Fractal Geometry :: Mathematics Math Geometric Essays

Fractal Geometry In the past, mathematics has been concerned largely with sets and functions to which the methods of classical calculus could be applied. Sets or functions that are not sufficiently smooth or regular have tended to be named as " pathological" and not worthy of study. They were regarded as individual curiosities and only rarely were thought of as a class to which a general theory might be applicable. However, in recent years this attitude has changed. Irregular sets provide a much better representation of many natural phenomena than do the figures of classical geometry. Fractal geometry provides a general framework for the study of such irregular sets. (Falconer) The word ‘fractal’ was coined in 1975 by mathematician Benoit B. Mandelbrot to describe an intricate-looking set of curves, many of which were not yet seen before the creation of the computer. Fractals often exhibit self-similarity, which means that various copies of an object can be found in the original object a t smaller size scales. This continues for many magnifications – like an endless nesting of Russian dolls within dolls. (Pickover) Fractals appear everywhere in nature, in galaxies and landscapes, in earthquakes and geological cracks, in aggregates and colloids, and even in the human body. Fractal geometry is an important tool in the analysis of phenomena, ranging from rhythms in music melodies to the human heartbeat and DNA sequences. Many professions including, mathematics, astronomy, physics, chemistry, engineering, and biology use fractal geometry. (Bunde) Waclaw Sierpinski was born on March 14, 1882, in Warsaw, Poland. Sierpinski attended the University of Warsaw in 1899, when all classes were taught in Russian. He graduated in 1904 and went on to teach mathematics and physics at a girl's school in Warsaw. He left teaching in 1905 to get his doctorate at the Jagiellonian University in Cracow. After receiving his doctorate in 1908, Sierpinski went on to teach at the University of Lvov. During his years at Lvov, he wrote three books and many research papers. These books were The Theory of Irrational numbers (1910), Outline of Set Theory (1912), and The Theory of Numbers (1912). In 1919, Sierpinski accepted a job as a professor at the University of Warsaw, and this is where Waclaw Sierpinski (The Mactutor) he would spend the rest of his life. Throughout his career, Sierpinski wrote 724 papers and an amazing 50 books. Sierpinski studied many areas of mathematics, including, irrational numbers, set theory, fractal geometry, and theory of numbers.