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Fractal Studio
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called Affine geometry, affine self-similar. Fractal geometry lies within the mathematical branch of measure theory. One way that fractals are different from finite geometric figures is how they Scaling (geometry), scale. Doubling the edge lengths of a filled polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mandel Zoom 00 Mandelbrot Set
Mandel is a surname (and occasional given name) that occurs in multiple cultures and languages. It is a Dutch language, Dutch, German language, German and Jewish surname, meaning "almond", from the Middle High German and Middle Dutch ''mandel''.''Dictionary of American Family Names''"Mandel Family History" Oxford University Press, 2013. Retrieved on 18 January 2016. Mandel can be a locational surname, from places called Mandel, such as Mandel, Germany. Mandel may also be a Dutch language, Dutch surname, from the Middle Dutch ''mandele'', meaning a number of sheaves of harvested wheat. Notable people *Alon Mandel (born 1988), Israeli swimmer *Babaloo Mandel (born 1949), American screenwriter *David Mandel (born 1970), American television producer and writer *Edgar Mandel (born 1928), German actor *Eli Mandel (1922–1992), Canadian writer *Emily St. John Mandel (born 1979), Canadian novelist *Naum Korzhavin, Emmanuil Mandel (1925–2018), Russian poet *Ernest Mandel (1923–1995), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal Curve
A fractal curve is, loosely, a mathematical curve whose shape retains the same general pattern of irregularity, regardless of how high it is magnified, that is, its graph takes the form of a fractal. In general, fractal curves are nowhere rectifiable curves — that is, they do not have finite length — and every subarc longer than a single point has infinite length. A famous example is the boundary of the Mandelbrot set. Fractal curves in nature Fractal curves and fractal patterns are widespread, in nature, found in such places as broccoli, snowflakes, feet of geckos, frost crystals, and lightning bolts. See also Romanesco broccoli, dendrite crystal, trees, fractals, Hofstadter's butterfly, Lichtenberg figure, and self-organized criticality. Dimensions of a fractal curve Most of us are used to mathematical curves having dimension one, but as a general rule, fractal curves have different dimensions, also see also fractal dimension and list of fractals by Hausdor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractals In Nature
Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes. Early Greek philosophers studied pattern, with Plato, Pythagoras and Empedocles attempting to explain order in nature. The modern understanding of visible patterns developed gradually over time. In the 19th century, the Belgian physicist Joseph Plateau examined soap films, leading him to formulate the concept of a minimal surface. The German biologist and artist Ernst Haeckel painted hundreds of marine organisms to emphasise their symmetry. Scottish biologist D'Arcy Thompson pioneered the study of growth patterns in both plants and animals, showing that simple equations could explain spiral growth. In the 20th century, the British mathematician Alan Turing predicted mechanisms of morphogenesis which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Fractals By Hausdorff Dimension
According to Benoit Mandelbrot, "A fractal is by definition a set for which the Hausdorff dimension, Hausdorff-Besicovitch dimension strictly exceeds the topological dimension." Presented here is a list of fractals, ordered by increasing Hausdorff dimension, to illustrate what it means for a fractal to have a low or a high dimension. Deterministic fractals Random and natural fractals See also * Fractal dimension * Hausdorff dimension * Scale invariance Notes and references Further reading * * * * External links The fractals on MathworldOther fractals on Paul Bourke's websiteFractals on mathcurve.com* [https://web.archive.org/web/20060923100014/http://library.thinkquest.org/26242/full/index.html Fractals unleashed] IFStile - software that computes the dimension of the boundary of self-affine tiles {{DEFAULTSORT:Fractals By Hausdorff Dimension Fractals, Hausdorff Dimension Fractal curves, Hausdorff Dimension Mathematics-related lists ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iteration
Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. In mathematics and computer science, iteration (along with the related technique of recursion) is a standard element of algorithms. Mathematics In mathematics, iteration may refer to the process of iterating a function, i.e. applying a function repeatedly, using the output from one iteration as the input to the next. Iteration of apparently simple functions can produce complex behaviors and difficult problems – for examples, see the Collatz conjecture and juggler sequences. Another use of iteration in mathematics is in iterative methods which are used to produce approximate numerical solutions to certain mathematical problems. Newton's method is an example of an iterative method. Manual calculation of a number's square root is a co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
