![]() There is not much incentive to look more closely at this mud, still less to draw the Julia sets, but they can actually be carried to look like this: Therefore a p of it is missing in the right-hand side, so that only the left-hand side gives information about the Julia sets: This Mandelbrot set is among those enclosed in several of the programs (and its formula is related to the solution of the third-degree equation z 3 = 1), but it is constructed from one and not from two different critical points. However, this is only true when the Mandelbrot set is constructed by means of two critical points. To each point in the plane is associated a Julia set, and the idea of the Mandelbrot set is to act as an "atlas" of the Julia sets, since the self-similar structure of a Julia set is a reflection of the local structure of the Mandelbrot set at the point. The bad understanding of the mathematics has had as consequence that almost all Mandelbrot sets, except the usual, are "false", that is, made without correct application of the concept critical point. ![]() Ought it not to be a matter of course nowadays that such a motif looked like this? ![]() When you buy a program that is able to make pictures like the two shown at the end of this, why give you the impression that the program is for pictures like this? ![]() However, they still haunt us, for it is solely these that the producers of fractal programs have recourse to in their presentation material. So these sets were slid in the background and not a single technically perfect picture was ever made. Countless have been invented, and they have all been easier to deal with than the Mandelbrot and Julia sets. And the exponents of the new and expanding branch of - "fractal " - did not inquire about research that could help them rectify with the defects that clearly encumbered their pictures besides they were busily engaged in finding new ways to create fractal patterns. The mathematicians specializing in the new subject did not care for the pictures when they needed illustrations in their papers, they were satisfied with somewhat home-made programming. This is possibly not difficult in cases where the iterations are towards infinity, but what if the function does not fulfil this condition? What then are the terminal positions of the iteration sequences? Is it (still) possible to colour perfectly? Won't the drawing time be intolerably long? These matters were up to the mathematicians to settle, while the computer people developed the computer.īut this settlement has not come off until the fractal project introduced by this. The method of getting all the near points visible, that is, of drawing the boundary, had to be generalized and a method of achieving smooth colouring had to be found. For the procedure, which in Mandelbrot's case only involved the simple function z 2, could obviously be generalized to any complex function, and what patterns could appear for so many different functions? Besides better computer technology, really beautiful pictures would, however, need some mathematical research. But there was certainly futurity in this pictures - for amazement and for decoration. However, at that time the progress was checked by the fact that the computers worked slowly and had too narrow a scope of colours. These computer-generated pictures displayed patterns that differed entirely from anything previously known and beyond anyone's imagination. I am sure that the mathematicians who made these pictures of the Mandelbrot set purposefully rejected colouring. There is an easy method of colouring the domain outside the black, namely by means of the iteration number, but if you use this method the colours will lie in bands with abrupt passages looking unaesthetic. Then it seemed that the Mandelbrot set was connected and that it looked as if it was dressed in a fur, the hairs of which could exhibit the most unbelievable shapes: However, some years later a method was invented that made it possible to plot out all the points lying near some cardoid or circle no matter what the size, and in this way, without unreasonable computation time, it was possible to look of the true form of the set. The figure had no connection whatsoever with beauty. There was nothing revolutionary about his first printouts, they only showed something looking like an infinite system of cardoids and circles of strongly varying size, lying outside each other and some touching. ![]() Mandelbrot first thought of his strange set. Presentation of the program package juliasetsĪnd evaluation of the program Ultra Fractal ![]()
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