Compasses Part 4

This is the fourth and final part of the guide to the Compasses fractal. The images produced with imaginary powers are much more complicated than the images we’ve seen so far for the Compasses fractal. The bailout condition (norm(z) > limit) is important as it has a great influence over the appearance of the fractal image.

The first image shows a distorted Mandelbrot in a tangled web of ribbons.

alpha = 1.75i, limit = 16000

The first image is coloured using iteration for outer colouring and black for inner colouring. To demonstrate the effect of changing the limit value we now zoom out and for the next 3 pictures we’ll use alpha = 2i, beta = 0 and vary the value of limit, the colouring methods stay the same.

limit = 16

limit = 1600

limit = 16000

Now for a change of scenery and a change in outer colouring, the following two pictures are the same except for the outer colouring method, there are no inner areas to colour. The image is a zoom in on an area above the horizontal line in the first segment containing a distorted Mandelbrot in the previous picture.

colour method = final magnitude (norm(z))

colour method = variance of angle

Finally an image with outer and inner colouring.

alpha = 1.75i, outer = variance of angle, inner = absolute log of minimum magnitude

I’ve only used two values for alpha 1.75i and 2i, these are good starting points for exploring the Compasses fractal with imaginary powers. I’m not going to cover complex powers, you can try them out yourselves, you will get similar results to the imaginary powers.

The Compasses fractal can be manipulated by applying various complex plane and formula transforms so there is a lot more to explore, that exploration will have to wait until I revisit this fractal in an other set of posts possibly to be called “Transformed Compasses” …

There are more fractal guides to come in the new year.

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