that is nonorientable in ordinary 3dimensional space. In fact, a Klein bottle can be constructed (conceptually) by "gluing" the edges of two Möbius strips together.
Did you know...
 ...that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type the complete works of William Shakespeare?
 ... that there are 115,200 solutions to the ménage problem of permuting six femalemale couples at a twelveperson table so that men and women alternate and are seated away from their partners?
 ... that mathematician Paul Erdős called the Hadwiger conjecture, a stillopen generalization of the fourcolor problem, "one of the deepest unsolved problems in graph theory"?
 ...that the six permutations of the vector (1,2,3) form a regular hexagon in 3d space, the 24 permutations of (1,2,3,4) form a truncated octahedron in four dimensions, and both are examples of permutohedra?
 ...that Ostomachion is a mathematical treatise attributed to Archimedes on a 14piece tiling puzzle similar to tangram?
 ...that some functions can be written as an infinite sum of trigonometric polynomials and that this sum is called the Fourier series of that function?
 ...that the identity elements for arithmetic operations make use of the only two whole numbers that are neither composites nor prime numbers, 0 and 1?
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