I received a query regarding the implementation of a fractal formula that didn’t produce the expected pictures. Here is such a picture. The implicated culprit was std::complex<T>, I’ve had trouble with this in the past leading to the use of my own version of the complex class in my software, it looks like it is entirely innocent.

I checked the formulae by trying them out in Saturn which also failed to produce the expected pictures. The program used for producing the fractals is called XoaS, the fractals in question aren’t part of XaoS but can be produced by setting the user formula.

An example of these formulae is shown below:

z_{n+1} = z_{n}^(1 – (1/(log(z) + 0.1))) + 0.3

where z_{0} = the location in the complex plane

This is what is produced by the Window’s version of Xaos (the image is centred on (0,0) and has a width of 3):

The Linux version of XaoS would be expected to produce the same image, it doesn’t (the image is centred on (0,0) and has a width 8):

Saturn produces this (the image is centred on (0,0) and has width 32):

Gnofract4d agrees with Saturn:

Ultra Fractal also agrees with Saturn and Gnofract4d, the much smaller circle produced by XaoS for Linux must be to do with its bailout function, the bailout value is 4 which is the same as Gnofract4d and Saturn where the absolute value of z is used (abs(z) in Saturn, cabs in Gnofract4d).

The picture for the Linux version of XaoS took several attempts, initially there was just featureless black, zooming out I got a simple circle, but it was only when I went back to XaoS on Linux to check the location and size of the image that the blobs appeared.

I’ve now found that Saturn produces the same image as XaoS on Linux when the bailout function is changed to norm(z).

I can only conclude that the fractal images produced by the Windows version of XaoS are in error, as XaoS, Gnofract4d and Saturn on Linux and Ultra Fractal on Windows all produce the same image.

The images produced by Saturn for fractals with similar formulae are not all interesting so fractals of this sort won’t be added to Saturn. The erroreous images produced by XaoS on Windows on the the other hand are definitely worth adding to any fractal program provided the error can be found and replicated, indeed Saturn already has several fractal types found due to programming errors.

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