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Justin Cook Fractals Cook 1 Fractals are a modern mathematical theory that radically departs from traditionally Euclidean geometry, fractal geometry describes objects that are self-similar, or scale symmetric. This means that when such objects are magnified, their parts are seen to bear an exact resemblance to the whole, the likeness continuing with the parts of the parts and so on and so on to infinity. Fractals, as these shapes are called, must also be devoid of translational symmetry. Instead a rough jagged quality is maintained at every scale at which an object can be examined.
The nature of fractals is reflected in the word itself, coined by mathematician Benoit Mandelbrot from the Latin verb flanger, to break and the related adjective facts, irregular and fragmented. Fractals are not exclusive to the realm of mathematics. If the definition can be broadened a bit a fractal can be found almost anywhere. The difference is that natural fractals are randomly, Statistically, or stochastically rathe than exactly scale symmetric. The rough shape revealed at one length scale bears only an approximate resemblance to that at another, but the length scale being used is not apparent just by looking at the shape. Moreover, there are both upper and lower limits to the size range over which fractals in nature are indeed fractal.
Above and below that range, the shapes are either rough or smooth in other words conventionally Cook 2 Euclidean. Whether natural or mathematical, all fractals have particular fractal dimensions. These are not the same as the familar Euclidean dimensions measured in discrete whole integers but are a different kind of quality. Usually non-integer, a fractal dimension indicates the extent to which the fractal object fills the Euclidean dimension in which it is embedded. A natural fractal or fractal dimension 2. 8, for example would be sponge like shape that is nearly three-dimensional in its appearance. A natural fractal or fractal dimension 2. 2 would be a much smoother object that just narrowly misses being flat.
Fractals give us the ability to generate lifelike images of complicated, highly irregular natural objects, such as the rugged terrains of mountains and the intricate branches of trees. Fractal simulations have been used to plot the distribution of galaxy clusters throughout the universe and to study problems related to fluid turbulence. There are many different sets by which fractals are generated, such as the Julia set or the Mandelbrot set. Julia studied the iteration of polynomials and rational functions throughout the early twentieth century.
Cook 3 Bibliography: 1. 1998 Groiler interactive encyclopedia 2. web 3. web 4. web
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