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Revision as of 21:09, 6 October 2009
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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| 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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Categories
Algebra | Arithmetic | Analysis | Complex analysis | Applied mathematics | Calculus | Category theory | Chaos theory | Combinatorics | Dynamical 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
Mathematics | History of mathematics | Mathematicians | Awards | Education | Literature | Notation | Organizations | Theorems | Proofs | Unsolved problems
Did you know...
- ...that a ball can be cut up and reassembled into two balls, each the same size as the original (Banach-Tarski paradox)?
- ...that it is impossible to devise a single formula involving only polynomials and radicals for solving an arbitrary quintic equation?
- ...that Euler found 59 more amicable numbers while for 2000 years, only 3 pairs had been found before him?
- ...that you cannot knot strings in 4 dimensions, but you can knot 2-dimensional surfaces, such as spheres?
- ...that there are 6 unsolved mathematics problems whose solutions will earn you one million US dollars each?
- ...that there are different sizes of infinite sets in set theory? More precisely, not all infinite cardinal numbers are equal?
- ...that every natural number can be written as the sum of four squares?
- ...that the largest known prime number is nearly 41 million digits long?
- ...that the set of rational numbers is equal in size to the set of integers; that is, they can be put in one-to-one correspondence?
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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Project pages Essays Subprojects Related projects
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Things you can do
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Things you can do
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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Project pages Essays Subprojects Related projects
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Things you can do
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Topics in mathematics
| General | Foundations | Number theory | Discrete mathematics |
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| Algebra | Analysis | Geometry and topology | Applied mathematics |
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Index of mathematics articles
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