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High Iteration and Deep Zoom Fractals

separator bar Nice Image: The Mandel-12 Fractal
The Mandel-12 Fractal separator bar

Now that FractInt has switched the integer precision for the loop iteration count to 32 bits, fractals can now iterate over 4 billion times per pixel.   Who can imagine wanting to?   Well, I can!   At least some day.   High Iteration Fractal My most audacious attempt to date still has the puny limit of 2 million iterations as a bailout value, but I am striving to find interesting regions at higher and higher iteration levels.   I like to think of it as a complimentary task to the deep zooming side of FractInt.   While deep zooming looks for fractal detail in areas that require more and more decimal places to differentiate between adjacent pixels, high iteration fractals require longer and longer iteration counts before the fractal detail will emerge.   Ultimately, both these techniques should be combined to push the fractal explorations to their furthest reaches, but I have that selfish human trait of wanting to see the results somewhere within a year, so I keep my searches on the quick side of the cross over from floating point into arbitrary precision math.

As well as the maximum iteration value, I think the minimum iteration value for an image is equally interesting.   High Iteration Fractal This image, (which I call "Tapestry" for obscure subconcious reasons that my therapist is not telling me), has a minimum iteration/pixel of 83,693.   Thats right!   Every pixel in the image looped at least that many times, while those at the mini-brot boundaries went up to 2 million iterations.   I find it incredible that this kind of detail and beauty lay dormant, waiting, for that many iterations, before it decided to unveil itself.   At this scale, right down at the limits of ordinary floating point precision, where the difference between one pixel and the next is of the order of 1x10e-18, it is hard to imagine that these small variations can still result in a complex pattern such as this, but there it is.   Furthermore, all indications show that things just get more complicated, more intricate, the longer that you iterate.   I like the texture invoked by the minute detail of the image.   It becomes so involved that there are not enough pixels on the screen to adequately represent it and the pattern becomes a mere hint of what might be there.

So where do you look for this type of image?   Well anywhere approaching the boundary of the Mandelbrot, or the boundary of any minibrot, you will find the iteration count must be set higher and higher to resolve the detail.   High Iteration Fractal How much patience do you have?   Compute times can become rather tedious when you iterate this long, but the rewards are there for those who look.   It is just a search process of ever lengthening intervals, to zoom in on these regions, pinpointing areas of interesting detail.   High Iteration Fractal   I like to jump from one region around a minibrot into another region of the next minibrot as I zoom slowly into the Mset, picking up characteristic detail from each level.   The influence of each minibrot I visit, is clearly evident in the hierarchy of detail that make up the whole image.   The colouring algorithm for the images on this page uses the log-palette option from the "X" menu of FractInt.   Can I say 'palettable' here without anyone groaning?   Maybe not.   Anyway, I find that log palettes give the most suitable distribution of colours across such a broad range of values.   Other techniques are certainly equally as valid.   I just happen to have my artistic preferences.   These images are all from the mandelbrot, but the use of high-iteration in other fractal formulae can give equally elaborate images.

You might ask me how I figured out the minimum iteration value for these images.   Well it is not too complicated.   One of FractInt's features when using log-palette mode, is an auto-detect function.   High Iteration Fractal If you are in the Basic Options Menu "X", scan down the left column until you see the Log Palette selection.   One of the options in this mode is 2 which puts it in auto mode.   When you go back to recalculate your image, you will notice that the first thing FractInt does, is to trace the entire border of the image.   This may be very quick or it may take some time to complete.   What it is doing is looking for the lower limit for an iteration count.   This works very well in the Mandelbrot because the set is connected, with no internal local minima and the lowest iteration count will always be somewhere on the image boundary.   This certainly is not true for all fractal types however, so be careful how you use this.   Once it has found this value and started it is standard image calculation, a trip back to the Basic Menu will show this number set to the lowest iteration value that it found.

Bengt Månsson's excursion, down a seahorse valley contains images that had maximum iteration counts of up to 3.2 million iterations.

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