The Fractal Universe, Benoit Mandelbrot
April 5th, 2005
Mandelbrot B., The Fractal Universe , from The Origins of Creativity , ed. Pfenninger K. & Shubik V., pp. 191-212, Oxford University Press, 2001
Mandelbrot starts his article with a quick outline of his education which, for a variety of small accidents, was highly visual in nature. He goes on to explain that during his “killer” entrance exams he was unable to answer many questions involving language and symbolic manipulation, but easily solved some of the hardest questions, “which no human can find algebraically in three hours under exam conditions,” by reasoning directly from the visual and sensory images that came to mind as he was posed the questions.
He goes on to discuss the shift of mathematical language from one of shapes and illustrations to one of written languages and formulas. One French high school mathematics text argued that “the artistic and sensual character of pictures would delude the reader,” so the text contains no illustrations at all.
Mandelbrot started to form his theories and develop his initial formulas when looking at the variations in the stock market, and other markets in economic history – invariance. He also describes the theories of self-similarity, such as in a cauliflower where the individual florets look the same as the whole cauliflower – apparently this continues down to microscopic proportions.
In Euclidian Geometry simple shapes have simple formula and complex shapes have complex formula. Mandelbrot introduces the new world of Fractal Geometry where simple formula describe very complex shapes. These new formula worked well to describe economic fluctuations, descriptions of coastlines and turbulence – but no one understood Mandelbrot’s theories and his work was ignored.
Mandelbrot now skips ahead and introduces some of the images that relate to his discoveries and formulas. Of particular interest is the realistic looks mountain shapes created by “piling” pyramids according to certain rules. The rendering of these “landscapes that never were” has no understanding or relationship to tectonic development, or the forces that shape the earths crust, or those that produce changes in the crust, yet the images are very realistic. In turn models and formulas that have been developed to represent the real forces often produce images which look nothing like the mountains they supposedly represent.
Mandelbrot’s formulae generate an extraordinary range of images by simply changing the single seed value of the formula. “Is it the creativity of mathematics, or the creativity of a formula? How can a one-line mathematical formula generate shapes of such far-ranging variety and complexity?”
The use of computer graphics gave Mandelbrot’s formulae a face and a visual proof for what they were. Mandelbrot and his collaborators focused on keeping the results pure with no artful interference – “we agreed not to involve our feelings of taste or beauty, only reveal the formula.”
Mandlebrot continues with a discussion of whether or not the images generated by simple formula can be considered art or even high-art. Comparing his processes and results with artists and photographers from history. He also makes the point that many of his images sell for much more than a lot of art sold in traditional galleries – which is often important in evaluating art.
Finally, Mandelbrot introduces Julia sets, and the Mandelbrot set – probably the most famous and well known output of his theories and work – before concluding with a story in which we move full circle back towards a “visual rhetoric”. Within this new visual rhetoric he asks of the new images created from his simple formula, “Are these images art? Perhaps no more than simple circles are art by themselves.“