(from mail exchange with Ken Shirriff)

(excerpt from a mail by Ken Shirriff)

>The easiest way to get symmetry for escape time Julia set fractals is to
>make your formula a function of z^n (for n-way symmetry).  I.e., you
>put pixel_value^3 into your formula instead of pixel_value and you
>automatically get 3-way symmetry.
>If you want to generate symmetric Mandelbrot fractals, i.e. z(0)=0 and the
>constant depends on the pixel value, it gets a bit trickier.  You can cube
>the pixel_value.  If you want to put constraints on the formula it is
>trickier; z^n is n-1 symmetric.  So for 3-way symmetry you could have a
>formula like:
>z = a*z^13+b*z^10+c*z^7+d*z^4+pixel_value
>because 13, 10, 7, and 4 are one more than multiples of 3.

Hi Ken,

Your mail gave me something to think about, for this reason I am answering after a long time. First of all I would like to be sure that I have understood your suggestion: you are speaking about complex maps. If this is the case,then I agree that it is very easy to get simmetry. Pickover has done a lot of work in this field with its biomorphs. Why it is so easy to have simmetry with complex maps? My feeling (I'm no mathematician) is that it depends from the fact that complex multiplication corresponds to a rotation in the plane.

Now what about 2D dynamical systems represented by the formula:

 x = f(x,y)
 y = g(x,y)

This is a more general case since it includes complex maps as a subset.

I feel more confortable speaking about pixels in a (viewing) window instead of points in a plane:let's say 1000*1000 pixels in the square

 -1 < x,y < +1

In which case we will have a set of colored pixels forming a symmetric pattern?

I will consider separately three subset of the 1000*1000 pixels/points

1 - The first one contains all pixels/points which are part of a limit set. We can have one or more limit sets. These sets can be or not chaotic attractors.
2 - The second one is the subset of pixels attracted by the limit sets in 1.
3 - The third one is the subset of pixels attracted by limit sets wich are outside the viewing window.

Something interesting is the following: the fact that we choose a square (so a highly symmetric) window has effect on the symmetry of the 3 subsets but for one exception: the shape of chaotic limit sets is indipendent from this square window. But the basin of attraction has a symmetry depending on the shape of the chosen window.

Symmetric chaotic attractors seem to be very difficult to find:Golubitsky(sp?) has found one formula. I have read his book and the impression is that it is very difficult to find symmetric attractors. (Unfortunately being no mathematician I cannot follow the way he discovered the formula).

With my brute force approach to finding new fractal formulas I have discovered only very few chaotic symmetric attractors with very simple bilateral simmetry. The simplest is:

    x = -y
    y = x^3-x-y

Inspecting these formulas and also the Golubitsky formula, it seems that the trick is more or less that when you apply this formula at once to all the points in the plane you should have a "stretch and fold" transformation that in space, should form a symmetric quartic in the viewing window. This said ,unfortunately I have no idea on how to build the symmetric attractor exploiting this idea.

Now, going to the other subsets: i.e. nonchaotic limit sets and the subsets 2 and 3; because of the window chosen it is very easy to have at least bilateral symmetry. A very common case is that a asymmetric chaotic attractor has a symmetric basin of attraction.See for example the "conchiglia" attractor.

In the case of non chaotic limit sets, it is very easy to produce symmetric shapes (in addition to the window effect): it is enough that a rotation is present in the formula. In this case you have spirals. That's why complex maps are often symmetric.

Well this is more or less my experience with symmetric attractors and fractals .

Giuseppe