This is the last part of the guide to Cczcpaczcp. There will be example pictures of Cczcpaczcp with non-integer, imaginary and complex powers and the use of a formula transform.
As it has been a while since part 6 here is a reminder of the formula:
zn+1 = c(αznβ + γznδ)
Example 1
α = 1
β = 2i
γ = 1
δ = -2
z0 = 0.623832938258 – 0.258400063681i
Bailout: norm(z) > 160000000
Example 1
Example 2
α = 1
β = 2i
γ = 1
δ = -2i
z0 = 1
Bailout: norm(z) > 1600000000000000
Example 2
Example 3
α = 4.2
β = -4.2
γ = 4.2
δ = 4.2
z0 = 1
Bailout: norm(z) > 16000000
Example 3
Example 4
α = -2
β = -2.25
γ = 1
δ = -4.5
z0 = 1
Bailout: norm(z) > 16000000
Example 4
Example 5
α = 1
β = 2i
γ = 1
δ = -2i
z0 = 1
Bailout: norm(z) > 16
Example 5
This example uses the same parameters as example 2 but with a greatly reduced bailout limit. The circle void present in example 2 and would’ve engulfed this Mandelbrot if the higher limit had been used. The colour map (gradient) and colouring method are also different.
Example 6
α = 1
β = 3 + 3i
γ = 1
δ = -3 – 3i
z0 = 1
Bailout: norm(z) > 1600000000
Example 6
Example 7
transform: z = ez
α = 1
β = 1
γ = 1
δ = -1
z0 = 0
Bailout: norm(z) > 160000
Example 7
This example is a little different as it is the first to have used a formula transform, the formula used becomes:
z ← c(αeβz + γeδz)
You’ll have noticed that I’ve used ← instead of z and subscripts for this formula, I’ve done this as HTML doesn’t really handle mathematical formulae. I need better way and when I find it there may even be blog post about it.
I used this particular transform for this example because the critical value was easily calculated and turned out to be zero. The are many other transforms available in Saturn and can be applied in combination and even even periodically changed as each location is calculated, mostly this will cause the critical value to be meaningless. The critical value does not need to be used, other values will mean the loss the Mandelbrot islands but that doesn’t mean the resulting pictures aren’t aesthetically pleasing.
This guide has shown that Cczcpaczcp has enormous scope regarding the variety of fractal pictures it can produce despite having a very awkward name.
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