Octics Part 9


The focus of this part on Octic fractals is the MC 8.6 formula:

z = αz8 + βz6 + c

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

f(z) = αz8 + βz6 + c
f'(z) = 8αz7 + 6βz5 + c

so

z5(8αz2 + 6α) = 0

so the solutions are 0 and the two square roots of -6β/8α.

Again the number of critical points has reduced by one, compare with MC 8.5. This time the pictures for the square root values are identical, so there are only two component part pictures. Using α = 1 and β = 1 the pictures are:

Critical point of zero

Critical point of zero

Critical points of the two square roots of -0.75

Critical points of the two square roots of -0.75

The component parts combined, ignoring overlapping outer colouring produce this:

Composite of both component pictures

Composite of both component pictures

The composite image shows regions of M2 Mandelbrot features and regions of M6 Mandelbrot features with the latter dominating. The three following pictures show the same area, one for each component and one for a composite using inner colouring other than black and averaging the colours of the two components:

Critical point: zero

Critical point: zero

Critical point: a square root of -0.75

Critical point: a square root of -0.75

Composite of the two component parts

Composite of the two component parts

As with the other formulae in the family of Octic fractals changing the values of α and β results in different pictures:

α = 1 β = -1

α = 1 β = -1

α = 1 β = -1.5

α = 1 β = -1.5

Detail using α = 1 and β = -1.2

Detail using α = 1 and β = -1.2

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