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Fractal Universe

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Fractal _________________________________________________________________________ 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 dimen
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  Fractal  _________________________________________________  _  __  __  ___________________  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. [7]:48[2]:15 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. [9][10]  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. [2]:405[6] 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. [11]  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. [2][3][4]  Fractals are not limited to geometric patterns, but can also describe processes in time. [1][5][12]  Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures and sounds   and found in nature. Article Source : http://en.wikipedia.org/wiki/Fractal ---------------------------------------------------------------------------------------------------------------------------  Fractals Are SMART: Science, Math & Art! www.FractalFoundation.org All contents copyright 2009 FractalFoundation   Fractal Pack 1 Educators’ Guide Fractals Are SMART: Science, Math & Art! www.FractalFoundation.org All contents copyright 2009 FractalFoundation   ....  Fractals Are SMART: Science, Math & Art! www.FractalFoundation.org All contents copyright 2009 FractalFoundation Contents Introduction 3Natural Fractals 4Geometrical Fractals 6Algebraic Fractals 7Patterns and Symmetry 8Ideas of Scale 10Fractal Applications 11Fulldome Animations: 12  Crystaloon Galanga Pleoria Morphalingus Featherino Peacock Geometric Fractals  Fractivities: Sierpinski Triangle Construction 16 Explore fractals with XaoS 18 Appendix: Math & Science Education Standards met with fractals 19

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Jul 23, 2017
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