Octics Part 8


The focus of this part on Octic fractals is the MC 8.5 formula:

z = αz8 + βz5 + c

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

f(z) = αz8 + βz5 + c
f'(z) = 8αz7 + 5βz4 + c

so

z4(8αz3 + 5α) = 0

so the solutions are 0 and the three cube roots of -5β/8α.

Again the number of critical points has reduced by one, compare with MC 8.4. This time the pictures for the component parts are unique, although two are mirror images of each other.

The pictures of component parts for α = 1 and β = 1 are as follows:

Zero critical point

Zero critical point

A cube root of -5/8 critical point

A cube root of -5/8 critical point

A cube root of -5/8 critical point

A cube root of -5/8 critical point

A cube root of -5/8 critical point

A cube root of -5/8 critical point

Combined, ignoring the outer colouring of overlapping areas, the fractal looks like this:

All components combined

All components combined

Fractals using this formula exhibit features of M2 and M5 Mandelbrots:

M2 Features

M2 Features

M5 features, somewhat distorted

M5 features, somewhat distorted

M5 features, somewhat distorted

M5 features, somewhat distorted

M2 and M5 features, combined components with merged inner and outer colouring

M2 and M5 features, combined components with merged inner and outer colouring

As with all the Octic formulae a cornucopia of images is possible by adjusting the parameters, in this case α and β.

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