The Octic formula I’m dealing with today is MC 8.1. Neptune has may versions of the Octic formula, this one has only two terms. It is a waste of resources to calculate the whole formula when all but two of the z terms are zeroed out especially as a picture is generated for each critical point.

The MC 8.1 formula is:

z = αz^{8} + βz + c

so

f(z) = αz^{8} + βz + c

The critical points are roots of f′(z) = 0, so:

8αz^{7} + β = 0

z^{7} = -β/8α

The solutions of this equation are the 7th roots of -β/8α which are neatly spaced on the circumference of a circle in the complex plane centred at the origin. Neptune automatically calculates the critical points.

The pictures produced by Neptune using this formula are made up of 7 pictures (one for each critical point). The colouring methods, selection and combination used are:

- Outer – absolute log of exponential sum of magnitude, scale = 120
- Inner – absolute log of fractal dimension of magnitude, scale = 200
- Selection – outer where norm(z) > 16
- Combination – outer areas: average colour data, inner/outer overlap – average inner and outer colour, inner areas: average colour data.

Now for some pictures:

α = 1 β = 1

All the MC 8.1 pictures exhibit the same characteristic, they are symmetrical. The picture for each critical point is the same as all the other critical points except for the orientation about the origin, their positions are equally spaced. Using just one critical point produces this picture:

One critical point

Two critical points produces the picture below. The chosen critical points show the images overlapping, they are effectively adjacent images, other pairs overlap in different ways.

Two critical points

Varying α just alters the size of the image, altering β significantly affects the resulting picture. The following pictures have positive real values for β.

β = 1.25

1.375

β = 1.5

β = 1.625

β = 0.75

β = 0.5

β = 0.25

Now for negative values of β.

β = -1

β = -1.125

β = -1.25

β = -1.5

Finally complex values of β.

β = -0.25 – 0.75i

-0.6 + 0.75i

These pictures show that the further away from the origin the value of β is the closer the images for each critical point is to the standard Mandelbrot set. The images are also smaller and further from the origin reducing overlap. The addition of an imaginary part to β appears to rotate image of each component part but it isn’t clear from the pictures illustrating this post.

I forgot to cover imaginary values for β so the next part will be a continuation of MC 8.1 dealing with imaginary values of β and it will also show how unlike a standard Mandelbrot the component images can be the closer to the origin they are.

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