Author: Lesmoir-Gordon, Nigel
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Plato sought to explain nature with five regular solid forms. Newton and Kepler bent Plato’s circle into an ellipse.
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Benoît Mandelbrot has established a discovery that ranks with the laws of regular motion.
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The word “fractal” was coined in 1975 by the Polish/French/American mathematician, Benoît Mandelbrot (b. 1924), to describe shapes which are detailed at all scales.
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Newton’s theory of infinitesimals is riddled with paradoxes.
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The methods of the calculus apply wherever a curve is smooth.
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His quest for absolute rigour led to his discovery of a nowhere differentiable continuous function: that
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His quest for absolute rigour led to his discovery of a nowhere differentiable continuous function: that is, a curve consisting completely of corners.
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Weierstrass and Cauchy developed a new branch of mathematics called Analysis.
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Analysis showed how all mathematics could be explained in terms of simple whole numbers.
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Sets can contain other sets but they cannot contain themselves, for this way lies madness.
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Georg Cantor, one of the pioneers of modern set theory, started out on a problem which continued to vex him for the rest of his life: the nature of the continuum. The continuum is the ideal infinitely divisible space conceptually required for a theory of continuous change.
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Cantor’s argument implied the existence of different types of infinity. He went on to develop a whole new theory of transfinite arithmetic, convinced that he had uncovered a powerful new principle of reality with profound physical and spiritual implications.
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Take a line, remove the middle third, leaving two equal lines. Likewise remove the middle thirds from each of these two lines. Repeat this process an infinite number of times, and you are left with the Cantor set.
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In technical parlance, it has “zero measure”.
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Dimension is a topological invariant. It cannot be altered by continuous deformation
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Hausdorff’s approach gave a fractional dimension. This allows for the intriguing possibility of one-and-a-half-dimensional objects.
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How can a shape lie in-between dimensions? By being fractal.
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Visually, it is apparent that the Cantor set consists simply of two small copies of itself. This property is known as self-similarity.
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In some sense the Cantor set and Koch curve lie on either side of the straight line.
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An n-dimensional shape is composed of mn 1/m-sized copies of itself.
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Logarithms are the reverse of exponents. If we know how many smaller copies of itself an object contains, and the relative size, logarithms allow us to calculate the dimension of the object.
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In the case of a straight line, halving the size of the squares requires twice as many squares to cover it. However, for a fractal, more than twice as many squares are needed.
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Henri Poincaré, mathematician, physicist and philosopher of science, showed that deep insights into the rather complicated behaviour of dynamical systems can be obtained from quite simple mathematical models.
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As Michael Barnsley has shown, if you add another screen, each screen now shows two screens, each showing two screens, etc.
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Mitchell Feigenbaum showed that the ratio of distances between successive bifurcations converged rapidly to a constant, 4.669201660910
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Physical instances were revealed in labs around the world. And when the rate of successive doubling was measured, experiment after experiment showed remarkable numerical agreement with Feigenbaum’s number,
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In 1685, the English mathematician John Wallis (1616–1703) hit upon the idea of representing complex numbers graphically, in a diagram.
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The boundary between the catchment areas was made up of repelling points, which pushed neighbouring points away. These boundaries turned out to be very complicated. They are now known as Julia sets.
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Often these transformations led directly to a solution in physics and chemistry. But where he could not employ geometry, he did not fare so well. Sitting these examination papers brought home again to Benoit that there was a “mathematics of the eye”, that visualization of a problem was as valid a method as any for finding a solution.
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Mandelbrot became interested in the Julia set of the simplest possible transformation: z→z2+c. This formula gives a rule for getting one complex number from another, or in other words, mapping the complex plane onto itself.
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The effect of this mapping is to cut the plane and wrap it around itself while stretching it away from the unit circle.
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Nevertheless, the fact that the Mandelbrot set exists and that it has an area is enough for mathematicians of the Everest school to attempt the challenge – because it’s there.
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The recurrence of the Mandelbrot set in a wide range of different fractals is due to the phenomenon of universality.
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This is the fundamental characteristic behind the growth of all complex organisms. The same forms recur in many different circumstances, in different materials both organic and inorganic, on a vast range of scales.
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once a tree is in flames, the fire easily spreads to neighbouring trees, and this process can now be modelled with iterative techniques.
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We are fractal. Our lungs, our circulatory system, our brains are like trees. They are fractal structures.
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Fractal geometry has revealed some remarkable insights into a ubiquitous and mysterious “three-quarter power law”. This particular power law models the way that one structure relates to and interacts with another. It is based on the cube of the fourth root.
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For example, complex protein surfaces fold up and wrinkle around towards three-dimensional space in a dimension that is around 2.4.
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All aspects of nature follow mathematical rules and involve some roughness and a lot of irregularity.
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Statistical models using fractal geometry are in use testing for stress loading on oil rigs and on aircraft in turbulence, with particular emphasis on the effects of very short gusts of wind.
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Corrosion reveals the fractal nature of the process, suggesting ways and means to alleviate the problem.
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take five points, arranged in a square with one in the middle. Replace each point with a scaled-down copy of the whole pattern. Don’t stop there! Keep going, replacing each point of this new diagram with a smaller duplicate of the whole shape. And so on … In the limit, you are left with a Fournier fractal.
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Although highly simplified, it shows that an infinite universe need not be equally bright in all directions, if it has a fractal structure.
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Benoît Mandelbrot published his most recent work in 1998. It is called Fractal and Scaling in Finance: Discontinuity, Concentration, Risk.
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Mandelbrot had always been convinced that his researches had shown that probability, statistics and fractal geometry could really help describe mathematically what goes on in markets.
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Fractal shapes were being expressed intuitively by artists long before they were recognized in science.
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Spectral analysis of music from classical to nursery rhymes has revealed a remarkable affinity with patterns in nature, in particular a fractal distribution called 1/f noise, which is found in the sound of a waterfall or waves crashing on a beach.
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The unfinished cathedral by Antonio Gaudí (1852–1926) in Barcelona is a stunning example of fractal architecture.
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It’s early days yet, but asked if fractals are going anywhere, Arthur C. Clarke retorted