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Justin Cook FractalsCook 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 frangere to break and the related adjective fractus 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 resembalance 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 conventially 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 quanity 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 28 for example would be sponge like shape that is nearly three-dimensional in its apperance Anatural fractal or fractal dimension 22 would be a much smoother
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