Natural objects that are approximated by fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, snow flakes, various vegetables (cauliflower and broccoli), and animal coloration patterns. It has a simple and recursive definition.īecause they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms).It has a Hausdorff dimension which is greater than its topological dimension (although this requirement is not met by space-filling such as the Hilbert curve).It is self-similar (at least approximately or stochastically).It is too irregular to be easily described in traditional Euclidiean geometric language.It has a fine structure at arbitrarily small scales.They are useful in medicine, soil mechanics, seismology and technical analysis.Ī fractal often has the following features. While fractals are a mathematical construct, they are found in nature, which has led to their inclusion in artwork. There are several examples of fractals, which are defined as portraying exact self-similarity, quasi self-similarity, or statistical self-similarity. Roots of the idea of fractals go back to the 17th century, while mathematically rigorous treatment of fractals can be traced back to functions studied by Karl Weierstrass, Georg Cantor and Felix Hausdorff a century later in studying functions that were continuous but not differentiable however, the term fractal was coined by Benoit Mandelbrot in 1975 and was derived from the latin fractus meaning “broken” or “fractured.” A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion. A fractal has been defined as “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,” a property called self-similarity.
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