Patterns in CA evolutions

[luanoz (Luan Ozelim)]

Hello everyone!
Todd R. and I were doing some experiments with 2D CAs. Our goal was to find something unusual.
After a few tries, by observing a slice of the evolution of 2D totalistic rule 169, we observed some “circular” patterns.
The code to get what we were analyzing is:

ArrayPlot[CellularAutomaton[{169, {2, 1}, {1, 1}}, {{{1}}, 0}, {1000,All, {{0}}}]]

And the output is:

Do you also see this “circular” shapes in the evolution above?
If these patterns actually happen to exist, this would be interesting since “circular” patterns do not tend to be common in this type of CA.
That is it =)
What is your opinion about it??

[rowland (Todd Rowland)]

How cool that you posted this. Is it an optical illusion? Is it real in some sense?

It would make a great example of curviness coming from discrete deterministic simple rules.

[Vitaliyk (Vitaliy Kaurov)]

Maybe image processing can help to find round structures? Below examples are too primitive to find some meaningful structures, but maybe someone can harness them to build more sophisticated filters.

img = Image[CellularAutomaton[{169, {2, 1}, {1, 1}}, {{{1}}, 0}, {1000, {{0}}}]];

ArrayPlot[ImageData[GradientFilter[img, 15]], ColorFunction -> "TemperatureMap"]

Binarize[GradientFilter[img, 15] // ImageAdjust, .09] // MorphologicalComponents // Colorize

GradientOrientationFilter[img, 60] // ImageAdjust

[rowland (Todd Rowland)]

This gets closer to what I was seeing, but I am still not sure if it is there or not.

ColorQuantize[Image[ArrayPlot[CellularAutomaton[{169, {2, 1}, {1, 1}}, {{{1}}, 0}, {1000, {{0}}}]]], 4]

Adding Blur brings out the curve a little better(?)

Curves in rigid lattices will always have some ambiguity. There is definitely something here, and at other scales too, but do they count as curves?