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The main catalyst for the development of chaos theory was the electronic computer. Much of the mathematics of chaos theory involves the repeated iteration of simple mathematical formulas, which would be impractical to do by hand. Electronic computers made these repeated calculations practical, while figures and images made it possible to ...
The Beauty of Fractals is a 1986 book by Heinz-Otto Peitgen and Peter Richter which publicises the fields of complex dynamics, chaos theory and the concept of fractals. It is lavishly illustrated and as a mathematics book became an unusual success. The book includes a total of 184 illustrations, including 88 full-colour pictures of Julia sets.
Chaotic maps and iterated functions often generate fractals. Some fractals are studied as objects themselves, as sets rather than in terms of the maps that generate them. This is often because there are several different iterative procedures that generate the same fractal. See also Universality (dynamical systems).
Chaos communications is an application of chaos theory which is aimed to provide security in the transmission of information performed through telecommunications technologies. For secure communications, one has to understand that the contents of the message transmitted are inaccessible to possible eavesdroppers.
The Fractal flame is an example of an IFS with nonlinear functions. The most common algorithm to compute IFS fractals is called the " chaos game ". It consists of picking a random point in the plane, then iteratively applying one of the functions chosen at random from the function system to transform the point to get a next point.
The paper is important because it is a "turning point" in Mandelbrot's early thinking on fractals. [15] It is an example of the linking of mathematical objects with natural forms that was a theme of much of his later work. A key property of some fractals is self-similarity; that is, at any scale the same general configuration appears. A ...
With the aid of the "chaos game" a new fractal can be made and while making the new fractal some parameters can be obtained. These parameters are useful for applications of fractal theory such as classification and identification. [3] [4] The new fractal is self-similar to the original in some important features such as fractal dimension.
The book sparked widespread popular interest in fractals and contributed to chaos theory and other fields of science and mathematics. Mandelbrot also put his ideas to work in cosmology. He offered in 1974 a new explanation of Olbers' paradox (the "dark night sky" riddle), demonstrating the consequences of fractal theory as a sufficient, but not ...