From Jacques Mandelbrojt via Roger Malina and the YASMIN list:


FRACTALS IN ART, SCIENCE AND TECHNOLOGY

Benoit Mandelbrot 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 which 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 Center.

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 gives 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 become more and more complex, its fractal dimension, which isn’t
necessarily a whole number, increases until it become 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 which 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


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