Mandelbrot Set - Formal Definition

Formal Definition

The Mandelbrot set is defined by a family of complex quadratic polynomials

given by

where is a complex parameter. For each, one considers the behavior of the sequence

obtained by iterating starting at critical point, which either escapes to infinity or stays within a disk of some finite radius. The Mandelbrot set is defined as the set of all points such that the above sequence does not escape to infinity.

More formally, if denotes the nth iterate of (i.e. composed with itself n times), the Mandelbrot set is the subset of the complex plane given by

As explained below, it is in fact possible to simplify this definition by taking .

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 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. See the section on computer drawings below for more details.

The Mandelbrot set can also be defined as the connectedness locus of the family of polynomials . That is, it is the subset of the complex plane consisting of those parameters for which the Julia set of is connected.