FRACTAL LOGIC

A New Paradigm

'Science did not come into this world of its own accord ... it was
proceeded by a number of failures.' (Subversion of the subject and the
Dialectic of Desire,
p.2, 1960)

Lacan's statement suggests that science, rather than impassively 'discovered' by the human subject, was actively 'invented' by it through intellectual rigour; the fractal was not discovered by cognitive processes, even though present in such common structures as trees and clouds. It was invented when Mathematician Benoit Mandelbro programmed a nonlinear equation into a mainframe computer. Ironically, the computer generated an image - the Mandelbro Set (Fig. 1) - which has enabled us to rethink our conception of natural phenomena. Without the calculating power of the computer, needed to reiterate the equation millions of times, the mathematics of fractal complexity would remain elusive.

The Mandelbro set,                                                                            
plotted through the equation:
xn+1=f(Xn)=xn2+c Fig. 1 The Mandelbro set,
plotted through the equation:
xn+1=f(Xn)=xn2+c

There are two major characteristics of a Mandelbro Set. Firstly, it occupies a finite space (you can ring the Set with a pen). Secondly, and paradoxically, its border is infinite. This is because Mandelbro's equation dealt in complex numbers rather than real numbers. In doing so, numbers that decide the nature of the border were giving conflicting commands, and were unable to decide a value on the graph, for example 0, 0.8322546. As the equation was reprocessed by the computer using a different input number (parameter) these border numbers too hovered infinitely never repeating themselves, but on a different scale. If you look at the Mandelbro Set's border through successively more powerful magnifications, each image would be similar to the previous, but not identical (Fig. 2). This is called self-similarity. The result of these 'border skirmishes' is a circumference of infinite complexity. Moreover a concept that does not conform to human rationale, although exercised by mathematical rigour. Self-similarity can be found in nature, from a single leaf (Fig. 3) to a mountain range. Indeed, unlike the 'straight line', it is 'the geometry of nature' (Chaos, Chapter title, 1988).

Fig. 2 Self-similarity of Mandelbro Set

For a dynamic example of this self-similarity
see An Active Mandelbro Set
 Self-similarity of Mandelbro Set.
 Self-similarity of leaf veins                   
carried through different scales. Fig. 3 Self-similarity of leaf veins
carried through different scales.

The new paradigm detaches itself from Euclidean geometry. The latter asserts a 'universe characterised by triangles, circles and other geometric figures' (Galileo, 1623). Unlike nonlinear geometry, it is unconcerned with scale; to magnify part of a circle will not yield new information about the object. But in Mandelbro's words, '[Euclidean] geometry is unable to describe a cloud, a mountain or a tree' (Fractals: Patterns of Chaos, 1992). Mathematicians have increasingly 'chosen to flee from nature' and in doing so have devised theories that are 'unrelated to anything we feel or see'. In agreement, I would argue that the discourse of Euclidean geometry is itself a simulacrum. Unable to signify what can be seen in nature, it only refers within itself in an endless (circular) chain of signification.

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