Fractional geometry and mathematical structures

Fractal Geometry is a somewhat new branch of mathematics that was developed in 1980 by Benoit B. Mandelbrot, a research mathematician in I.B.M.'s Thomas Day Watson laboratory in New York. Mandelbrot was experimenting with the theories of Gaston Julia, a French mathematician when he discovered the fractal set was discovered.

Julia dedicated his life to the study of the iteration of polynomials and rational functions. Around the 1920s, Julia published a paper on the iteration of a rational function, which brought him to fame. However, after his death, he was all but forgotten...until the 1970's when Mandelbrot, who was inspired by Julia's work, revived his work.

By using computer graphics, Mandelbrot was able to show the first pictures of the most beautiful fractals known today.

Mandelbrot, who is now Professor of Mathematics at Yale, made the discovery of fractal geometry by going against establishment and academic mathematics -- going beyond Einstein's theories to reveal that the fourth dimension includes not only the first three dimensions, but also the gaps or intervals between them, which are known as the fractal dimensions.

Fractal geometry is most widely recognized as pictures used as backgrounds on computer screens but it is really much more complicated than that. According to Mandelbrot's theory, the majority of nature's physical systems and human artifacts are not regular geometric shapes and cannot be found in standard geometry. Fractal geometry focuses on this, offering countless methods of describing, measuring and predicting these natural phenomena.

Fractal geometry has caused quite a stir in the mathematical world, as people are fascinated both by the pretty images of fractals and the extension beyond the typical perception of mathematics as a body of complicated, boring formulas.

Fractal geometry is a combination of mathematics and art that demonstrates that equations go far beyond just a collection of numbers. Fractals are the optimal existing mathematical descriptions of various forms in nature, including coastlines, mountains or parts of living organisms.

Fractal geometry, which was created nearly entirely by Mandelbrot, is now recognized as the true geometry of nature. This fractal method has replaced Euclidian geometry, the dominant theory behind mathematical thinking for centuries. Today, mathematicians view Euclidian geometry as pertaining only to the artificial realities of the first, second and third dimensions, which are imaginary. Mandelbrot introduced the fourth dimension, which is real.

Before the discovery of fractal geometry, the academic world was dominated by arithmetics. Geometry was placed in a secondary position and mathematics was rather detached from the real world, particularly nature. Fractal geometry changed this entirely.

Mandelbrot was born into the atmosphere of academic math, as his uncle was a distinguished member of a well-known French mathematician group in Paris. However, Mandelbrot received no formal education. He was never taught the alphabet and never learned multiplication tables past fives. Still today, Mandelbrot says it is difficult for him to recite the alphabet.

However, he had the mind of a genius and enrolled in Paris universities when he was older. In college, his mathematical mind, which was unlike anything ever seen in the academic world, helped him to obtain his doctoral degree. His mind was a visual one, a geometric mind, yet he was not taught this way.

Mandelbrot claims he could not do algebra well yet managed to receive the highest grades by translating the questions mentally into pictures. When he finished school, he came to the United States, where IBM gave him the freedom to pursue his mathematical interests, as he deemed worthy.

Mandelbrot's research led to a huge breakthrough summarized by a simple mathematical formula: z -> z^2 + c. This formula is now called the Mandelbrot set. It is important to understand that this formula, and the Law of Wisdom which it represents, could not have been discovered without computers. Many say that this mathematical breakthrough, which occurred in the research laboratories of I.B.M., is the greatest in twentieth century mathematics.

The Mandelbrot set is a dynamic calculation based on the iteration of complex numbers with zero at the beginning. The order behind the chaotic production of numbers created by the formula can only be seen by the computer calculation and graphic portrayal of these numbers on computers. Otherwise the formula appears to generate a totally random and meaningless set of numbers. It is only when millions of calculations are mechanically performed and plotted on a computer screen that the hidden geometric order of the Mandelbrot set is seen. The order is of a beautiful yet different kind, containing self-similar recursiveness over an infinite scale.

Euclidian geometry was unable to describe the shape of a cloud, coastline, a hill or a tree. As Mandelbrot says in his book the Fractal Geometry of Nature:

Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line."