If ‘Spot On’ is about the big in the small and vice versa then sooner or later Mandelbrot and fractals had to be mentioned. Unfortunately it is sooner than expected as Benoit Mandelbrot died 14 October.

The obituary below was written by his nephew. Also watch Benoit Mandelbrot: Fractals and the art of roughness.

FRACTALS IN ART, SCIENCE AND TECHNOLOGY

Benoit Mandelbrot, the French and American mathematician who passed away two nights ago, was born in Warsaw. He and his family fled away from Hitler to France in 1936 where he was greeted by his uncle, the mathematician Szolem Mandelbrojt, professor at Collège de France. After having been a student at Ecole Polytechnique he did linguistics and proved the Zipf law.

He was an extremely original scientist who with the invention of fractals created a new branch of mathematics that has applications in numerous fields of science and art. His unconventional approach was fully encouraged when he came to IBM. He was Sterling Professor Emeritus of mathematical sciences at Yale University and IBM fellow emeritus at IBM T.J. Watson Research Centre.

The concept of fractals, as Benoit Mandelbrot, liked to emphasize, unites and gives a solid mathematical framework to ideas, which artists, scientists and philosophers of art have often felt more or less clearly.

Let me start with this very striking quotation from Eugene Delacroix’s Journal in 1857 (1):

“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…. Without being such a great observer of nature I realized this 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. I am convinced that we could find many such similarities. A feather is made up of million of small feathers…”.

This description by Delacroix corresponds to what will become clearly defined in the concept of fractals.

Similarly René Huyghe in his book “Formes et Forces”(2) (Shapes and Forces) makes a distinction between art based on shapes, actually shapes which can be described by Euclidian geometry such as are encountered in Classical art, and art based on the action of forces, for instance shapes which are encountered in waves, in tourbillions etc; these shapes correspond to Baroque art. These shapes also appear in several of Leonardo da Vinci’s drawings. With the discovery or invention of the concept of fractals (about the same year Huyghe’s book was published) we could now assert that both Classical and

Baroque art can be described geometrically, the first one by Euclidian geometry, the second one by fractal geometry.

In sciences as Benoit Mandelbrot mentioned, both the mathematician Henri Poincaré, and physicist Jean Perrin pointed out the fact that many fundamental phenomena cannot be given a proper causal description because of their complexity. Here again fractals give an adequate framework to these phenomena, just as it is the appropriate framework for describing chaos.

Fractals give a precise mathematical framework to complex phenomena, and in particular to the description of complex curves. A simple usual curve when looked at one point from very close, can be identified to its tangent, in other words to a straight line. Other more complicated curves look the same from very close or from afar, this is called self-similarity, and it corresponds to fractal curves, an example being the coast of Brittany. These curves are very complex looking and their degree of complexity is defined by their fractal dimension (or Hausdorff dimension): A usual plane curve has fractal dimension 1, and as it becomes more and more complex, its fractal dimension, which isn’t necessarily a whole number, increases until it becomes 2.

With technology, fractal shapes surprisingly sometimes appear on the screen of computers. Benoit Mandelbrot was the first one to be surprised when he saw the shapes of what was to become the Mandelbrot set, appear as resulting from an equation. This is the origin of fractal art that has become a main branch of computer art.

Thus fractals have two different domains in art: traditional art which can be described by fractals, as I mentioned in René Huyghe’s book, and art which is made to be fractal, generally by using computers.

To conclude I would suggest that the universal appeal of fractals might correspond to the fact that it can subconsciously imply that the small part of the world that we are, is an image of the whole world, in other words that we are a microcosm.

Jacques Mandelbrojt, 16th of October 2010

(1) Delacroix E. Journal, Paris, Plon 1986

(2) Huyghe R. Formes et Forces, Paris, Flammarion, 1971

Source: YASMIN, Arts Science Mediterranean International Network