A HYPERPOLAR IMAGE OF THE MANDELBROT SET 139 A Hyperpolar Image of the Mandelbrot Set

Non-random complicated motions can exhibit a very rapid growth of errors and, despite perfect determinism, they inhibit the pragmatic ability to render accurate long term predictions. Technically, such non-random motions are termed “chaotic. ” The geometry of chaotic behavior is known as “fractal geometry.” The term “fractal ” was coined by Benoit Mandelbrot to mean “fractional dimension.” Fractals are structures which are elaborated upon at smaller and smaller scales differently at each point of an object. In 1979 a computer experiment made it possible to create the diagram of a certain object called the Mandelbrot set. It is a “limit fractal ” that contains many other fractals. As such it is of great importance in fractal geometry. In this article we aim to map the Mandelbrot set from the normal complex plane into the hyperpolar complex plane and present a computer generated hyperpolar image of this “limit fractal.” In 1918 Gaston Julia (1893-1978) published a 199-page masterpiece entitled Memoire sur l’iteration des fonctiones rationelles. Pure et Appl. 8. (1918) 47-245. This paper deals with the iterations of complex polynomials like z2 + c, z3 + c,..., zn+c, where c is a parameter such that for any fixed value of c, a sequence of complex numbers is generated when we choose some value for z and then start an iteration process. For example, if we have the polynomial z2+ c for fixed c, iterating means that we choose some value for z and obtain z2+ c. Next we substitute this resulting complex number into the polynomial obtaining (z2+ c)2 + c. Proceeding in this manner, we generate the sequence z 0 = z, z 1 = z 2 + c, z2 = (z 2 + c)2 + c, z3 = z2 + c 2 +