This is the final part of the series of posts on Octic fractals as currently implemented in Neptune and Triton. The formulae featured so far have been tailored to allow the critical points to be found relatively easily and to optimise the calculation of the fractals. The final formula is called MC 8.7.6.5.4.3.2.1 i.e. all terms of the polynomial are in play.

The formula is:

z = αz8 + βz7 + γz6 + δz5 + εz4 + ζz3 + ηz2 + θz + c

The critical points are found by solving f'(z) = 0:

f(z) = αz8 + βz7 + + γz6 + δz5 + εz4 + ζz3 + ηz2 + θz + c
f'(z) = 8αz7 + 7βz6 + 6γz5 + 5δz4 + 4εz3 + 3ζz2 + 2ηz + θ

Solving f'(z) = 0 involves finding the roots of a complex number polynomial, the mathematics used to solve these equations is complicated and I am not going to detail how the solutions are derived. The necessary numerical routines for finding the roots of complex number polynomials don’t appear to be readily available for C++ so I had to translate Fortran 90 routines. As with all critical points Neptune and Triton handle them automatically so you don’t have to. My other fractal programs Saturn and Titan also feature a general octic formula called over there Octic, version 5.0.0 will include critical point solving but only a single critical point can be used to produce a fractal picture.

Setting all parameters to 1 produces the following component and composite fractals:

Critical Point 1

Critical Point 2

Critical Point 3

Critical Point 4

Critical Point 5

Critical Point 6

Critical Point 7

Composite

The general formula allows combinations of powers of z that aren’t provided by the 3 and two term versions of the Octic formulae. I could have implemented every single possible combination of powers available as octic formulae but that would have led to a great deal more than nine versions.

Cancelling out the fourth power of z by setting ε to zero produces this composite:

ε = 0

Changing ε to 2.25 produces this:

ε = 2.25

Changing a sign of a parameter also has a significant affect on the appearance of the fractal, all parameters set to 1 except β which set to -1 produces this composite:

β = -1

Now with two parameters set to -1:

β = -1 δ = -1

And three parameters set to -1:

β = δ = ζ = -1

And four parameters set to – 1:

β = δ = ζ = ι = -1

A different set of four parameters set to -1, the other parameters set to 1:

α = γ = ε = η = -1

Now with five:

α = γ = ε = η = ι = -1

And back to four:

γ = ε = η = ι = -1

So that’s shown the great variety of weird and wonderful variations possible with the general octic formula mostly by just changing the signs of parameters.

To conclude the series of posts:

• I’ve only scratched the surface of what’s possible with Octic fractals.
• The number of critical points varies from 2 to 7.
• M2 Mandelbrot features always appear in the fractals.
• One of M3 to M7 Mandelbrot features appear depending on the number of critical points, 6 critical points have M3 features, 5 critical points have M4 features and so on.
• Where there is only one critical point the fractal is a standard M8 Mandelbrot and as such is not a multiple critical point fractal.

Development of Neptune and Triton continues, version 2.0 will allow parameters to be modified by multiplying or dividing by the location in the complex plane, where c is explicitly used in a formula it’ll be replaced with a parameter. The rule is that at least one parameter MUST be modified by multiplying or dividing by the location in the complex plane otherwise the Mandelbrot algorithm can’t be used. The effect of these changes will be that already immense possibilities will be enhanced still further.