The NKS-style question to ask here is: what computations are these doing? Totally unique-unto-themselves computations? Potentially useful computations? Computations analogous to familiar human ones? How would we know? Can we know? Do we run into the limits of undecidability?
The cyclic boundary conditions somewhat 'spoil' things, though. Maybe a particular rule was "destined to multiply input by 5" or "calculate log-2 of input" or something (given suitable encoding of input on the tape), but the computation is foiled as soon as the machine wraps around and starts interfering with itself.
Then again, the self-interference is what makes many of these patterns do the cool things they do.
For this particular canvas, no, as the canvas is finite.
Given enough time or space we can exhaust all states of the canvas and catch cycles. This applies to any finite canvas.
I would guess these are equivalent to regular languages because of the finite state. You can treat the different canvas states as states of a finite automaton. A finite automaton is a canvas turing machine by ignoring the canvas.
It is obvious, yeah. I wasn't implying you didn't know it was or anything, just explaining the first sentence to someone who might not have studied complexity theory.
http://wry.me/hacking/Turing-Drawings/#4,3,0,1,1,0,2,1,0,2,1...
At one point, right before it goes over the "entropy cliff" it creates what looks like a lot of Sierpinski triangles:
http://wry.me/hacking/Turing-Drawings/#4,3,2,1,0,1,2,3,2,2,2...
I found it quite interesting that one as complex as this could stabilize:
http://wry.me/hacking/Turing-Drawings/#4,3,0,1,3,1,1,2,0,2,3...
This seems to end up like a frozen lightning bolt: http://wry.me/hacking/Turing-Drawings/#4,5,3,3,3,2,4,0,2,2,3...
This creates a pattern almost like hair in the wind: http://wry.me/hacking/Turing-Drawings/#4,5,2,3,0,2,4,3,0,3,1...
Seems as though it's many cats standing on each others heads: http://wry.me/hacking/Turing-Drawings/#4,5,0,3,1,0,3,1,2,1,0...
Like waves on a beach: http://wry.me/hacking/Turing-Drawings/#4,5,3,4,2,1,2,2,0,4,0...
Some otherworldly data wind: http://wry.me/hacking/Turing-Drawings/#4,5,0,3,3,3,3,1,0,3,2...
Eventually, all clouds fade away: http://wry.me/hacking/Turing-Drawings/#4,4,1,1,1,1,2,1,2,3,3...
In and out of phase: http://wry.me/hacking/Turing-Drawings/#4,4,2,2,0,3,1,2,3,1,0...
Like sand dunes moving: http://www.wry.me/hacking/Turing-Drawings/#4,3,1,1,3,3,1,0,2...
Saws on parade: http://www.wry.me/hacking/Turing-Drawings/#4,3,0,2,0,2,2,3,1...
Scooty lightning: http://www.wry.me/hacking/Turing-Drawings/#4,3,3,2,0,3,2,2,3...