The Man, The Myth, The Mandelbrot Theory
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Benoit Mandelbrot was a mathematician who discovered the Mandelbrot set, a set of complex numbers calculated by repeatedly applying a simple formula. The Mandelbrot set has an infinite boundary and self-similar patterns that occur regardless of scale. While seemingly simple, the Mandelbrot set exhibits complex behavior and has connections to real-world patterns seen in coastlines, stock price fluctuations, and other natural phenomena.
The Man, The Myth, The Mandelbrot Theory
- 1. Introducing ... The Man... The Myth... The Mandelbrot Theory ... Starring Benoit Mandelbrot Al Urasek CS 131 March 31, 2010
- 2. The Man The Myth Quick Biography : Born November 20th, 1924. 1944: Ecole Polytechnique, graduated in 1947 1948: M. S. in Aeronautics from the California Institute of Technology 1952: Ph.D. in Mathematical Sciences from the University of Paris 1949-1957: CNRS. 1955-57: Professor of mathematics in Geneva 1958: IBM 1987: professor of mathematics at Yale University. IBM Fellow Emeritus, (Physical Sciences) at the IBM T. J. Watson Research Center, Professor of the Practice of Mathematics at Harvard. 1995: Professeur de l'Académie des Sciences de l'École Polytechnique. Distinctions: "Barnard Medal for Meritorious Service to Science" (1985) Franklin Medal for Signal and Eminent Service in Science (1986) Alexander von Humboldt Prize (1988) Charles Proteus Steinmetz Medal (1988) Science for Art" Prize (1988) Harvey Prize for Science and Technology (1989) Nevada Medal The Wolf Prize in Physics The Honda Prize (1994) Click text box for source.
- 3. The Mandelbrot Theory The rekursive formula applied to real numbers: The mandelbrot-set is based on a fairly simple formula: z = z² + c This is a so-called rekursive function. When we want to calculate values of this function, we have to choose a value for the constant c first, for example c = 1. Now we start with z = 0 and calculate z² + c ("first iteration"), in our example this would be 0² + 1 = 1. We use this as the new value of z, and put it back into the Ite ra tio n : c = -2 .1 c = -2 c = -1 c = -0 .5 c = 0 .5 c = 1 formula ("second Iteration"): z² + c = 1² + 1 = 2. 1 -2 .1 0 -2 -1 -0 .5 0 0 .5 0 1 This process is repeated several times: 2 2 .3 1 2 0 -0 .2 5 0 .7 5 2 2² + 1 = 4 + 1 = 5 5² + 1 = 25 + 1 = 26 3 3 .2 4 2 -1 -0 .4 4 1 .0 6 5 26² + 1 = 676 + 1 = 677 4 8 .3 7 2 0 -0 .3 1 1 .6 3 26 Definition of the mandelbrot-set: 5 6 8 .0 0 2 -1 -0 .4 0 3 .1 5 677 We notice: For some c the value of z increases quite fast (e. g. c = -2.1; c = 0.5; c = 1), we say "z approaches infinity". For other values of c this doesn't happen: z doesn't approach infinity, but stays between certain boundaries (e. g. c = -2; c = -1; c = -0.5). But how do we know, that with c = 0.5 the value of z doesn't decrease after the fifth iteration? It is possible to prove, that z approaches infinity, if it gets greater than 2. That's why with c = 0.5 z has to approach infinity, because in the fifth iteration it is greater than 2. Thus the calculation can be stopped if z gets greater than 2, because then we know that z approaches infinity. If z doesn't get greater than 2 after a sufficiently c=-1+i c = - 0.5i c = 0.5 + 0.5i big number of iterations (e. g. 100), we can assume, that z doesn't approach Iteration: infinity. z |z| z |z| z |z| Now we investigate the behaviour of z for as many c as possible. In order to 1 -1+i 1.41 - 0.5i 0.50 0.5 + 0.5i 0.71 record the results we mark each c with a black dot, if z doesn't approach infinity. These coloured numbers form the mandelbrot-set: 2 -1-i 1.41 - 0.25 - 0.5i 0.56 0.5 + i 1.12 We see: Every number between -2 and 0.25 is part of the mandelbrot-set. 3 - 1 + 3i 3.16 - 0.19 - 0.25i 0.31 - 0.25 + 1.5i 1.52 The rekursive formula applied to complex numbers: 4 - 9 - 5i 10.30 - 0.03 - 0.41i 0.41 - 1.69 - 0.25i 1.71 This time we choose c as a complex number, e. g. c = - 1 + i. We start again with z = 0: 5 55 + 46i 106.33 - 0.16 - 0.48i 0.51 3.29 + 1.34i 3.55 1st Iteration: z = z² + c = 0² - 1 + i = - 1 + i 2nd Iteration: z = z² + c = (- 1 + i)² - 1 + i = (-1)² + 2·(-1)·i + i² - 1 + i = 1 - 2i - 1 - 1 +i=-1-i 3rd Iteration: z = (- 1 - i)² - 1 + i = 1 + 2i - 1 - 1 + i = - 1 + 3i 4th Iteration: z = (- 1 + 3i)² - 1 + i = 1 - 6i - 9 - 1 + i = - 9 - 5i 5th Iteration: z = (- 9 - 5i)² - 1 + i = 81 + 45i - 25 - 1 + i = 55 + 46i Besides we need the absolute value of z (written: |z|), that is |z|² = (Re(z))² + (Im(z))². Example: |-9-5i|² = 9² + 5² = 81 + 25 = 106, therefore |-9-5i| = 10.3 The mandelbrot-set: There are again certain values for c, where |z| approaches infinity (e. g. c = -1+i older c = 0.5+0.5i), and others, where |z| doesn't approach infinity (e. g. c = 0.5i). As for real numbers, we also know for complex numbers: If |z| gets greater than 2, then |z| approaches infinity for sure. With c = 0.5+0.5i for example |z| approaches infinity, because in the fifth iteration |z| = 3.55 is greater than 2. Now we mark again every number c, for which |z| doesn't approach infinity. We cannot plot complex numbers on a (one-dimensional) line, so we get a two- dimensional picture.
- 4. Beyond the Mandelbrot You have just seen the theory in all it‟s mathematical glory. You may still be wondering though, “What the heck does it mean and what the heck is „i‟?” The letter „i‟ represents the „imaginary‟ number equal to the square root of -1 because we can‟t calculate that value using „real‟ numbers or integers. By plugging „i‟ into the simple equation z2+c, where c is a constant value, and repeating this equation over and over using the previous result as the variable, we get values that are very extreme and other values that are normal. When the results are put on a graph, they look like the shape you saw at the end of the last slide. By adding color to the different numbers outside of the equation values, and magnifying the picture indefinitely, you get these funky looking designs like the backgrounds in these slides. So, you say “What‟s the purpose of this besides drawing neat little pictures?” Mandelbrot discovered that these patterns can be found in many examples throughout the real world. For example, if you were to measure the coastline of a section of land, it would get longer and longer as you measured it in greater and greater detail because of the extremely small irregularities in the shape of the land. Another example is in the way stock market prices fluctuate. We often see graphs of stock prices as they rise and fall, but if you measure the prices in infinitely shorter time intervals the graph will get more and more complex.
- 5. Odd facts about the Mandelbrot set: The area of the Mandelbrot set is unknown, but it's fairly small. The length of the border is known - it's infinite! The barnacle covered pear shape that you see occurs an infinite number of times in the Mandelbrot set. Rotated, distorted and shrunken, but quite recognizable. All of the black areas of the Mandelbrot set are connected together. Every band of colour around the Mandelbrot set goes all the way around, without breaking, and without crossing any other colour bands. Think about that when looking at some of the more complex areas! The Mandelbrot set can be used as a very inefficient way to calculate PI. The Mandelbrot set is named after its discoverer, Benoit B. Mandelbrot. Without computers, the Mandelbrot set would be invisible. Even a very low-resolution image of the Mandelbrot set requires millions of calculations. The Mandelbrot set has infinite detail - you can The Mandelbrot Set Crop Formation keep zooming in forever. The Mandelbrot Set appeared as a crop circle on August 11, 1991 near Cambridge University. Ironically, Mandelbrot had taught at Cambridge. In the years following 1991 many crop circle formations related to the science of chaos have appeared, such as the Koch Snowflake Fractal, and Julia Set types.