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In part 1 I introduced the Cczcpaczcp fractal as produced with the default values and the initial z value set to the location in the complex plane. The final picture showed four Mandelbrots arranged in a ring, changing the powers so that both values are 2 and above (ignoring signs) leads to five, six, seven and more Mandelbrots in a ring. Using the location in the complex plane as the initial value makes it very difficult to determine the values of alpha and gamma that will produce well formed Mandelbrots, the solution is to use the same initial value for all calculated positions. The initial value used for the standard Mandelbrot is zero which happens to be be its critical value, using any other value will distort and degrade the shape of the Mandelbrot. It isn’t a bad thing in itself as the resulting “perturbed” Mandelbrot can have some very pretty or striking structures.

So after mentioning the critical value an explanation of how it is derived is in order. As is the nature of fractals Mathematics is now going to intrude. The critical value is found by differentiating the fractal function.

For fractals the following general formula is iterated:

z_n+1 = f(z_n)

The critical value is found by solving the following general equation:

f'(z) = 0

NOTE: in the following formulae c is a constant representing the location in the complex plane, and * indicates multiplication.

For the Mandelbrot:

f(z) = z² + c
f'(z) = 2z

so

f'(z) = 0

is

2z = 0

So the critical values is 0.

For Cczcpaczcp:

f(z) = c(alpha*z^beta + gamma*z^delta)
f'(z) = c(beta*alpha*z^beta-1 + delta*gamma*z^delta-1)

to find the critical value:

c(beta*alpha*z^beta-1 + delta*gamma*z^delta-1) = 0
c*beta*alpha*z^beta-1 = -c*delta*gamma*z^delta-1

c cancels out so, for Cczcpaczcp, a value of z that satisfies the following equation is the critical value:

c*beta*alpha*z^beta-1 = -c*delta*gamma*z^delta-1

For

alpha = 1
beta = 2
gamma = 1
delta = -2

we get

2*1*z = -(-2)*1*z^-3

which boils down to

z⁴ = 1

So the critical value is 1.

For

alpha = 1
beta = 3
gamma = 1.5
delta = -2

The critical value is again 1, different values could be used for alpha and gamma but you’ll end up a power of z equaling some value other than one and hence a different critical value. The size of the image will also be different but the structure of the fractal will be identical. The resulting picture is a ring of 5 Mandelbrots:

Swaping the signs for beta and delta results in a different 5 Mandelbrot ring:

Maintaining a critical value of 1 the following:

alpha = 1
beta = 3
gamma = 1
delta = -3

will result in a 6 Mandelbrot ring.

A different 6 Mandelbrot ring can be produced using

alpha = 2
beta = 2
gamma = 1
delta = -4

maintaining the critical value at 1 and this ring:

Swapping the signs for beta and delta produces a third version of the ring:

Where beta and delta have opposite signs then, ignoring signs, the sum of beta and delta will determine the number of Mandelbrots in the ring provided beta and delta are 2 and above. This is why this fractal will not produce rings of this type containing two or three Mandelbrots.

If one of beta and delta is 1 or -1 the resulting fractals are completely different. Those fractals will be the subject of part 3.