I found an interesting fractal formula on fractal forums in this thread (reply 111). The formula is:

z = abs(z/c1) + c2

Where c1 = 0.5 + 0.5i, c2 = -0.8 + 0.2 and the initial value of z is the location in the complex plane. This class of fractals are coloured using “inner” methods, no bailout is used and the the number of iterations is small.

Saturn doesn’t explicitly have this formula but “Single Function” formula can be configured to duplicate the formula. Saturn’s formula is:

z = f1(αz) + β

The parameter α is multiplied to z so it needs to be inverted to match the original formula. The parameters for Saturn are:

f1 = abs,abs

α = 1/(0.5 + 0.5i) = 1 – 1i

β = -0.8 + 0.2

No bailout

80 iterations

Inner colouring: absolute log of exponential inverse change sum of change in ratio of abs(z) divided by abs(real(z)*imag(z)), scale = 250

Using an image size of 640×640 this is what I got:

Usually I expand images and then scale them to a suitable size before posting them on this blog. The picture above is the raw file seed file saved from Saturn. The reason is because when I expanded the image to 12000×12000 I got a very unexpected result, the expanded image was significantly different. The picture has been scaled to 640×640.

My initial thought was that there was a bug in Titan, but as it now uses code that is shared with Saturn it would follow that Saturn would also fail to reproduce the seed file correctly when it was loaded back into Saturn. The seed file was correctly reproduced when loaded back into Saturn, so it should also be correctly reproduced by Titan and indeed it is when the output dimensions are the same as input dimensions. So what’s going on?

Initially I thought it was one off but I soon discovered that it wasn’t. I used a second example using the same formula but a different value for β to investigate further.

β = -2 – 0.4i

The following pictures start with a 640×640 seed file followed by expansions scaled down to 640×640.

This particular fractal is unusual in that its colours change as the dimensions are changed, I haven’t yet come across any other fractal that behaves this way. Usually there are some differences between a seed image and the full size expanded image (typically 12000×12000 for square pictures) but usually it just that the details are finer but the colours are much the same.

The most notable feature of these fractals is that many of the colour methods produce numerous parallel lines of varying colours and that feature is the root of the phenomenon, the greater the dimensions the greater the number of parallel lines, the width of which, are relatively narrower than the in the original seed file and often of slightly different colours, the overall effect is to alter the apparent colour of the areas packed with parallel lines. The phenomenon is more apparent when the colour map has large differences in colour from one index to the next. It should be noted that areas not filled with parallel lines retain their correct colours.

So it isn’t after all a bug.

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