fractals

Fractals have been called the geometry of chaos as well as the geometry of nature.
Here nature is "in the details" but the details occur at all scales.

The classic example used by Benoit Mandelbrot to introduce the fractal geometry of Nature is the question: How long is the coastline of Britain? Imagine measuring the coastline with a meter-long stick. This act of measurement would make an approximation of the coastline consisting of a finite number of meter-long straight lines and would give us one result. If we went back with a 50cm long stick and remeasured, we would get a total more than twice the first. As our unit of measure became smaller, the total length would increase without bounds. Our answer to the question would be that the length is scale dependent and increases without bound as the scalar unit becomes smaller. However, the "roughness" of the coastline seems to remain constant at every scale. Fractals have the same degree of irregularity at all scales of measurement. This roughness can be calculated as the log-log plot between the measured length and the reciprocal of the measuring unit, between "count" and "step" (see fractal dimension)

Sierpinski gasket and M.C. Escher's studies of patterns on the twelfth century pulpit of the Ravello Cathedral.


Geometric fractals are generally obtained by a construction process consisting of recursive functions, and they usually exhibit scaling self-similarity. (note: scaling describes a relation between two variables whose logarithms have a linear relationship) For example, to generate a Cantor set, remove the middle third of a line of finite length. The Cantor dust is the result of recursive iterations of this process. Geometric figures such as the Sierpinski gasket display linear self-similarity, in that any part of the object is exactly like the whole. But the most important fractals, such as the Mandelbrot set, are not this rigidly self-similar. Nonetheless, pattern and form in fractals, because of self-similarity, are largely independent of scale. According to Benoit Mandelbrot fractal forms have dimensions that are not integers. Their dimensional ambiguities arise from some of their unusual aspects. For instance, the Peano curve is a line that can be shown to fill a plane. We would usually assign 1 dimension to a line and two dimensions to a plane, but if this line can fill a plane whose area is calculated in two dimensions, it would seem that it cannot be a one-dimensional line (even if infinitely long). In describing the fractal as a way of translating the strange data of smooth space into a striated one, Deleuze and Guattari emphasize the way in which the space and that which occupies it become identified, rather that one containing the other.

Leonardo da Vince Deluge

There are limitation to which fractals can be used as models for natural shapes. Natural objects that display self-similarity, such as a cauliflower or a fern, do so only through a limited number of scales. At some point the fractal process stops. "Big whorls have little whorls, which feed on their velocity, And little whorls have lesser whorls, and so on to viscosity."
Broccoli Romanesco: a cross between cauliflower and broccoli. Overall view and detail

The geometric fractal is really a limit structure. Any approximation of it does not exhibit the perfect properties of the fractal such as self-similarity. The degree of resolution of the graphic device in the computer is analagous to this limit. In the study of clouds, some windblown sands, impact craters on Mars, river patterns in relation to drainage basin areas, porous media, and caves in limestone, it appears that when the driving process for the phenomenon can be scaled and it operates in a single medium, the products of the process may exhibit strong fractal properties. These properties are often better modelled by stochastic fractals, in which scaling and self-similarity are not exact patterns but statistical ones, such as Brownian fractal surfaces.