Theory.org » intro » 2ndary » Secondary Fractals / Fractal Secondaries

These two fractals are identical except for the use of slightly different palettes. They represent a Julia set for a c-value taken inside a secondary Mandelbrot set. A secondary Mandelbrot is a mandelbrot that occurs inside a dendrite coming off the main Mandelbrot. A ternary Mandelbrot would be inside the dendrite coming out of the secondary Mandelbrot; etc.

Each of the inner circles --- seen inside the dendritic branches below --- could be thought of as a secondary Julia. The vacuole at the bottom of the images is the primary Julia vacuole.

Is there a relationship between a secondary (ternary, etc.) Julia and a structure in the Mandelbrot set (say, ternary (quadrary, etc.) ) Mandelbrots? Use XaoS to find out.

-0.000060 < x < +0.000050 :: p = -0.15652
+0.083936 < y < +0.084046 :: q = -1.03225

f(z) = z2 + c z = x + i*y c = p + i*q
c-value - A c-value is a function parameter.
For example, let's take the function: x2 - c.
We will define x to be initially zero and we will start out with our c-value, our paramater, to be one.
Lets, iterate this function! 
x0  =   02 - 1  =  0 - 1  =  -1
x1  =  -12 - 1  =  1 - 1  =   0
x2  =   02 - 1  =  0 - 1  =  -1
x3  =  -12 - 1  =  1 - 1  =   0
x4  =   02 - 1  =  0 - 1  =  -1
Apparently this paramater, or c-value, gives us an oscillator.
What about c=3?
x0  =     02 - 3  =        0 - 3  =       -3
x1  =    -32 - 3  =        9 - 3  =        6
x2  =     62 - 3  =       36 - 3  =       33
x3  =    332 - 3  =     1089 - 3  =     1086
x4  =  10862 - 3  =  1179396 - 3  =  1179393
Looks like this is going to head out to infinity. Thus, different c-values can give different behaviors.
What would happen for c=2?
If you have any questions, don't hesitate to email me!
Jeremy Avnet : : brainsik(at)theory.org
Revised: 22nd March 1999
Created: 7th August 1998