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Force-System Resultants and Equilibrium
Published in Richard C. Dorf, The Engineering Handbook, 2018
There is no better conclusion to a chapter on chaos, fractals and Julia sets than to introduce a concept that blends together features of all these concepts in a beautiful structure known as the Mandelbrot set. The Mandelbrot set is a subset M of the complex plane M⊂C and it consists of all those complex numbers c with the property that the orbit of c (or of the critical point 0) under the quadratic map fc(z)=z2+c is bounded. It is easy to decide, using a computer, whether the iterates of c under fc stay bounded or diverge. If one checks in this manner a comprehensive array of values of c around the origin, and plots those c values for which the iterates of 0 under fc are bounded, one discovers an amazing shape. Equivalently the Mandelbrot set is the set of points c, so that the Julia set of fc is connected. Mandelbrot ran into this set, apparently by accident, and at first he thought that it was the result of some computer mulfunction. The boundary of the set has a fractal look, with intricate detail, and all its geometric features recur on all scales. The main cardioid of the Mandelbrot set corresponds to the set of all c that correspond to a simple closed curve Julia set. The Mandelbrot set is a wonderful discovery that reflects the beauty of fractals and the order of chaos.
Faster, Further
Published in José Guillermo Sánchez León, ® Beyond Mathematics, 2017
Often, the Mandelbrot set is represented using the escape-time coloring algorithm based on applying different colors depending on the number of iterations required to prove if a series belongs to the set. Darker colors (in our example bright red) indicate that only a few iterations are needed while lighter ones mean that many iterations were required. There’s always a limit for the maximum number of iterations allowed. Since Mathematica 10 the build-up function MandelbrotSetPlot is available, however, we will build a function step by step to represent the Mandelbrot set.
Fractals and chaos
Published in A. W. Jayawardena, Environmental and Hydrological Systems Modelling, 2013
The Hausdorff–Besicovitch dimension of the boundary of Mandelbrot set has been mathematically proven to be the integer 2 (Shishikura, 1998), which is unusual for a fractal. Mandelbrot (1967) is also known for his paper entitled ‘How long is the coast of Britain? Statistical self-similarity and fractional dimension’, which showed that the length of a coastline varies with the scale of the measuring instrument, has self-similarity at all scales, and is infinite in length for an infinitesimally small measuring device.
Non-functional biomimicry: utilizing natural patterns in order to provoke attention responses
Published in International Journal of Design Creativity and Innovation, 2018
Bryan G. Young, Andrew Wodehouse
There are recurring patterns that manifest throughout nature, one example of this is the Fibonacci sequence. The Fibonacci sequence is a sequence where N is the sum of the two preceding numbers, i.e. 0, 1, 1, 2, 3, 5, 8 …, It is not only abundant in nature, but it has already formed the basis for many esthetic designs (Figure 1). Teuscher (2004) explains mathematician Alan Turing believed that such patterns are a result of living matter’s ability to self-organize. Turing used nonlinear differential equations to create a computer model of nature’s hypothesized ability to self-organize. In 1979 mathematician Benoit Mandelbrot created the Mandelbrot set (Figure 2), a fractal set of points which demonstrates the creation of complex self-similar structures from simple mathematical rules.