Trees, spirals (logarithmic spirals, coastlines, mountains, galaxies, clouds, rivers, weather pattern, brains, lungs, turbulence, heartbeats. What do all these natural wonders have in common? They are dynamic systems that can be best explained by fractal geometry. If we look closer at these dynamic systems, we can distinguish two groups: a) Stable systems : trees, shells, mountains, blood vessels b) Chaotic systems : clouds, turbulent flows Recognizing and identifying the transition from stability to chaos and vice versa has always been intellectually and experimentally challenging. The recent contribution of fractal geometry has brought some insight into this phenomenon.

In our presentation we will, briefly, try to describe how "Science" (mainly an analytic approach) and "Art" (mainly an intuitive approach) experience this natural fractal world.

 

1. THE SCIENTIFIC POINT OF VIEW by BERNARD METAIS

A) Stable systems

 

A fractal object can be defined as a geometric figure in which an identical motif repeats itself on an ever diminishing (or increasing) scale. A typical & simple example of a fractal figure is this logarithmic spiral. Magnifying (or rotating) a logarithmic spiral always reproduces the same self-similar figure. Because of its fractal quality this spiral is common in nature (see the ammonite of the previous figure).

Because a spider constructs its web according to the law of self-similarity of a logarithmic spiral, it can easily repair any broken segment (A<B) of the web by keeping the same angle? between the radius and the tangent to the web during the spinning.

Self-similar geometrical figures of great complexity can be obtained by repeating simple geometric transformations. Around 1880, Georg Cantor created one, " the Cantor's comb", by removing the middle third of a linear segment, then repeating the same operation on the remaining two segments and so onŠ.Another one, "the snowflake of Koch", can be created by adding a scaled down (by 1/3) equilateral triangle to each side of a equilateral triangle and so on. These "monster figures" with ever increasing perimeters scared away some of the greatest mathematicians, such as Henri Poincaré (early 1900), who, otherwise, contributed greatly to the field of systems instability, by being the first one to demonstrate that an enclosed non-linear system could show instability.