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- Fractal abstract "A fractal is a mathematical set that typically displays self-similar patterns. Fractals may be exactly the same at every scale, or, as illustrated in Figure 1, they may be nearly the same at different scales. The concept of fractal extends beyond self-similarity and includes the idea of a detailed pattern repeating itself.Fractals are distinguished from regular geometric figures by their fractal dimensional scaling. Doubling the edge lengths of a square scales its area by four, which is two to the power of two, because a square is two dimensional. Likewise, if the radius of a sphere is doubled, its volume scales by eight, which is two to the power of three, because a sphere is three-dimensional. If a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a number which is not an integer. A fractal has a fractal dimension that usually exceeds its topological dimension and may fall between the integers.As mathematical equations, fractals are usually nowhere differentiable. An infinite fractal curve can be conceived of as winding through space differently from an ordinary line, still being a 1-dimensional line yet having a fractal dimension indicating it also resembles a surface.The mathematical roots of the idea of fractals have been traced throughout the years as a formal path of published works, starting in the 17th century with notions of recursion, then moving through increasingly rigorous mathematical treatment of the concept to the study of continuous but not differentiable functions in the 19th century, and on to the coining of the word fractal in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 21st century. The term "fractal" was first used by mathematician Benoît Mandelbrot in 1975. Mandelbrot based it on the Latin frāctus meaning "broken" or "fractured", and used it to extend the concept of theoretical fractional dimensions to geometric patterns in nature.There is some disagreement amongst authorities about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as "beautiful, damn hard, increasingly useful. That's fractals." The general consensus is that theoretical fractals are infinitely self-similar, iterated, and detailed mathematical constructs having fractal dimensions, of which many examples have been formulated and studied in great depth. Fractals are not limited to geometric patterns, but can also describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures and sounds and found in nature, technology, art, and law.".
- Fractal thumbnail Mandelbrot-similar-x1.jpg?width=300.
- Fractal wikiPageExternalLink Shlomo%20Havlin%20books_d_r.php.
- Fractal wikiPageExternalLink Shlomo%20Havlin%20books_f_in_s.php.
- Fractal wikiPageExternalLink Shlomo%20Havlin%20books_fds.php.
- Fractal wikiPageExternalLink index.html.
- Fractal wikiPageExternalLink fractals.
- Fractal wikiPageExternalLink www.fractalexplorer.com.
- Fractal wikiPageExternalLink fractauk.html.
- Fractal wikiPageExternalLink hunting-hidden-dimension.html.
- Fractal wikiPageExternalLink benoit_mandelbrot_fractals_the_art_of_roughness.html.
- Fractal wikiPageExternalLink watch?v=7Pf6jZWguCc.
- Fractal wikiPageExternalLink watch?v=VB-XUoDqYfs.
- Fractal wikiPageID "10913".
- Fractal wikiPageRevisionID "605913828".
- Fractal align "right".
- Fractal caption "86400.0".
- Fractal caption "Figure 1a. The Mandelbrot set illustrates self-similarity. As the image is enlarged, the same pattern re-appears so that it is virtually impossible to determine the scale being examined.".
- Fractal caption "Figure 1b. The same fractal magnified six times.".
- Fractal caption "Figure 1c. The same fractal magnified a hundred times.".
- Fractal direction "vertical".
- Fractal group "notes".
- Fractal hasPhotoCollection Fractal.
- Fractal image "Mandelbrot-similar-x1.jpg".
- Fractal image "Mandelbrot-similar-x100.jpg".
- Fractal image "Mandelbrot-similar-x2000.jpg".
- Fractal image "Mandelbrot-similar-x6.jpg".
- Fractal width "200".
- Fractal subject Category:Digital_art.
- Fractal subject Category:Fractals.
- Fractal subject Category:Mathematical_structures.
- Fractal subject Category:Topology.
- Fractal comment "A fractal is a mathematical set that typically displays self-similar patterns. Fractals may be exactly the same at every scale, or, as illustrated in Figure 1, they may be nearly the same at different scales. The concept of fractal extends beyond self-similarity and includes the idea of a detailed pattern repeating itself.Fractals are distinguished from regular geometric figures by their fractal dimensional scaling.".
- Fractal label "Fractal".
- Fractal label "Fractal".
- Fractal label "Fractal".
- Fractal label "Fractal".
- Fractal label "Fractale".
- Fractal label "Fraktal".
- Fractal label "Fraktal".
- Fractal label "Frattale".
- Fractal label "Фрактал".
- Fractal label "هندسة كسيرية".
- Fractal label "フラクタル".
- Fractal label "分形".
- Fractal sameAs Fraktál.
- Fractal sameAs Fraktal.
- Fractal sameAs Φράκταλ.
- Fractal sameAs Fractal.
- Fractal sameAs Fraktal.
- Fractal sameAs Fractale.
- Fractal sameAs Fraktal.
- Fractal sameAs Frattale.
- Fractal sameAs フラクタル.
- Fractal sameAs 프랙털.
- Fractal sameAs Fractal.
- Fractal sameAs Fraktal.
- Fractal sameAs Fractal.
- Fractal sameAs m.02yfd.
- Fractal sameAs Q81392.
- Fractal sameAs Q81392.
- Fractal wasDerivedFrom Fractal?oldid=605913828.
- Fractal depiction Mandelbrot-similar-x1.jpg.
- Fractal isPrimaryTopicOf Fractal.