I have always been fascinated by fractals. But most of all I like the Symmetric Icon fractals.

enter image description here

There is a nice book about these fractals, written by Michael Field, called Symmetry in Chaos.

I'm trying to understand the formulas given in the book, however I am not very familiar with iterative formulas.

The formula for fractals such as the one shown above, is given as:

$$F(z) = \{\lambda + \alpha z \bar{z} + \beta\ \text{Re}(z^n) + \delta\ \text{Re} \left( [z/|z|]^{np} \right)\}z + \gamma \bar{z}^{n-1} $$

Where $n$ gives the fractal a $D_n$ symmetry.
$\alpha$, $\beta$, $\delta$ and $\gamma$ $\in \mathbb{R}$
$n$ and $p$ $\in \mathbb{N}$

A fractal is generated by creating a pixelgrid. Then calculating the location $F(z)$ and increasing the corresponding pixel value by 1 (where the center of the picture is the center of the complex plane).

A while ago I wrote a piece of software that generates these fractals. In the software all the parameters can be changed, but apart from symmetry it doesn't seem possible to predict what is going to happen when changing parameters.

Can someone explain me a bit what the effect of the different parameters is?

I'm also very curious if it is possible to see if $F(z)$ stays near the center or explodes for a given set of parameters.

Some more information about the parameter $p$.
This is what happens to the fractal when changing the parameter (from left to right $p={0,1,2,3}$).


  • 1
    $\begingroup$ Others may find this obvious, but could you specify from which sets the parameters are chosen? Is it that for each fractal we have specified constants $\lambda,\alpha,\beta,\delta,\gamma\in\mathbb C$ and $n\in\mathbb Z$ and then $p\in\mathbb N$ some prime? And then we produce the fractal as the image $F(\mathbb C)\subset\mathbb C$? $\endgroup$ – String Jan 27 '15 at 9:19
  • $\begingroup$ @String, I added the sets for each parameter. $\endgroup$ – Douwe66 Jan 27 '15 at 9:31
  • $\begingroup$ Thank you. So you do not mean that $p$ should be prime as I understand it, right? $\endgroup$ – String Jan 27 '15 at 9:33
  • $\begingroup$ According to the book, $P$ is an integer and $\geq 0$. $\endgroup$ – Douwe66 Jan 27 '15 at 9:39

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