Benoit Mandelbrot on his Set

Benoit Mandelbrot relates some thoughts on the eponymous Mandelbrot Set he investigated and popularised, in a foot-note (note # 146, p 293) to his commentary The MisBehaviour of Markets :

The Mandelbrot set starts with an old problem... The problem concerns so-called iterated functions, a kind of mathematical feedback loop that keeps operating on its own outputs again and again.

[ Definition of the Fractal Generator - An Iterated Function ]

z_1 = (z_0)^2 + c

where

z_0 is the starting value of the process
c is a constant
z_1 is the first output

Then repeat the operation,

z_2 = (z_1)^2 + c

and

z_3 = (z_2)^2 + c

If you keep doing this forever with starting numbers c like 3 or 4, the sequence (irrespective of z_0) will soar off into infinity. But if you say z and c are complex numbers, with the imaginary i [= square root of -1] in them, then the story gets more interesting... Sometimes the series will veer off into infinity, but sometimes it will not. And the precise pattern is exquisitely intricate.

With the Mandelbrot set, you start off by setting z_0 equal to 0, and then see what happens to the sequence when you try different values of c. If the sequence does not run off to infinity, then [the point] c is said to be in the Mandelbrot set. If it does, then c is not in the set.

Black and white illustrations of the set typically assign a computer-screen pixel to every value of c being tested, then paint it black if the pixel is inside the Mandelbrot set, and a variety of other colours if it is not. Different colours are often used to denote how quickly the series soars to infinity.

The surprising thing is that as you look at smaller and smaller scales - say, zoom in on values of c in a tenth of the screen, rather than the whole screen - you find the pattern of what is in the set and what is not becomes far more complicated than it first appeared. Zoom again, and yet more fine detail emerges. You can do this forever, and at each stage get an entirely different picture...

[Chaotic Nature]

The Mandelbrot set belongs to both fractal geometry and chaos theory. A chaotic system, far from being disorganised or non-organised, starts with one particular point, and cranks it through a repeating process, the outcome is unpredictable if you do not know the process - and it depends heavily on the starting point...

The basic idea is that if you stand a pencil on its point, and let it fall through the force of gravity, exactly where it lands depends on where it began, and whether it was leaning infinitesimally on one direction or another.