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Mathematics is the study of numbers, quantity, space, pattern, 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. It is used for calculation and considered as the most important subject. 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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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. Fractals, when zoomed in, will keep showing more and more of itself, and it keeps going for infinity. ( Full article...)

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three lines connecting corresponding vertices of a larger triangle on the left and a smaller one on the right converge at a point further to the right called the "center of perspectivity"
Credit:  User:Jujutacular, based on an original by User:DynaBlast
In projective geometry, Desargues' theorem states that two triangles are in perspective axially if and only if they are in perspective centrally. Lines through the triangle sides meet in pairs at collinear points along the axis of perspectivity. Lines through corresponding pairs of vertices on the triangles meet at a point called the center of perspectivity.

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General Foundations Number theory Discrete mathematics
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Algebra Analysis Geometry and topology Applied mathematics
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