The Mathematics Portal
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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- ... that more than 60 scientific papers authored by mathematician Paul Erdős were published posthumously?
- ... that mathematician Daniel Larsen was the youngest contributor to the New York Times crossword puzzle?
- ... that in the aftermath of the American Civil War, the only Black-led organization providing teachers to formerly enslaved people was the African Civilization Society?
- ... that a folded paper lantern shows that certain mathematical definitions of surface area are incorrect?
- ... that people in Madagascar perform algebra on tree seeds in order to tell the future?
- ... that Latvian-Soviet artist Karlis Johansons exhibited a skeletal tensegrity form of the Schönhardt polyhedron seven years before Erich Schönhardt's 1928 paper on its mathematics?
- ... that mathematics professor Ari Nagel has fathered more than a hundred children?
- ... that owner Matthew Benham influenced both Brentford FC in the UK and FC Midtjylland in Denmark to use mathematical modelling to recruit undervalued football players?
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- ... that there are 115,200 solutions to the ménage problem of permuting six female-male couples at a twelve-person table so that men and women alternate and are seated away from their partners?
- ... that mathematician Paul Erdős called the Hadwiger conjecture, a still-open generalization of the four-color 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 14-piece 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?
- ...that as of April 2010 only 35 even numbers have been found that are not the sum of two primes which are each in a Twin Primes pair? ref
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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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