# Portal:Mathematics

## The Mathematics Portal

**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.*

## Selected article

Fractals arise in surprising places, in this case, the famous Collatz conjecture in number theory. Image credit: Pokipsy76 |

A **fractal** is "a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole". The term was coined by Benoît Mandelbrot in 1975 and was derived from the Latin *fractus* meaning "broken" or "fractured".

A fractal as a geometric object generally has the following features:

- It has a fine structure at arbitrarily small scales.
- It is too irregular to be easily described in traditional Euclidean geometric language.
- It is self-similar (at least approximately or stochastically).
- It has a Hausdorff dimension which is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve).
- It has a simple and recursive definition.

Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms). Natural objects that approximate fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, and snow flakes. However, not all self-similar objects are fractals—for example, the real line (a straight Euclidean line) is formally self-similar but fails to have other fractal characteristics.

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## Selected image

This image illustrates a failed attempt to comb the "hair" on a ball flat, leaving a tuft sticking out at each pole. The **hairy ball theorem** of algebraic topology states that whenever one attempts to comb a hairy ball, there will always be at least one point on the ball at which a tuft of hair sticks out. More precisely, it states that there is no nonvanishing continuous tangent-vector field on an even-dimensional *n*‑sphere (an ordinary sphere in three-dimensional space is known as a "2-sphere"). This is not true of certain other three-dimensional shapes, such as a torus (doughnut shape) which *can* be combed flat. The theorem was first stated by Henri Poincaré in the late 19th century and proved in 1912 by L. E. J. Brouwer. If one idealizes the wind in the Earth's atmosphere as a tangent-vector field, then the hairy ball theorem implies that given any wind at all on the surface of the Earth, there must at all times be a cyclone somewhere. Note, however, that wind can move vertically in the atmosphere, so the idealized case is not meteorologically sound. (What *is* true is that for every "shell" of atmosphere around the Earth, there must be a point on the shell where the wind is not moving horizontally.) The theorem also has implications in computer modeling (including video game design), in which a common problem is to compute a non-zero 3-D vector that is orthogonal (i.e., perpendicular) to a given one; the hairy ball theorem implies that there is no single continuous function that accomplishes this task.

## Did you know…

- ... that according to Kawasaki's theorem, an origami crease pattern with one vertex may be folded flat if and only if the sum of every other angle between consecutive creases is 180º?
- ... that, in the Rule 90 cellular automaton, any finite pattern eventually fills the whole array of cells with copies of itself?
- ... that, while the criss-cross algorithm visits all eight corners of the Klee–Minty cube when started at a
*worst*corner, it visits only three more corners on average when started at a*random*corner? - ...that in senary, all
**prime numbers**other than 2 and 3 end in 1 or a 5? - ... if the integer
*n*is prime, then the*n*th Perrin number is divisible by*n*? - ...that it is impossible to
**trisect a general angle**using only a ruler and a compass? - ...that in a group of 23 people, there is a more than 50% chance that two people share a birthday?

*Showing 7 items out of 73*

## WikiProjects

The **Mathematics WikiProject** is the center for mathematics-related editing on Wikipedia. Join the discussion on the project's **talk page**.

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Algebra | Arithmetic |Analysis | Complex analysis |Applied mathematics | Calculus |Category theory |Chaos theory | Combinatorics | Dynamic systems |Fractals |Game theory | Geometry |Algebraic geometry |Graph theory |Group theory |Linear algebra | Mathematical logic |Model theory |Multi-dimensional geometry | Number theory |Numerical analysis |Optimization |Order theory |Probability and statistics | Set theory |Statistics |Topology |Algebraic topology |Trigonometry |Linear programming

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## Topics in mathematics

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## Index of mathematics articles

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