fractals
Bringing Order to Chaos Fractals, a word that the vast majority of people don’t know. Of those that do, very few really understand its meaning, let alone the impact fractals have on everyday life. Yet they have come to help us understand some of the most complex problems that exists today. So what are they and more importantly, where did they come from? Jovan Nedic finds out “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightening travel in a straight line”
Benoit B. Mandelbrot
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f you were to look at the world around you, you would quickly realise that fractals appear in nature more often than you might think. Take for example a tree, which has one thick trunk, that then splits into thinner branches, each of which have their own, slightly smaller branches, who in turn have their own even smaller branchlets. This self-repeating pattern, where each iteration becomes progressively smaller than the previous one, is what is referred to as a fractal pattern. In a similar vein, the river networks around the world, vegetables such as the romanesco brocolli, the snowflakes that fall from the sky and even the lungs in our bodies all have this same, self repeating pattern, showing that fractals really do exist everywhere we look. Benoit B. Mandelbrot, a Polish-born mathematician, was the first to coin the word fractal in 1975 to describe an object which exhibits similar characteristics, no matter what the magnification. Initially shunned by the academic world, Mandelbrot carried on his research, expanding his fractal theories to the natural world in his 1982 paper entitled the Fractal Geometry of Nature and eventually into the financial world. He warned in his 2005 book, The (Mis)Behaviour of Market, that traders were under the impression that a financial market is predictable and could withstand large fluctuations. Four years later, he was proved right when markets across the globe began to crash. Mandelbrot was a rare type of scientist, one that could not only communicate his views exceptionally well to the rest of the scientific community, but also to the general public, often giving popular lectures on the subject. Following the news of his death in October 2010, there was a sudden media explosion with feature articles on the “father of fractal geometry”. Quite rightly so! But the vast majority of these articles and features failed to fully comprehend the impact his work had on all walks of life. In simple terms, Mandelbrot’s work brought order to chaos, and found a method to describe the unpredictable. His work and theories have expanded into numerous fields from cosmology to geography, as well as engineering and medicine, each time giving an insight into phenomena that people thought too unpredictable to model. Take turbulence for example, a phenomena that is inherently unpredictable and unsteady yet occurs in the vast majority of fluid dynamic problems. The wind blowing against your face as you walk
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down the high street, the water flowing around objects in a river, even the air flowing over cars and aircraft, are all turbulent. For the past century, fluid dynamists’ have been using a regular mesh grid to generate turbulent flows in their wind tunnel experiments and have then developed models to describe the effect turbulence would have on a variety of examples. But is the air that buildings, cars or aircraft experience accurately modelled by this simple mesh? Going back to the tree example, which we have already shown to be fractal in nature, it is not too difficult, therefore, to imagine that the air passing through these trees would also have fractal properties. The realisation of this is both simple and frightening at the same time; our turbulence models could be wrong! Although it may appear that the last 100 years of work have all been for nothing, the reality is that the solutions are incomplete and by applying fractal theory, we expand our understanding of the problem. This work is in fact being carried out right here at Imperial, under the supervision of Professor Vassilicos from the Department of Aeronautics. Understanding the fractal nature of turbulence has led to several important discoveries, with the work by Professor Vassilicos and his team suggesting that a new class of turbulence should be defined. It has even led to patents of several fractal applications, including mixers in chemical plants, ventilation systems, aircraft spoilers and even improving combustion levels in engines. In other scientific disciplines, fractals have been used for fracture mechanics in materials science, studying the failure modes of materials which can be highly irregular. They have also been used for signal and image compression, whereby an image is analysed to search for self-repeating patterns that are saved as codes, instead of saving the individual pixels, thus saving space. Outside the realms of science, fractals have been found to exist in music and have been used to generate some stunning artistic images, including works by Jackson Pollock. Even though the Father of Fractals is no longer with us, it is clear that his ideas have had an affect on a wide range of subjects and applications, with many scientists and engineers taking up the challenge of fractals. Mandelbrot spent most of his life expanding our understanding of the natural world through his love of fractals and it is not surprising that when once asked what the B in Benoit B. Mandelbrot stood for, he simply replied “Benoit B. Mandelbrot”.
“Mandelbrot spent most of his life expanding our understanding of the natural world through his love of fractals”
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