Notion of Fractal Dimension

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You may wonder what help the notion of fractal dimension can be to disentangle the relationship between fractal pictures and Art. Indeed this page was included only to explain a few points in the comments about fractalist paintings, and "non-Euclidean" pictures.

In his book Les objets fractals Benoît Mandelbrot proposes several definitions for this quantity, which lead to the same result ("generally", he says). The simplest argument for the layman consists in starting from a seemingly simple problem, namely, what is the length of a fractal line? For instance, in concrete terms, what is the length of a rocky coast?

The surprising result is that this length depends on the rule with which it was measured: the smaller the rule, the larger the length. loupe In fact, we must first analyse the measuring process. If we consider the smooth curves of the classical geometry (top-left), under a magnifying glass, the more enlarged the curve is, the more flattened it looks and the more easily it can be followed with a correspondingly small rule. On the opposite, if we consider a fractal curve (bottom left), the enlargement does not smooth down the irregularities and the rule is always poorly matched to the curve, however large the magnification and however small the rule. We are obliged to accept this point. We go on by stating that we thus build a polygonal approximation of the curve (in blue on the left) and we content ourselves with measuring the length of this polygonal line.

Don't think that this polygonal approximation is a disaster. Imagine that the small picture above is continued over one meter and step back 10-20 m away, then look at it again : very likely, you will confuse the blue polygonal line and the black fractal curve, because its details will be blurred by the distance). In the same way, when a fractal line is "drawn" on the screen or by a printer, what is really drawn is rather a polygonal approximation where the elementary side corresponds to adjacent pixels.

Once the first polygonal approximation is obtained and its length is measured, the rule is halved and the whole process is done again, indefinitely. One thus gets a sequence of "lengths". In the case of classical curves, the result stabilizes fastly around a finite value, which is called the "length" of the curve. On the opposite, in the case of fractal curves, this polygonal length grows up to infinity, more precisely as L / L**D (L to the power D in the denominator – I am sorry, but it is no possible to always hide the mathematics!), where L is the side of the polygone and where D is a number characteristic of the analyzed fractal, lying between 1 and 2 for non self-crossing fractals. For those severely allergic to formulas, let us say that D characterizes the speed of the growing towards infinity; the larger D, the faster the growing.

This parameter can be easily derived for Von Koch curves, which are just defined from a sequence of such polygonal approximations. In the case sketched below, mathematicians find
D = log 2 / log (2 cos a / 2), which leads to the values displayed on the right.

courbes Von Koch
From left to right, the figure successively shows the elementary indentation scheme, the plot after 3 or 4 iterations and the "final" plot, together with the D value. The interesting point is as follows: one gets values close to 1 for nearly smooth fractals (top line), and one goes closer and closer to 2 as the fractal is more indented and fills more densely the surface over which it is plotted. In the limiting case a = 90°, the curve fills the triangle completely and one gets D = 2. The same value is also obtained for the Peano curve, which fills a square completely.

Within classical geometry, a curve is regarded as a one-dimensional space just because one can measure distances along it, so that any point on the curve can be defined with a single number, its abscissa, i.e. the distance from an arbitrary point taken as origin. In the same way, classical surfaces are two-dimensional spaces because any of their points can be defined with 2 coordinates. This integer number 1 or 2 is the Euclidean dimension of the curve or the surface. It is impossible to localize points in a fractal since it is impossible to measure lengths along it. A fractal curve drawn in a plane belongs to this plane, of dimension 2, but it cannot be attributed any Euclidean dimension at all, and specially not the dimension 1. However one has the number D, a real number close to 1 when the fractal is nearly smooth, which increases when the fractal becomes more indented, and which reaches the value 2 when the fractal fills a surface in the plane, i.e. when the fractal "curve" looks like a filled surface. To some extent, fractal curves allow us to carry out a kind of transition between the classical curves, of dimension 1, and the surfaces, of dimension 2, while the number D just goes from 1 to 2. Because of this analogy, this number is called the fractal dimension of the fractal curve.

The curious reader will find quite more involved pages in the book of Benoît Mandelbrot, specially about the different ways to define the fractal dimensions. If he looks for an easier reading, though more accurate than this page, with some mathematics but not too much, he can link to the website of Glenn Elert.

 

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[ Return to home page ]    [ Art and Fractals Contents ]

Discovering the fractal world:
Introduction - Mandelbrot Exploration - Lyapounov Exploration - Von Koch Curves - IFS Fractals - Fractal Dimension - Mandelbrot Relatives - Finest Fractal Pictures - Software - Biblio and Links

Fractals and mysticism:
Introduction - The Mysticism of Infinite - Non-Euclidean Art?

A new Art?
Introduction - Fascination of Fractals - Fractals and Photography - Definitions of Art - The Colour Choice - Other Colour Choices - Fractalists Painters - Compositions with Mandelbrot - Put a pretty girl - Algorithmic Art - Beyond the "Fractal" Art