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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Banach–Tarski Paradox Image credit: Benjamin D. Esham 
The Banach–Tarski paradox is a theorem in settheoretic geometry which states that a solid ball in 3dimensional space can be split into a finite number of nonoverlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball. The reassembly process involves only moving the pieces around and rotating them, without changing their shape. However, the pieces themselves are complicated: they are not usual solids but infinite scatterings of points. A stronger form of the theorem implies that given any two "reasonable" solid objects (such as a small ball and a huge ball) — solid in the sense of the continuum — either one can be reassembled into the other. This is often stated colloquially as "a pea can be chopped up and reassembled into the Sun".
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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 in graph theory, a pseudoforest can contain trees and pseudotrees, but cannot contain any butterflies, diamonds, handcuffs, or bicycles?
 ...that it is not possible to configure two mutually inscribed quadrilaterals in the Euclidean plane, but the Möbius–Kantor graph describes a solution in the complex projective plane?
 ...that the six permutations of the vector (1,2,3) form a 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 the Rule 184 cellular automaton can simultaneously model the behavior of cars moving in traffic, the accumulation of particles on a surface, and particleantiparticle annihilation reactions?
 ...that a cyclic cellular automaton is a system of simple mathematical rules that can generate complex patterns mixing random chaos, blocks of color, and spirals?
 ...that a nonconvex polygon with three convex vertices is called a pseudotriangle?
 ...that the axiom of choice is logically independent of the other axioms of Zermelo–Fraenkel set theory?
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