Mandelbrot set

The Mandelbrot set is a two-dimensional set that is defined in the complex plane as the complex numbers ${\displaystyle c}$ for which the function ${\displaystyle f_{c}(z)=z^{2}+c}$ does not diverge to infinity when iterated starting at ${\displaystyle z=0}$, i.e., for which the sequence ${\displaystyle f_{c}(0)}$, ${\displaystyle f_{c}(f_{c}(0))}$, etc., remains bounded in absolute value.

It is a famous example of a fractal in mathematics and is named after Benoît Mandelbrot, a Polish-French-American mathematician. The Mandelbrot set is important for chaos theory, as points in its equation can be very chaotic. The boundary of the set shows a self-similarity, which is perfect, but because of the minute detail, it looks like it evens out.

• Moreover, the complete details of the complication of the structure of Mandelbrot's set cannot really be fully comprehended by anyone of us, nor can it be fully revealed by any computer. It would seem that this structure is not just part of our minds, but it has a reality of its own. ... The computer is being used in essentially the same way that the experimental physicist uses a piece of experimental apparatus to explore the structure of the physical world. The Mandelbrot set is not an invention of the human mind: it was a discovery. Like Mount Everest, the Mandelbrot set is just there!
  • Roger Penrose. The Emperor's New Mind, 1989 Penguin edition, ISBN 0140145346. Ch. 3, Mathematics and reality, p. 95.