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I wonder if the 3D version exhibits behaviors similar to the 2D version, including periodic patterns.

I also have to link my favorite pattern, XKCD's homage to John Conway: https://xkcd.com/2293/

Certain rules do, though the exact analog rule of Conway's Game of Life is a bit boring as i recall. In general it was a lot easier to find interesting behavior in Brian's Brain rulesets.

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