Guide to Cczcpaczcp Part 3

This part deals with the fractals generated where the powers have opposite signs and, ignoring signs, one of the powers is one and the other is two or greater. Before I continue, I’ll mention the case where the powers are 1 and -1, which is the start of the sequence of a family of fractals where one of the powers is 1, the sequence where one of the powers is -1 produces a completely different family of fractals.

A reminder of the fractal formula:

z_n+1 = c(alpha*z_n^beta + gamma*z_n^delta)

For the fractals in this part the initial value is set to the “critical value”, for a description of how the critical value is calculated see part 2.

So starting with

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

the critical value or z₀ = 1

The resulting fractal is enclosed in a circle with the two main buds merging into each other. I’ve been using a very high bailout condition of norm(z) > 16000000, so the image is made up of a multitude of dots and a high number of iterations is required, in this case 12000.

An even higher higher number of iterations is required to reveal the structure of the fractal in greater detail, the form will become apparent as the negative power is decreased, the value of alpha is adjusted to maintain the critical value at 1, for delta = -2 alpha is 2, for -3 alpha is 3 and so on.

The following sequence of pictures delta is set to -2, -3, -4 and finally -5.

While preparing this part I discovered that these particular forms of Cczcpaczcp are sensitive to the bailout condition, the results are more pleasing, but that will have to be the subject of the next part of the guide, pushing back what would’ve been in part 4 to part 5.

At the start of this part of the guide I mentioned that a completely different sequence of fractals is produced where one of the powers is set to -1 instead of 1 and that the other power was positive and greater than 2.

So using

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

z₀ remains at 1. The critical value can be kept at 1 by setting gamma to be the same value as beta and keeping alpha and delta to 1 and -1, you can of course use different values which will require z₀ to be calculated so that the images produced match the following examples starting with beta equal to 2 and increased by one for each following image.

Like the rings of Mandelbrots in part 2 these forms of Cczcpaczcp, the number of their main features, be it buds or clover like leaves, matches the sum of the powers, ignoring signs. In what would’ve been the next part I’ll show a form of this fractal where the number of main features does not match the sum of the unsigned powers. That subject will have to wait to part 5.

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