Math Research Paper On Fractals

2574 words - 10 pages

Fractals Usually when one speaks of fractals to an audience, the audience refers to a piece of art that consists of repeating shapes and self-similar patterning ("same as near as far"). However, they do not realize that fractals were made and recognized due to a mathematical concept that was developed by many people such as Gottfried Leibniz, Georg Cantor, Waclaw Sierpiński, Gaston Julia, and Benoit Mandelbrot. Mandelbrot was the one to coin the term "fractal", but the other mathematicians paved the way for Mandelbrot's findings. Also, some people may think that there is just one way to make a fractal, which is by using math. This is not totally true, as there ways to make the same fractal in a different approach, such as by using the L-system.A fractal created with the Julia set under constant -.74434 - .10772i.The mathematics behind fractals, such as the Koch Curve, is rather simple, yet it becomes a complicated, yet ingenious method when referring to the Mandelbrot Set. In order to create a fractal, you will need to be acquainted with complex numbers. Complex numbers on a graph are characterized by the coordinates of (x,y) in which the x-coordinate is any rational number, whereas the y-coordinate contains an imaginary number, denoted by "i". An imaginary number is the square root of a negative number. An example of a complex number can be: (5 + 4i).Complex numbers look ambitious, but they are not. In order to add complex numbers, you just need to add like terms:. In order to multiply complex numbers, you need to use the distributive law:.If you are faced with i2, please note that it is equal to -1!We shall start with Mandelbrot's set. The Mandelbrot set was popularized due to its aesthetic appeal to many, and the simple rules that were applied in order to generate a complex structure.The Mandelbrot set is generated by iteration, which is the repetition of a process multiple times. The process for fractals is the application of mathematical functions. In the Mandelbrot set, the function is z2 + z0, where z0 is a constant value.In order to start the iteration, you need to start with a starting value, or seed, that takes the form of z0. Thus, iterating this starting value with the function yields:z1 = (z0)2 + z0z2 = (z1)2 + z0z3 = (z2)2 + z0z4 = (z3)2 + z0z5 = (z4)2 + z0and so on. The numbers generated by this process have a name, called the orbit of z0being iterated by z2 + z0. During the years, mathematicians wondered the "fate" of the orbits, whether they converge or diverge or perform a cycle of patterns or just do not create a pattern and behave chaotic.With Mandelbrot's set, we can determine the fate of the orbits. For example, when we substitute the constant, z0, as 1, and keep the seed of the iteration as 0:z1 = 02 + 1 = 1z2 = 12 + 1 = 2z3 = 22 + 1 = 5z4 = 52 + 1 = 26and we see that this orbit tends to infinity, therefore 1 is not included in the MandelbrotSet.If, however, the z0 valued 0 and the seed was 0, then the orbit would...

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