Julia Sets. (Back) (Root)

A generalizedJulia set

This article assumes you've read the Julia set article under real and complex fractals. It covers a simple generalization of the algorithm given there. In spite of its simplicity, this variation opens up a lot of territory for making Julia sets with appearances you cannot get by simply manipulating the Julia parameter.

With a simple quadratic Julia set we check membership by starting with a given point and then repeatedly squaring it and adding in the Julia parameter. The variation lets the constant added in after squaring depend on the iteration number. The generalized Julia set at the top of the page is very similar to standard Julia sets except that on odd numbered iterations we add in mu_1=-0.1+0.3i while on even ones we add in mu_2=-0.05+-0.26i.

Let's take another look at this new way of doing things. For a normal Julia set we add in, on each iteration:

mu, mu, mu, mu, mu, mu,...
for the set at the top of the page we add in:
mu_1, mu_2, mu_1, mu_2, mu_1, mu_2, ...
One could specify any sequence of mu_n's in any order. For the nonce we will only look at alternating families of two or three parameters. We will give the new membership algorithm in its full generality.

A generalized Julia set membership algorithm.

Choose a sequence of parameters mu_n=a_n+b_ni,
  (at least as many as the iteration cutoff limit).

Pick an iteration cutoff limit.

For every point (x,y) in your view rectangle 
  Let z=x+yi
  Set n=0
  While(n less than limit and |z|<2)
    Let z=z*z+(mu_n)
    Increment n
  End While
  if(|z|<2) then z is a member of the approximate 
    Julia set, plot (x,y) in the Julia set color
  otherwise z is outside the Julia set, 
    plot (x,y) in the outside color.
End for

If you've read the article on how Mandelbrot sets index the Julia set then you already know that the Julia parameter mu determines if a Julia set is connected or not. If the Julia parameter is a member of the Mandelbrot set then the Julia set is connected and can have positive area. If the Julia parameter is not a member of the Mandelbrot set then the Julia set is a disconnected dust of points covering no actual positive area. The two parameter Julia sets can fail to be connected while having components that do include positive area. The fractal below with mu_odd=0.3+0.46i and mu_even=-0.3+-0.46i is approximated with enough iterations that increasing the number of iterations adds no pixels. There are reigons with positive area that are not connected to one another. This demonstrates that this generalization is non-trivial, in other words it adds in Julia sets we haven't been able to create before.

It is worth noting that we did not lose anything when we generalized in this fashion. If we choose all the mu_n to be the same number mu then we get a standard quadratic Julia set. Let's conclude by taking a look at a couple of generalized Julia sets that cycle through three parameters so that the sequence is mu_1, mu_2, mu_3, mu_1, mu_2, mu_3, etc. The first uses: mu_2=-0.5+0.7i, mu_3=0.15+0i, mu_3=0-0.16i. The second uses: mu_1, mu_2, mu_3. The first uses: mu_2=0+0i, mu_3=0+0i, mu_3=0.235-0.8034i.