Octics Part 4   Leave a comment


On to MC 8.2. These fractal have the following formula:

z = αz8 + βz2 + c

To determine the critical points f′(z) = 0 needs to be solved:

f(z) = αz8 + βz2 + c

so

f′(z) = 8αz7 + 2βz = 0

which is

z(8αz6 + 2β) = 0

So the solutions are z = 0 and the six sixth roots of 2α/7β.

Critical points for α = 1 and β = 1 shown in Neptune's settings window.

Critical points for α = 1 and β = 1 shown in Neptune’s settings window.

Here is an MC 8.2 Octic fractal:

MC 8.2 Octic α = 1 and β = 1

MC 8.2 Octic α = 1 and β = 1

Ignoring outer colouring where inner and outer areas overlap the same fractal looks like this:

Ignoring outer colouring for inner/outer overlap

Ignoring outer colouring for inner/outer overlap

This fractal looks odd when compared against the standard Octic or M8 Mandelbrot (see part1). I have a tendency use Heptabrot (based on the number of primary buds) to refer to M8 Mandelbrots. The following pictures show each of the component pieces, one for each critical point.

Critical point 1

Critical point 1

Critical point 2

Critical point 2

Critical point 3

Critical point 3

Critical point 4

Critical point 4

Critical point 5

Critical point 5

Critical point 6

Critical point 6

Critical point 7

Critical point 7

From the above pictures it is clear that there are only four distinct components: critical point 1 is unique but the points 2, 3 and 4 are duplicated as 5, 6 and 7. I’ve come across cases where there are identical components before which is why I added a facility to Neptune to ignore critical point(s) as unnecessary calculation can be avoided.

There is a clear difference between MC 8.1 Octics and MC 8.2 Octics. MC 8.1 Octics have just one component in regard to its shape but has seven different locations. MC 8.2 Octics have four distinct components in regard to their shapes and three locations are shared. I’ve checked MC 8.2 with different values for β and they all have four distinct shapes and three locations are indeed shared. I suspect this is always the case, I do not have a mathematical proof and if I had it would be too technical for these observations on Octics.

Changing the value of β changes the appearance of the fractal. All the values of β used for the following pictures are on a unit circle with its centre at the origin of the complex plane (0 + 0i). Outer colouring is ignored where inner and outer areas overlap.

β = -1

β = -1

β = 0 + 1i

β = 0 + 1i

β = 0 - 1i

β = 0 – 1i

β = 1/√2 + 1i/√2

β = 1/√2 + 1i/√2

β = -1/√2 + 1i/√2

β = -1/√2 + 1i/√2

β = 1/√2 - 1i/√2

β = 1/√2 – 1i/√2

-1/√2 - 1i/√2

-1/√2 – 1i/√2

The next part will continue with MC 8.2 with values of β that have absolute values less than and greater than 1. There will also be detail showing how the component parts overlap and the presence of M2 Mandelbrot islands (copies of the standard Mandelbrot, often distorted).

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