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This is the fifth part of my guide to a fractal type I’ve called Cczcpaczp for want of a better name. The formula is:

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

So far I’ve dealt with the images that can be generated when:

• the initial values is the location in the complex plane.
• the initial value is the critical value and how to calculate it.
• the powers have opposite signs and the absolute powers are 2 or greater and the initial value is the critical value.
• one power is 1 and the other is an inverse power of 1 or greater, i.e. -1, -2, -3 …
• one power is -1 and the other power is 2 or greater.
• when one power is 1 and the other is an inverse power of 1 or greater and how they are affected by the bailout value.

In this part I’m dealing with positive values for both powers where one power is always 1, to maintain a critical value of 1 the values of alpha and gamma should have opposite signs and that the absolute products of alpha and beta, and gamma and delta are equal. The values of beta and delta should never be equal as the terms will cancel out and the result of each iteration will be zero regardless of location. I’m skipping the first pair of values 1 and 2 because the resulting image does not match the pattern starting with 1 and 3. The image is centred on the origin for all these fractals and the number of main buds or bulbs starts at 2 and increases by one every time the power is increased by one.

So for

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

the result is a circular central section with radial buds, there are only two of the largest buds:

For

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

there are three of the largest buds:

and so on …

Now back to the case where one power is 1 and the other is 2, the position of the centre of this fractal is at -0.5 + 0i and not 0 + 0i and there is no central circle with radial buds. This particular fractal has a name all of its own, it is called the Lambda fractal.

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

and z₀ = 1 as it is for all the pictures in this post.

Part 6 will deal with the fractals generated when both powers are positive or negative and their absolute values are all 2 or more.