Controlled Chaos - click to enlarge.

Image Details

There are a couple of interesting things going on in this image. First, and perhaps most obviously, the image is centered on a wonky, lop-sided minibrot. We have previously seen these miniature copies of the Mandelbrot set,[1][2][3] yet in each of these previous examples the copy has appeared as a visually-perfect copy of the main body of the set. In this case, the minibrot is clearly and strikingly imperfect. Given the symmetry found in other parts of the Mandelbrot set, I find this asymmetry to be worthy of comment.

Next, note that the colored region outside of the Mandelbrot set is a patchwork of purples and greys. This image was not colored using the smooth coloring algorithm, but rather the basic escape time algorithm described earlier. I could have used the smooth coloring algorithm, but the results would not have been much better, and it would have required somewhat more computational time. The smooth coloring algorithm is useful because it can estimate the color of an individual pixel outside of the set without needing to compute many iterations of the function that generates the Mandelbrot set. This is fine if we only care about coloring the region outside of the set. Unfortunately, in the image above, much of that region requires many iterations to find in the first place, which means that many iterations are required, thus negating the benefits of the smooth coloring algorithm. Because of this, there are some notable bands of color. On the other hand, if I were a better programmer, the smooth coloring algorithm could be made to provide a more aesthetically pleasing image without a loss of fidelity.

Finally, as a general note, this image is centered near \(0.257+0.001i\), and measures less than 0.00002 units across. This is near the end of elephant valley. If you look very carefully, you may be able to spot a few elephants, though the color scheme I chose does a fair job of camouflaging them.

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