(notes by Roberto Bigoni)

For each complex number *c*, we can build the sequence
formed by the numbers that, given *z _{0} = 0*, are obtained by adding

The set of the roots *c* for which *M _{c}* does not diverge,
that is, such that all the terms

It can be shown that if, while developing a sequence *M _{c}*, we get a term

Since, given a root *c*, it is physically impossible to calculate all the terms *z _{i}* of

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

- For each root
*c*we call*duration*the index of the first of the terms*z*of_{i}*M*with absolute value ≥ 2._{c} - We establish the minimum duration that a root
*c*should have to not be excluded from the representation of*MS*. - In the complex plane, inside the square with vertices 2 + 2i, -2 + 2i, -2-2i, 2-2i, we choose a suitably dense matrix of evenly spaced points and we represent each of these points with a pixel of a graphic window.
- We calculate the duration of each of these points. If the duration is less than the established minimum, the point does not belong to
*MS*and is represented by a black pixel; otherwise most probably the point belongs to*MS*and is represented by a white pixel. - The representation is more reliable the higher the minimum predetermined duration.
- We can get more phantasmagoric graphics by assigning different colors to each point depending on his duration, reserving the white to the probable elements of
*MS*.

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

- The frontier of
*MS*is not a continuous line, such as those studied by classical geometry (polygons, circles, conic sections, algebraic, trigonometric, exponential and logarithmic curves, etc.). In fact, it has a non-integer dimension: while in classical geometry lines have dimension 1, surface have dimension2, volumes have dimension 3, etc. this line has a dimensione between 1 and 2. *MS*has an internal homothety: whenever you zoom in on its details you will find figures of the same shape; inside*MS*there are many tiny*MS*, inside each of which other more tiny MS, and so on without end.

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