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Mathematics is the study of representing and reasoning about abstract objects (such as numbers, points, spaces, sets, structures, and games). 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. ( Full article...)

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truncated icosahedron with black pentagonal faces and white hexagonal faces, beside a similar-looking 1970s soccer ball
truncated icosahedron with black pentagonal faces and white hexagonal faces, beside a similar-looking 1970s soccer ball
Here a polyhedron called a truncated icosahedron (left) is compared to the classic Adidas Telstar–style football (or soccer ball). The familiar 32-panel ball design, consisting of 12 black pentagonal and 20 white hexagonal panels, was first introduced by the Danish manufacturer Select Sport, based loosely on the geodesic dome designs of Buckminster Fuller; it was popularized by the selection of the Adidas Telstar as the official match ball of the 1970 FIFA World Cup. The polyhedron is also the shape of the Buckminsterfullerene (or "Buckyball") carbon molecule initially predicted theoretically in the late 1960s and first generated in the laboratory in 1985. Like all polyhedra, the vertices (corner points), edges (lines between these points), and faces (flat surfaces bounded by the lines) of this solid obey the Euler characteristic, VE + F = 2 (here, 60 − 90 + 32 = 2). The icosahedron from which this solid is obtained by truncating (or "cutting off") each vertex (replacing each by a pentagonal face), has 12 vertices, 30 edges, and 20 faces; it is one of the five regular solids, or Platonic solids—named after Plato, whose school of philosophy in ancient Greece held that the classical elements (earth, water, air, fire, and a fifth element called aether) were associated with these regular solids. The fifth element was known in Latin as the "quintessence", a hypothesized uncorruptible material (in contrast to the other four terrestrial elements) filling the heavens and responsible for celestial phenomena. That such idealized mathematical shapes as polyhedra actually occur in nature (e.g., in crystals and other molecular structures) was discovered by naturalists and physicists in the 19th and 20th centuries, largely independently of the ancient philosophies.

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A homotopy from a circle around a sphere down to a single point.
Image credit: Richard Morris

The homotopy groups of spheres describe the different ways spheres of various dimensions can be wrapped around each other. They are studied as part of algebraic topology. The topic can be hard to understand because the most interesting and surprising results involve spheres in higher dimensions. These are defined as follows: an n-dimensional sphere, n-sphere, consists of all the points in a space of n+1 dimensions that are a fixed distance from a center point. This definition is a generalization of the familiar circle (1-sphere) and sphere (2-sphere).

The goal of algebraic topology is to categorize or classify topological spaces. Homotopy groups were invented in the late 19th century as a tool for such classification, in effect using the set of mappings from a c-sphere into a space as a way to probe the structure of that space. An obvious question was how this new tool would work on n-spheres themselves. No general solution to this question has been found to date, but many homotopy groups of spheres have been computed and the results are surprisingly rich and complicated. The study of the homotopy groups of spheres has led to the development of many powerful tools used in algebraic topology. ( Full article...)

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  1. ^ Coxeter et al. (1999), p. 30–31; Wenninger (1971), p. 65.