A FIRST APPROACH OF FRACTALS

The idea of FRACTAL geometry has been developed by a French mathematician Benoit MANDELBROT. The main part of its career has been made at the IBM company. The FRACTAL geometry has been confirmed during the seventy years. It is linked to many notions as the fractional dimensions, the theory of complexe functions, the vortex of fluids and generaly speaking to all that it is close to the chaotic phenomenas...etc. Related on this matter, I recommend to the reader the following book: The Science of FRACTAL Images with, as one of the authors, B.B. Mandelbrot. Springer-Verlag Editions, Berlin, New York, Paris.

In matter of computing programs for FRACTAL on PC, it is a very good one made by an americain reseach team, and I suggest you to see their www pages at FRACTINT. On this site a project of FRACTINT on MACINTOSH is also explained and it is indicated many links for FRACTAL softwares. For the french side, there is also a very good site made by a member of the "université de Bordeaux" : Jean-Pierre LOUVET and very extensive for pictures as well as for links. It is possible to found on this HYPERFRACTALS site the list of the main URL for available Softwares/Freewares.

At the beginning, the FRACTAL theory has been developed for the complexe function: F(Z) = Z2. Herafter, it is supposed that the reader know the use of complex numbers, complex plane and complex functions. HYPERFRACTALS is useful for many functions as polynomial functions of Z, sinusoidal functions, hyperbolic funtions (See the §4 of the documentation.). Other functions, subject to request, can be realised by the autor.

A FRACTAL drawing can be obtained by the following process, applied to each point of the complexe plane,: ZN = F(ZN-1) + W as long as N < NITER ou |F(Z| < LIMIT, According to the position of the starting point, interior or exterior to the FRACTAL line of the function under computation, the process of the iteration will be different. The drawing in the margin show a very clear exemple of convergent/divergent trajectories of computation. Usually, the surface where the function is convergent is coloured uniformly. In other hand, the multi-coloured FRACTAL drawings, are situated outside of this surface. For the computation, there are various algorithms, called MANDELBROOT, JULIA...and so on. Five different algorithms, giving various FRACTALS, are proposed in HYPERFRACTALS.

For the colouring, there is a very important notion which is the latence. As first approach, this one may be define as the number NITER of prefixed iteration. This latence is divide in K intervals, each of them being coloured of different manners. In this case, it has been defined, for the point under computation, a latence to K colours (The word "latence" that I use is very close to the French. If you know a better translation, please let me know). HYPERFRACTAL include also the possibility of ZOOM which one allow you to go more and more further in the details of the drawings, but require more and more time of computation .