For the layman, the adventure of fractal pictures begins with the Mandelbrot Set. Splendid pictures of it were revealed as soon as computers were available with graphic outputs worthy of the name. A first exhibition in 1984 was a great success and it was continued into a book with an explicit title, The Beauty of Fractals, by H.O. Peitgen and P.H. Richter (Springer-Verlag, Berlin, 1986). A few years later (86-90), the improvement of personal computers popularized the access to these pictures.
Very hard mathematics are hidden behind the Mandelbrot Set, but you can completely ignore them if you are interested only in the beauty of pictures. Many user-friendly programs now exist that allow you to get pictures comparable to the one above. If you never run any of them, click here for a first guided tour in the Mandelbrot Set. Don't worry, there will be no mathematics at all! By the way, you will also get a first contact with a basic property of the fractal pictures, the self-similarity, which we shall meet again many times in the following.
A similar walk can be done across the Markus-Lyapunov pictures if you have the relevant explorer software for this kind of fractals.
In fact, it seems that these bursts of colours, so fascinating for the layman, were a pleasant surprise for the pioneering mathematicians in the 1970-80s. In their mind, the Mandelbrot Set was mainly the figure on the right and they were specially analyzing the complicated structure of its outline. Benoît Mandelbrot himself recounts this adventure in his book "Les objets fractals" (Flammarion, Paris, 3rd edition, 1989). The Set then was a kind of "simplifying" key to understand and to classify the behaviour of other fractal objects. According to the Mandebrot's book, the main problem by this time was to define tools meant to simulate natural shapes such as mountain landscapes or rocky seashores. The pratical purpose was to get complex shapes with as few parameters as possible; look at the explanations in the page on the Von Koch curves. This page also gives another example of fractal, together with the simplest illustration of the concept of self-similarity. We shall use the Von Koch curves again in the following, to explain the notion of fractal "dimension".
The IFS fractals ("Iterated Function Systems") also play a great part in the saga of fractals. They have been developed by Michael Barnsley in a book with a provocative title, Fractals Everywhere (Academic Press, New York, 1988), also with a lot of pretty illustrations but make no mistake, the book is plain mathematics! In fact, Barnsley was specially successful in his program of "fractals everywhere" since he succeeded in demonstrating that any picture can be represented with a "small" number of these IFS fractals and he succeeded in embodying this theoretical result into the practical compression scheme commercially known as fractal compression. Beyond this practical application, these results have a clear philosophical importance fractals are everywhere, but they are of little interest in the artistic domain since no drawing software was ever written allowing to make arbitrary pictures through IFS fractals.
But what is a fractal?
Maybe the reader could feel entitled now to expect a clear, comprehensive, definition of a fractal. I shall refrain from giving any attempt. After all, one can read the whole book of Mandelbrot without coming across this definition!
So do not expect any accurate statement. After the preceding examples of curves, duly stamped "fractals" by the real scientists, we shall remain with the somewhat vague idea of very irregular curves (or surfaces), with a more or less obvious self-similarity, which means that special patterns can be seen in the object in several places and at every scale from the macroscopic to the infinitesimal. The frond of fern on the right is an obvious example since the whole fern can be seen in reduction in all its branches, in all the leaflets along these branches, and so on. In the above picture of the Mandelbrot Set, disks can be seen everywhere along the outline, carrying smaller disks, which themselves carry even smaller disks, etc... but it is also known that tiny replicas of the whole Set exist, that can be seen when the figure is strongly enough enlarged. This self-similarity was also clearly visible in the Von Koch curves. Of course, when random parameters are introduced in the fractal construction, the strict self-similarity is replaced with a simple resemblance.
Also, it seems that a fractal object must be characterized with a fractal dimension to be worthy of the name, but we shall not need this notion in the following except once, when we shall discuss about the label of "non-Euclidean" art.
Fractals for the Artist
For the "artist" we shall discuss the label later on there are two domains where fractals are of interest.
There is a boundless variety of fractal pictures.
The layman specially knows the pictures from the Mandelbrot Set, and, paradoxically, he sometimes reject them because he saw too many of them. I came across this reaction several times with image people and I explain it with the large spreading of free software to explore the Set; too many people tried and stopped too rapidly, obviously on nearly the same pictures, which would have been astounding ten years before, when The Beauty of Fractals was just published, but which now are hackeneyed. There is however much more to see even in the Mandelbrot Set, provided the exploration is persued on.
By way of examples, I add a few very simple pictures, obtained some years ago with an Amiga computer, and a few far more advanced pictures, signed by some of the best specialists on the Net. The difference in quality should be undisputable and this might awake a few more vocations...
Click here to get basic information on the main available programs, together with a few useful links.
Click here to get the references of the various books, articles and web links that I used to prepare these pages.
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