Multiple Critical Points Part 1   Leave a comment


This post is as a result of reply 5 on this thread in FractalForums.

The Mandelbrot set, and its higher power cousins, only has one critical point (at 0 + 0i). The critical point is used as the starting value for iteration the Mandelbrot formula.

The general form of the standard Mandelbrot set and its higher power cousins is:

zn+1 = znN + c

where N is an integer greater than or equal to 2, c is the location in the complex plane.

Adding in extra terms to the formula alters the critical value from 0 + 0i and increases their number.

Adding a z term to the standard Mandelbrot set just alters the critical point, using the new critical point produces the Mandelbrot in a new location. It gets more interesting when adding z squared and z to the cubic Mandelbrot as two critical points exist, in general adding extra terms to the general multi-power Mandelbrot formula results in a number of critical points that is one less than the highest power in the formula, so for z squared it is 1, or z cubed it is 2, for z to the power 4 is it 3 and so on.

To illustrate I’ll use a 4th power formula:

zn+1 = zn4 + zn3 + zn2 + zn + c

it has the following critical points:

z0 = -0.605829586188268 + 0i
z0 = -0.072085206905866 – 0.638326735148376i
z0 = -0.072085206905866 + 0.638326735148376i

The values are good approximations and produce the following pictures:

z0 = -0.605829586188268 + 0i

z0 = -0.605829586188268 + 0i


z0 = -0.072085206905866 - 0.638326735148376i

z0 = -0.072085206905866 – 0.638326735148376i


z0 = -0.072085206905866 + 0.638326735148376i

z0 = -0.072085206905866 + 0.638326735148376i

The pictures look somehow incomplete, there is the budding that is found on the standard Mandelbrot set but it breaks down in parts. To fully complete a fractal that has multiple critical points it has to viewed as a compound of images using all the critical points.

Saturn and Titan can not produce such pictures, so to crudely illustrate a compound image I’ve used GIMP to overlay the three images, the top two images have reduced opacity so that the lower layers can be seen, the result is this:

Compound image using all critical points

Compound image using all critical points

The above picture show the overall form of this fractal, where the budding is missing from one image it is provided by one of the other images. An overlap image can be produced by altering the way the image is constructed, calculation at each location should be completed for each critical point then the inner (or black part) of the fractal can be determined by whether any of the calculations do not escape. Doing so would lose the features of the individual parts, so some method of combining the colouring of inner and outer parts where they overlap is required. As Saturn and Titan can’t do this and it would be difficult to incorporate such features into that software I’ll be producing some new software to handle multiple criticial point fractals provisionally to be called Neptune in line with my naming convention based on planets for the explorer programs, its sister program for producing high resolution pictures would then be named after one of Neptune’s moons.

To show why a means of combining inner and outer colouring where they overlap is worthwhile there follows an example of a highlighted area of a different multiple critical point fractal.

The formula:

zn+1 = zn4 + zn2 + c

Critial points:

z0 = 0 + 0i
z0 = 0 – 0.707106781186548i
z0 = 0 + 0.707106781186548i

It is only necessary to calculate two layers for this fractal because critical points 2 and 3 are the same. The two layers produced are:

z0 = 0 + 0i

z0 = 0 + 0i


z0 = 0 - 0.707106781186548i

z0 = 0 – 0.707106781186548i

Two completely different images but there are areas in the two images that look like they coincide and indeed they do:

Images combined

Images combined

Treating the layers such that the inner part (black in this case) takes precedence would result in an image that is only slighty different to the image for critical points 2 and 3 and the features of the image from the first critical point would be lost.

One curiosity of the two formulae used in this post is that they contain Mandelbrot islands that match the 2nd power or standard Mandelbrot set, there are no islands that match the 4th or 3rd power Mandelbrots. This is not always the case but that’s a topic for a future post.

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Posted 29 January 2014 by element90 in Art, Fractal, Mathematics, Software

Tagged with , , , ,

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