Today we can generate beautiful complex self-similar fractal figures with the help of computers. They can, for instance, simulate the growth of trees or a cauliflowerŠ

B) From simplicity to complexity: Chaotic systems

The study of chaotic systems started in the 1840s, when Pierre François Verhulst refined the negative feedback model of growth of the population of a species. If Xn is the number of flies in the year n and C is the reproduction rate of the flies during the year n, the number of flies in the year (n+1) will be Xn+1=CXn(1-Xn). The factor (1-Xn) takes into account the realistic fact that the number of flies cannot grow indefinitely. This equation has been extensively studied, especially since the advent of the computer:

The growth of Xn depends on its initial value X0 and on its growth rate C. It was found that: · If C is smaller than 3.0 (point 2), the growth is regular · When C reaches 3.0 there is a bifurcation point (pt 2): depending on the starting point, Xn can take either one of two values located on two different curves. · When C reaches 3.45 (pt 3) there is again a doubling of the possible values for Xn (4 cycles) · When C reaches 3.54, there are again bifurcations (8 cycles) · When C reaches 4.0, there is complete chaos A small change in the value of C can tremendously impact the behavior of X. This is called the "butterfly effect" from the article by Edward Lorenz "Predictability: does the flap of a butterfly's wings in Brazil set off a tornado in Texas" (1979)

In 1977, Mitchell Feigenbaum found out that the value of C atconsecutive bifurcations points decreases by about the same factor F = 4.669Š. This is a universal number to be found each time there is a repeated period doubling (i.e. phase transition). It has been said that: "Seemingly pure mathematical research, computer experiments, and physical reality are in fact intimately related".

Already in the 1960s, Benoit Mandelbrot was evaluating the fractal dimension of coastlines (the value of the a coastline depends on the length of the yardstick) and was studying the fractal figures mentioned above, especially the ones from the Julia set, which corresponds to a simple geometric transformation. The set had been developed by Gaston Julia in early 1900, but could not be thoroughly analyzed until Benoit Mandelbrot was able to use the new computer techniques of the 1960s.