Fractal Geometry



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Fractal Geometry
Fractals have been around for millions of years, but were not given a name until 1973. At this time, mathematician Benoit Mandelbrot was studying a type of geometry that did not seem to correspond with any of the existing categories of geometry. Mandelbrot grew up Poland and France, and after completing two years at a French college, Mandelbrot received a scholarship to attend the California Institute of Technology. Mandelbrot earned a master’s degree in aeronautics before returning to France. Mandelbrot happened to read a book written by George Zipf, which addressed naturally occurring distributions that did not fit the bell-shaped curve. This inspired Mandelbrot to write his thesis in 1952 at the University of Paris. Mandelbrot wrote on Zipf’s ideas, which eventually led him to try using mathematics to answer questions in other areas of study. Mandelbrot taught at several European universities before moving with his wife and children to New York, where he joined the IBM Thomas J. Watson Research Center.
Mandelbrot began studying the stock market and found that the daily price variations were similar to monthly price variations, and therefore became more predictable. While studying another problem dealing with random noise along wires, Mandelbrot yet again detected a self-similar pattern and was able to use the Cantor set, developed in the nineteenth century, as a model. In the 1960’s Mandelbrot became interested in the problem of determining the length of coastlines. The question he addressed is known as “How Long is the Coast of Britain?” Mandelbrot found that the answer depended on the length of the ruler being used, and as the scale became smaller, the measured length became infinitely large. Mandelbrot also came up with computer programs to determine the wiggliness of curves and began looking at these patterns in nature. Finally, in 1973 Mandelbrot was asked to give a talk at the College de France, and he realized that his research had led him to a new kind of geometry. Mandelbrot coined the term fractal from the Latin word fractus, meaning fragmented or irregular. Mandelbrot’s definition of a fractal states that “a fractal is a rough or fragmented geometric shape that can be subdivided into parts, each of which is (at least approximately) a reduced-size copy of the whole” [3]. In 1982 Mandelbrot published his book The Fractal Geometry of Nature. This book laid down the foundations for fractal geometry, stated the new vocabulary used in fractal geometry, and discussed the existence of fractals in nature.
It is only fitting to begin with the most famous fractal named after the creator himself, the Mandelbrot set. Mandelbrot discovered this set after studying the work of French mathematician Gaston Julia. This set is generated by squaring a complex number and then adding the original number to the answer. That result is then plugged back into the original formula. The formula follows this format:

c, c2 + c, (c2 + c) 2 + c, [(c2 + c) 2 + c] 2 + c, . . . where c is a complex number.



For c to be a member of the Mandelbrot set, the sequence must be bounded for the complex number c. The Mandelbrot set is nonlinear, and under magnification, it is evident that each of the outgrowths is self-similar to the original shape. Another fractal set is the Triadic Cantor Set which is generated by starting with a line segment, dividing it into three equal parts, and then removing the middle third, leaving its endpoints. This process is repeated with the two remaining parts and so on, producing very short segments, giving this set the name “Cantor dust”. This result is typical of a fractal set because as the numbers of repetitions grow, the total length of the set goes to zero, while the number of segments goes to infinity. The snowflake curve is another example of a fractal set. This curve was first defined in 1904 by Helge von Koch, a Swedish mathematician, thus it is also known as the von Koch curve. Starting with an equilateral triangle with side length a, the curve is produced by successively adding a new equilateral triangle to the middle third of each side. The sides of the new triangles are of length a/3. This process is repeated, and as a result the perimeter of the curve goes to infinity. Looking at the area of the curve, however, gives a surprisingly different result. The area is contained within the circumcircle of the original triangle and is calculated to be 8/5 of the original area. Thus the area of the curve is finite. There are many other fractal constructions including the Sierpinski triangle and the Sierpinski carpet.
Fractals play a part in the daily lives of every living creature and are literally everywhere. It should be noted that fractals in nature are slightly different than those that are created mathematically. “. . . no object in nature can be magnified an infinite number of times and still present the same shape of every detail in successive magnifications. . .” [1]. The main reason for this is that molecules and atoms are both of finite size. One example of a fractal surface found in nature is the brittle fracture. This type of fracture occurs in materials such as silicon, glass, or steel. Many coastlines and river networks seem to have fractal properties. They boast self-similar traits and appear to have a perimeter that goes to infinity. Clouds are also examples of fractals in nature. If a cloud were to be magnified, it would seem to be made of many smaller pieces similar to the whole cloud. Other fractal structures in nature include craters, fault systems, lightning, frost, coral, polymers, and post- percolation coffee ground clumps. Fractals have also been used to make models to analyze fluid mechanics, linguistics, economics, biological tissue, and population growth.
Almost thirty years ago the word fractal was born, and since that time fractals have come to have great meaning to people of many different professions. The applications of fractal geometry seem endless although the study of this geometry has only just begun. As mathematicians and scientists learn more about this new branch of knowledge, I am sure there will be many more important discoveries regarding fractals in our universe.
Sarah Rochleau, 2007
References:
1. Gullberg, Jan. Mathematics – From the Birth of Numbers. New York: Norton, 1997.
2. O’Connor, J.J., and E.F. Robertson. “Benoit Mandelbrot”, 1999,


3. Sornette, Didier. Critical Phenomena in Natural Sciences. Berlin: Springer, 2000.
4. Young, Robyn V. Notable Mathematicians. Detroit: Gale, 1998.

Mandelbrot Set




Triadic Cantor Set



Snowflake Curve




Sierpinski Triangle


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