- Benoit Mandelbrot and Fractals in Art, Science and Technology
Benoit Mandelbrot, recently deceased French and American mathematician, was born in Warsaw in 1924. In 1936 he and his family fled the Nazis and in France joined his uncle Szolem Mandelbrojt, professor of mathematics at Collège de France. After attending Ecole Polytechnique, he studied linguistics and proved Zipf's law. He was an extremely original scientist who with the invention of fractals created a new concept with applications in numerous fields of science and art. His unconventional approach was well accepted when he came to IBM in 1958. He was also a professor at Yale University (1999–2005). In 1981 he published an article in Leonardo 1.
The concept of fractals unites and gives a solid mathematical framework, as Mandelbrot liked to emphasize, to ideas that artists, scientists and philosophers of art have sometimes intuited more or less clearly.
Let me start with this striking quotation from Eugene Delacroix's Journal in 1857:
Swedenborg asserts in his theory of nature that each of our organs is made up of similar parts, thus our lungs are made up of several minute lungs, our liver is made up of small livers.… I realized this a long time ago: I often said that each branch of a tree is a complete small tree, that fragments of rocks are similar to the big rock itself, that each particle of earth is similar to a big heap of earth.… A feather is made up of millions of small feathers 2.
This description by Delacroix corresponds to what would become clearly defined with fractals.
Perhaps most natural shapes are fractals, unlike shapes corresponding to human-made objects, and so fractals can be considered a way of freely escaping the artificial constraints of Euclidian shapes. I recently heard someone suggest that fractal shapes in animals are within their bodies. I would say that animals have interior fractal shapes, whereas trees for instance have exterior fractal shapes, because the former interact with the outside world with their interior shapes—to breathe with their lungs—whereas trees breathe with their exterior shapes, and an interaction surface is larger when it is fractal.
A simple plane curve appears straighter as one approaches it. Other more complicated curves look the same from very close or from afar—this self-similarity corresponds with fractal curves; an example can be seen in the coast of Brittany. Fractal curves appear very complex; their degree of complexity is defined by their fractal dimension, which is 1 for simple curves and between 1 and 2 for more complex curves.
Concerning art, René Huyghe in Formes et Forces 3 makes a distinction between art based on shapes describable by Euclidian geometry, such as are encountered in Classical art, and art such as Baroque art, based on the action of forces—for instance shapes that are encountered in waves, in vortices, etc. We could assert that both Classical and Baroque art can be described geometrically: the former by Euclidian geometry and the latter by fractal geometry.
In the sciences, ranging from physics at all scales to economics, fractals give new insights and a suitable framework for understanding chaos or phenomena that have been outside the mainstream of science due to their complexity.
Concerning technology, Benoit Mandelbrot was the first to be surprised when he saw weird and complex shapes appear on his computer screen as the result of an equation. This was to lead to the Mandelbrot set and is the origin of fractal art, a main branch of computer art.
Fractals can subconsciously suggest that each of us is a microcosm, an image of the whole world, hence their strong appeal.
Benoit Mandelbrot, whose ideas apply to so many domains, is one of the very few true scientists who is known and appreciated far beyond the scientific community.