Octics Part 3   Leave a comment


Continuing on from part 2. More MC 8.1 fractals mostly with imaginary values for β. As a reminder the MC 8.1 formula is:

z = αz8 + βz + c

α = 1 β = 0 + 1i

α = 1 β = 0 + 1i

α = 1 β 0 + 0.5i

α = 1 β 0 +0.5i

α = 1 β = 0 + 0.5i One critical point

α = 1 β = 0 + 0.5i One critical point

α = 1 β = 0 +0.25i

α = 1 β = 0 +0.25i

While β is less than 1i most of the areas of the component parts (one for each critical point) overlap. Above 1i the component parts look more like standard Mandelbrots and the overlapping areas are reduced in size until there is virtually no overlap at all.

α = 1 β = 0 + 1.25i

α = 1 β = 0 + 1.25i

Once the fractal has split apart the colouring of the image is affected: inner areas are entirely overlapped by multiple outer areas so that inner colouring has a greatly diminished effect on the final picture. Inner colouring only can be used for areas where at least one critical point produces inner colouring, any overlapping outer areas are ignored.

Inner colouring ignoring outer overlaps

Inner colouring ignoring outer overlaps

Using negative imaginary values produce the same images as for the positive values except that they are mirror images of the positive value when rotation is taken into account. This behaviour is markedly different to real values for β where the negative value produces a completely different picture to the corresponding positive value. All the pictures above have the origin (0 + 0i) of the complex plane at theirs centres.

The following two pictures show how unlike a standard Mandelbrot a component picture (1 critical point) can be.

α = 1 β = -1

α = 1 β = -1

α = 1 β = -1.125

α = 1 β = -1.125

Finally detail from the centre of an MC 8.1 fractal.

α = 1 β = -1.125

α = 1 β = -1.125

The next part will focus on MC 8.2.

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Posted 1 October 2014 by element90 in Fractal, Mathematics

Tagged with , , , , , ,

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