Octics Part 7   Leave a comment


The focus of this part on Octic fractals is on the MC 8.4 formula:

z = αz8 + βz4 + c

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

f(z) = αz8 + βz4 + c
f'(z) = 8αz7 + 4βz3 + c

so

z3(8αz4 + 4α) = 0

so the solutions are 0 and the four fourth roots of -β/2α.

There are five critical points one fewer than MC 8.3. The pictures of each component show that that there are only two different pictures, the pictures for all the non-zero critical point components are identical.

The component parts look like this:

Zero critical point

Zero critical point

Non-zero critical points

Non-zero critical points

Combined the resulting picture looks like this:

All critical point components combined

All critical point components combined

This fractal features M4 and M2 Mandelbrots which are illustrated in the following pictures:

M2 features on the left, M4 features on the right

M2 features on the left, M4 features on the right

M4 features

M4 features

M2 features

M2 features

I’ve tried various values for α and β and all result in identical pictures for the non-zero critical points.

The next part will deal with the M8.5 formula.

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Posted 19 November 2014 by element90 in Art, Fractal

Tagged with , , , , , ,

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