Iterated functions in the complex plane were investigated in the late 19th and early 20th centuries by Henri Poincaré, Felix Klein, Pierre Fatou and Gaston Julia. Georg Cantor also gave examples of subsets of the real line with unusual properties-these Cantor sets are also now recognized as fractals. The idea of self-similar curves was taken further by Paul Pierre Lévy, who, in his 1938 paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole described a new fractal curve, the Lévy C curve.

Wacław Sierpiński constructed his triangle in 1915 and, one year later, his carpet. In 1904, Helge von Koch, dissatisfied with Weierstrass's abstract and analytic definition, gave a more geometric definition of a similar function, which is now called the Koch curve. It was not until 1872 that a function appeared whose graph would today be considered fractal, when Karl Weierstrass gave an example of a function with the non-intuitive property of being everywhere continuous but nowhere differentiable. The mathematics behind fractals began to take shape in the 17th century when the mathematician and philosopher Gottfried Leibniz considered recursive self-similarity (although he made the mistake of thinking that only the straight line was self-similar in this sense). They are useful in medicine, soil mechanics, seismology, and technical analysis. 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 Benoît Mandelbrot in 1975 and was derived from the Latin frāctus meaning 'broken' or 'fractured.' A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion.

Many points reach a value that exceeds 2 after only a few iterations.Ī 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. Fortunately, there is a very useful criteria for detecting divergence - it can be demonstrated mathematically that if during the iteration process the number sequence exceeds the absolute value of 2, then the sequence will diverge to infinity and the original number C is definitely outside of the set. So instead, the process is applied several hundred or several thousand times, and if the sequence remains relatively small during these iterations, then it is assumed that C is probably a member of the set. Of course in practice we cannot apply the iteration process an infinite number of times to be sure that C is a member of the set. If the sequence converges, drifts chaotically, or cycles periodically, then C is in the set. If the resulting numbers generated during the iteration process grows ever and ever larger, then the original complex number C is not in the Mandelbrot set. In other words, to generate the set, take a complex number, multiply it by itself, and add it to the original number take that result, multiply it by itself, and add it to the original number and so on.

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