The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and for being a complicated structure arising from a simple definition, and it is one of the most well-known examples of mathematical visualization. Benoît Mandelbrot and others worked hard to communicate this area of mathematics to the public.
The Mandelbrot set shows more intricate detail the closer one looks or magnifies the image, usually called "zooming in". The following example of an image sequence zooming to a selected c value gives an impression of the infinite richness of different geometrical structures, and explains some of their typical rules.
The Mandelbrot set is self-similar under magnification in the neighborhoods of the certain points (Misiurewicz and Feigenbaum points). The Mandelbrot set in general is not strictly self-similar, but it is quasi-self-similar, as only small slightly different versions of itself can be found at arbitrarily small scales.
The Mandelbrot set is a compact set, contained in the closed disk of radius 2 around the origin. The area of the Mandelbrot set is estimated to be 1.50659177 ± 0.00000008.
Mathematically the Mandelbrot set can be defined as the set of complex values of c for which the orbit of 0 under iteration of the complex quadratic polynomial
remains bounded. That is, a complex number, c, is in the Mandelbrot set if, when starting with and applying the iteration repeatedly, the absolute value of never exceeds a certain number (that number depends on c) however large n gets.
For example, letting c = 1 gives the sequence 0, 1, 2, 5, 26, etc. which tends to infinity. As this sequence is unbounded, 1 is not an element of the Mandelbrot set.
On the other hand, c = i (where i is i² = -1) gives the sequence 0, i, (−1 + i), −i, (−1 + i), −i…, which is bounded and so i belongs to the Mandelbrot set.
When computed and graphed on the complex plane, the Mandelbrot set is seen to have a boundary which does not simplify at any given magnification: we can always see even more details after further zooming in (hence the boundary is a fractal).
Mathematically, the Mandelbrot set is just a set of complex numbers. A given complex number, c, either belongs to M or it does not. A picture of the Mandelbrot set can be made by colouring all the points c which belong to M black, and all other points white. The more colourful pictures usually seen are generated by colouring points not in the set according to how quickly or slowly the sequence diverges to infinity. As with the boundary, these pictures also does not simplify at any given magnification. In simple implemented viewers, pictures and boundaries will simplify when the magnification reaches the limits of the floating point format of the particular machine (arbitrary precision arithmetic must be used for deep views).