Fractal Geometry

Fractal Geometry

How would you like to take a class called geometry of chaos? Probably doesn’t sound too thrilling. A man named Benoit Mandelbrot is responsible for creating the geometry of chaos. The geometry of chaos is considered to be the fourth-dimension. It is considered to be the world in which we live in, a world where there is constant change based on feedback, an open system where everything is related to everything else. It is now recognized as the true geometry of nature. The geometric system the can describe the simple shapes of the world (Lauwerier).

Fractal geometry is a structure that provided a new key for the study of non-linear processes (Lauwerier). Benoit Mandelbrot explained that lines have a single dimension, plane figures have two dimensions and that we live in a three dimensional spatial world (Fractals Useful Beauty). In a paper published in 1967, Mandelbrot investigated the idea of measuring the length of a coastline. Mandelbrot explained that the shape of a coastline defies conventional Euclidean geometry and that rather than having a natural number dimension, it has a “fractional dimension.” The coastline is an example of a self-similar shape, which is a shape that repeats itself over and over on different scales (Fractals).

Benoit Mandelbrot was born in Warsaw in 1924 to a Lithuanian Jewish family and grew up there until they moved to Paris in 1936 (Fractals). Benoit had never received formal education and was never taught the alphabets; to this day he still doesn’t know them from memory. Benoit’s mind was a visual geometric mind, he had a tremendous gift in math in which he would take the problems from his work and translate them mentally into pictures. Benoit’s incredible mind took him all the way to the United State in 1958 to pursue his own way of doing math (Barnsley). Mandelbrot was offered a job at IBM’s research center in New York and was allowed free reign to pursue his mathematical interests as he wished. They proved to be more diverse, eclectic and far reaching than anyone could have imagined.

His now famous study in the field of economics concerned the price of cotton, the commodity for which we have the best supply of reliable data going back hundreds of years. The day to day price fluctuations of cotton were unpredictable, but with computer analysis an overall pattern could be seen. The pattern that Mandelbrot found was both hidden and revolutionary. Mandelbrot discovered a pattern where in the tiny day to day unpredictable fluctuations repeated on larger, longer scales of time. He found symmetry in the long term price fluctuations with the short term fluctuations (Barnsley). This was surprising, and to the economists - and everyone else - completely baffling. Even to Mandelbrot the meaning of all this was still unclear. Only later did he come to understand that he had discovered a "fractal" in economic data demonstrating recursive self similarity over scales (Fractals).

Mandelbrot’s research led him to what some consider being the greatest mathematical breakthrough of the twentieth century. The law of wisdom that it represents could not have been discovered without the use of computers. Benoit composed a simple equation; z -> z^2 + c (Fractals). The order behind the chaotic production of numbers created by the formula z -> z^2 + c can only be seen by a computer calculation and graphic portrayal of these numbers. It is only when millions of calculations are mechanically performed and plotted on a two dimensional plane that the hidden geometric order of the Mandelbrot set is revealed.

Mandelbrot's fractal geometry replaces Euclidian geometry which had dominated our mathematical thinking for thousands of years, we now know that Euclidian geometry pertained only to the artificial realities of the first, second and third dimensions. These dimensions are imaginary. It could not describe the shape of a cloud, a mountain, a coastline or a tree. Only the fourth dimension is real. Before Mandelbrot, mathematicians believed that most of the patterns of nature were far too complex, irregular, fragmented and amorphous to be described mathematically. But Mandelbrot conceived and developed a new fractal geometry of nature based on the fourth dimension and Complex numbers which is capable of describing mathematically the most amorphous and chaotic forms of the real world (Barsnley).

Mandelbrot discovered that the fourth dimension of fractal forms includes an infinite set of fractional dimensions which lie between the zero and first dimension, the first and second dimension and the second and third dimension. He proved that the fourth dimension includes the fractional dimensions which lie between the first