Maths & art in fractals
The form of a cloud or the population dynamics of locusts threatening to reach plague proportions are both examples of complex systems at the heart of chaos theory. Reducing such systems to mathematics has been the holy grail of chaos theorists for decades. Fractal geometry, pioneered by Benoit Mandelbrot, was a paradigm leap but it has had some shortcomings.
MSI mathematician Professory Michael Barnsley has pushed chaos theory to a new level with superfractals theory – a description of complex systems which is a closer approximation to the real world than is afforded by conventional fractal geometry.
Chaotic systems are driven by apparently simple deterministic laws and, theoretically, can be reduced to mathematical equations. “At each stage, what happens next is determined by what happened before,” says Barnsley, one of the world’s leading theoreticians, who is also gaining recognition for his works of art generated from the mathematics of chaos.
However, they are sensitively dependent on the starting point – the initial conditions.
“You can’t specify the initial conditions precisely enough to be able to predict what’s going to happen,” says Barnsley.
The upshot is that chaotic systems appear to be random, so Barnsley has introduced statistical theory into conventional chaos theory. The work increases the theory’s range of applications which range from climate modelling, through biology, to economics.
He has published two authoritative books on the subject, Fractals Everywhere and Superfractals: Patterns of Nature. Barnsley made his debut on the art scene with an exhibition at the Huw Davies Gallery in Canberra. He says mathematics and nature give him “a sense of the pristine, the perfect”.