Screaming Stone Design

Fractals Revisited Again

Saturday, 24th of October, 2015

Having conquered the Mandlebrot set and the Julia set (at least in my own opinion) I decided to expand upon the image mapping possibilites available to me and attempt to create fractals with higher powers of n where z = zⁿ+c. I thought I would be able to use the same distortion correction measurements which had worked for the basic Mandlebrot set but I was SO wrong.

As you can see from the following 4 images, the distortion just becomes more and more intensified.

Multibrot variation of Mandelbrot set, z = z³+c

Image tessellated inside a Multibrot Set #1, z = z³+c

Multibrot variation of Mandelbrot set, z = z⁴+c

Image tessellated inside a Multibrot Set #2, z = z⁴+c

Multibrot variation of Mandelbrot set, z = z⁵+c

Image tessellated inside a Multibrot Set #3, z = z⁵+c

Multibrot variation of Mandelbrot set, z = z⁶+c

Image tessellated inside a Multibrot Set #4, z = z⁶+c

Because of the increasing distortion new anti-distortion measurements would need to be created for each new exponent of z, which is a very time consuming process. Another problem is that with each successive exponent of z the mapped images become more and more squashed.

I measured the distortion for the z = z³+c Multibrot and with a little bit of jiggery and pokery I managed to stretch the mapped image back to a more-or-less square shape. As you can see the distances between all of the white grid lines in each test card image are evenly spaced, so things are looking good already.

Image tessellated inside a Multibrot Set #5, z = z³+c

I continued on and measured the distortion for the z = z⁴+c Multibrot. Again the distances between all of the white grid lines in each test card image are evenly spaced but despite using the same anti-squashing as for the z = z³+c Multibrot the mapped images are getting squashed again.

Image tessellated inside a Multibrot Set #6, z = z⁴+c

The horizontal part of each mapped image is based upon the decomposition of the final angle of z once it has escaped past the bailout distance. Removing half of the images and showing only the ones where z is on the positive side of the X-axis you can see that the width of each mapped image above is 180 degrees, or 1 radian.

The problem is that the decomposition is based upon where the final z lands, not on where it starts or on any other factor, and all mapped images must begin and end between 0 and 2 radians. In the image below in the second largest band it looks as if there are 32 radians all the way around the image but in fact it is the same 2 radians repeated 16 times.

Image tessellated inside a Multibrot Set #7, z = z⁴+c

As a temporary fix I can double the anti-squashing so that each mapped image takes up the full 2 radians. To avoid the images looking stretched I can use a wider non-square source image and this is exactly what I have done here.

Photograph tessellated inside a Multibrot Set #8, z = z⁴+c

Unfortunately I can't rely on this solution because with each increase in the exponent of z the source images would need to be narrower and narrower. It looks as if z = z⁵+c might be the furthest I take this mapping system.

It looks like it is time to go back to the drawing board....