Last week I did attend to a “Flower Festival”, in the city of Joinville. I had the opportunity to admire several of the most beautiful flowers, orchids, and decoration plants. After few hours of contemplation, I realized I never had thought on flowers to be fractals.

Fractals attract lots of interest, especially because of their structures and combinations made by simple equations. But what exactly makes something a fractal?

A fractal is defined as an object that displays self-similarity on all scales. In fact, it doesn’t need to have exactly the same structure at all scales, but the pattern must be visible or recognizable any how.

## Mandelbrot

The structure or the equation that defines a fractal is most of the time very simple. For instance, the formula for the famous Mandelbrot is $z_{n+1}=z_n^2+c$.

### How can we start?

We start by plugging in a constant value for c, z can start at zero. After one iteration, the equation gives us a new value for z; then we plug this back into the equation at old z and iterate it again, it can proceed infinitely.

As a very simple example, let’s start this with c = 1.

$z_{1} = z_{02} + c= 0 + 1 = 1$ $z_{2} = z_{12} + c = 1 + 1 = 2$ $z_{3} = z_{22} + c = 4 + 1 = 5$

Graphing these results against n would create an upward parabolic curve because the numbers increase exponentially (to infinity). But if we start with c = -1 for instance, z will behave completely different. That is, it will oscillate between two fixed points as:

$z_{1} = z_{02} + c= 0 + -1 = -1$ $z_{2} = z_{12} + c = 1 + (-1) = 0$ $z_{3} = z_{22} + c = 0 + (- 1) = -1$ $z_{4} = z_{32} + c = 1 + (- 1) = 0$

And this movement back and forth will continue forever as we can imagine. I figured out, that the Mandelbrot set is made up of all the values for z that stay finite, thus a solution such as the first example for c = 1 is not valid and will be thrown out because z in those cases goes to infinity and Mandelbrot lives in a finite world. The following is an example of such set.

### The script for this set:

#reproducible