The self-similarity and the vertigo of the infinite
A real mysticism surrounded the birth of fractal pictures, mainly
due to their self-similar features which invite the reader to a
dive to the infinite. The concept of the infinite was always
fascinating. For the layman, such an invitation materialized
in a picture
is far more impressive than an abstract philosophical discussion.
In a nutshell, there are patterns in these pictures which appear
here and there, at every scale. In the classical view of the
Mandelbrot set on the left, you can see a kind of bean in the
center (a cardioid, if you prefer), which carries quite a variety
of disks along its outline, and you also see or you can guess
that all these disks carry other smaller disks, which themselves...
And as you know, when a part of the main set outline is zoomed in,
one can see
of the whole set, together with the central bean.
And one can enlarge again and again, endlessly...
In the same spirit, but more prosaically, there is a well known French cheese, "La vache qui rit", the label of which displays a smiling cow with the same cheese box as earrings; on a smaller scale, these earrings display the same smiling cow with the same earrings; on an even smaller scale, these small earrings...and so on.
details from the original Benjamin Rabier advertising
Theoretically, this sequence of successive zoomings which always exhibit the same (or similar) patterns can go on endlessly, which leads to the first key word, infinity. So far, we only mentioned the infinitesimal, but let us be a bit imaginative, revert the sequence, forgive the initial point, and we are sailing towards the infinitely great. The gods are not far, maybe God Himself? On the other hand, since the initial Set is wholly replicated in each of these smaller and smaller enlargements, would not you agree that this is a striking illustration of everything is in everything, the great mystical Eastern formula?
Let us keep our sanity. Fractal structures, from the infinitesimal to the infinitely great, are mathematical abstractions, exactly as the ideal straight line with its zero width is an abstraction. Sometimes in real life objects exist which can be approximately depicted with straight lines, along some distance, provided their width is ignored. The act of deciding which features are negligible and which must be taken into account is fundamental when one attempts to represent the world. Other objects in real life can be conveniently depicted with fractals, to some extent. In his celebrated book Les objets fractals, Benoît Mandelbrot quotes the example of a rocky coast on a map, which displays the same general pattern on the scale of the whole country or of a small town. But this would not be true on the scale of a beach or a pebble! An approximation with a fractal is valid over a limited space and over a limited range of scales, neither too large nor too small. Since Mandelbrot and his rocky coast, other physical systems were proposed fractal representations, such as liquefying gases or polymerized aggregates, or more simply particular plants such as the fern on the right.
When ones makes a drawing of a fractal, like for the fern on the right, one gets a material thing which is not this ideal fractal, exactly as a pencil stroke is not the straight line of mathematicians. If the drawing is sharply printed, the famous self-similar details can be seen through a magnifying glass. But through a microscope, nothing can be seen other than the paper fibers or the screen pixels. Nevertheless, everybody will perceive the fractal through the drawing; if necessary, the drawing will be completed by the mind for the details which cannot be printed. Actually, one is really invited to do this. When similar patterns can be seen on different scales to the naked eye, large ones, medium ones, small ones, and other hardly visible, one is tempted to guess that others exist, too small to be noticeable but which must be there. And there is no reason to stop.
As a conclusion, nobody has to fall into mystical delirium, but the temptation of vertigo does exist.
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