A Short Introduction to the Pictures
The Mandebrot Set is a very complex mathematical object, which is still the subject of highly involved researches, but, for most of us, it is merely a mine of fascinating pictures. These pictures are obtained without worrying about the underlying theory through Mandelbrot explorers, i.e. specialized pieces of software. There exist many of them, such as my own MandelTour, for the Amiga computer, or FractInt, for PC computers.
These pictures can be understood simply. Have a look at the general view on the left. The Set itself is the black central shape, but the most interesting part is the coloured pattern around it.
Let us assume that the Set can be cut out of a metal plate and that this plate is heated. The temperature arises all around and this can be visualized with a thermography, which means that points where the temperature lies within a given range are rendered with a given color, whereas points with a temperature inside another range are rendered with another color, and so on... Roughly, the temperature varies all the more rapidly (in other terms, there are all the more colors) as one is closer to the Set boundary, and this is the key point: the Set boundary is fantastically, marvelously complicated. You cannot really sense this from this figure. You can see disks in all sizes and you can admit that others exist that are too small to be seen, due to the poor resolution of the figure. But there is more. The Set can be continued with myriads of invisible lines that connect it with myriads of microscopic replicas of the main set. These lines are invisible but, within our thermography interpretation, they make the temperature arise around them and this makes fantastic structures appear. Now, in order to view them, you must watch very tiny details, very close to the Set boundary; you need a powerful microscope. The software just supplies it.
In most of Mandelbrot explorers, one frames the area to be enlarged with the mouse, and the software computes it again at the size of the full screen. For instance, the white rectangle in the general view above leads to the image on the right, and enlarging the white square in this picture leads to the image below. The colors of each picture are arbitrary; the user can adjust them as he wants -more or less easily, however, according to the software.
In this third picture, one notices that the black shape looks
strikingly like the Main Set in the general view. This is a general
peculiarity of the Mandelbrot Set. Whatever the enlarging ratio, the
complicated areas worth being detailed exhibit tiny replicas of the Main
Set. This is a kind of signature of the Mandelbrot Set.
Mathematicians claim that all these tiny replicas are connected by
The whole game thus consists in examining every picture and in guessing what could be worth enlarging further, then computing new pictures and ajusting their colors at best. As a rule, the most enlarged pictures are the most interesting, but it also happens that pictures become too complex for the resolution of the screen...-a point for experts. As an example, here is an interesting image, very highly enlarged, from the arrow on the left of the Main Set.
Assuming that this picture would be 10 cm wide on your screen, the Main Set on the same scale would be 10E33 cm wide (10 to the power 33), i.e. 10E15 light years, an astounding figure desperately out of prehension. Our Milky Way is only 100,000 light years across...
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