Benoit Mandelbrot: The Father Of Fractals

CHAPTER 1 :

INTRODUCTION

Benoit Mandelbrot (1924-2010), scientist and mathematician who also worked at IBM, is known as the father of fractal geometry. Mandelbrot coined the word fractal in the late 1970s. Before the invention of computers, Fractals have come up as an important question. Fractal is a set, which is self–similar under magnification.[27] It is, however, remarked that many of the fractals and their similes go back to traditional mathematics and mathematicians of the past like George Cantor(1872), Giuseppe Peano(1890), David Hilbert(1891), HelgeVon Koch(1904), Waclaw Sierpinski(1916), Gaston Julia (1918), Felix Hausdorff (1919) and others.

“… that they are representations of relatively simple yet extremely powerful mathematical

2.4 Dimensions in Fractals

Fractals provide us with facility of having a non-integer dimension (Hausdorff-Besicovitch dimension) over traditional Euclidean dimension having integer dimension (with which we are more familiar till now). A definition According to Mandelbrot [3], ‘A fractal is a set whose Hausdorff-Besicovitch dimension strictly exceeds its topological dimension’.

Before studying fractal geometry, we were only able to understand the topological or ‘usual’ integer dimension illustrated by Euclidean geometry. For example consider, a line is one dimension object, a square is two dimension object and, a cube is three dimension object.

Said E.Al-Khamy[33], defines the fractal dimension D as the measure of the complexity or the space filling ability of the fractal shape. The fractal dimension (a positive real number) is either equal or greater than the topological dimension (a positive integer number). Various formulation and methods exists to find the fractal dimension of fractal (self-similar) structure. One such method which is most popular is formulated as- ,

Where, D is fractal dimension, N number of parts contained in a self-similar object and, r is the ratio of

As quoted by Devany [10], iteration of function Ac=z’2+c, using the Escape Time Algorithm, results in many strange and surprising structures. Devany [10] has named it Tricorns and observed that f(z’), the conjugate function of f(z), is antipolynomial. Further, its second iterates is a polynomial of degree 4. The function z’2+c is conjugate of z’2+d, where d=e2πi/3, which shows that the Tricorn is symmetric under rotations through angle 2π/3. The critical point for Ac is 0, since c=Ac(0) has only one pre-image whereas any other w ϵ C, has two preimages.

3.4 Iteration Techniques and Classification Iteration means to repeat a process again and again. Starting with the initial value, the output is fed back to the process. The procedure is repeated until the result or goal is reached.

Based on the steps, which has to be followed for completing one feedback circuit, the iteration techniques or the iteration processes can be classified as, one step iteration and two steps iteration. The Mann’s Iteration is an example of one step iteration. The Ishikawa iteration is an example of two step iteration.

3.4.1 Mann’s Iteration : One Step