The Mandelbrot set

(notes by Roberto Bigoni)

For each complex number c, we can build the sequence Eqn001.gif formed by the numbers that, given z0 = 0, are obtained by adding c to the square of the previous term. We shall say that c is the root of the sequence Mc.

The set of the roots c for which Mc does not diverge, that is, such that all the terms zi of Mc have finite absolute value, is called the Mandelbrot set (MS), named after the mathematician Benoit B. Mandelbrot.

It can be shown that if, while developing a sequence Mc, we get a term zi with absolute value ≥ 2, the sequence diverges, then the root c does not belong to MS. This finding implies that, in the complex plane, all of the numbers of MS lie inside a circle of radius 2 centered at the origin. We can call this circle as Mandelbrot Circle (MC).

Since, given a root c, it is physically impossible to calculate all the terms zi of Mc, if a root c is represented in the complex plane by a point inside MC, it is impossible to be certain that c belongs to MS. It can, however, be stated that it does not belongs to this set, if, while developing the sequence Mc, we get soon enough a term zi with absolute value ≥ 2.

So, in order to achieve a quite reliable graphical representation of MS, we can proceed in the following way.

The representations that are obtained in this way show interesting properties of MS compared to those of most known figures.

Mandelbrot called fractals the sets that have these properties.

The following JS application allows you to obtain graphic representations of MS. Clicking on one of them you can select a square and then clicking on the button + you can zoom its content.

last revision: 30/05/2016