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-- Slow growing chaotic r=2, k=2 1d CAs. (http://forum.wolframscience.com/showthread.php?threadid=1892)

Slow growing chaotic r=2, k=2 1d CAs.

Since the Summer School, I've been searching for slow growing chaotic CAs in the k=2, r=2 ruleset. Thus far, I've been focusing on a portion of the ruleset that I identified as having a large proportion of slow growing CAs, which has yielded some interesting results. The slowest class 3/4 I've found so far is only 41,423 cells wide after one million steps. I'm working on refining my searching methods in order to do an exhaustive search of the entire rulespace.

I noticed that Stephen did a live experiment ~6 years ago that touched on this problem:

http://forum.wolframscience.com/sho...hp?threadid=753

I didn't find find anything else posted on the forum on this topic, does anyone know if someone else has followed up on this question?

constant rate of growth?

Interesting topic.

does the growth appear to be linear, or can it slow down?

It's irregular, but overall it's linear.

I've attached a plot of the growth rate over the first 200,000 steps and some images of the evolution.

Incredible rule. Definitely a live one.

I see it comes close to dying out, is it actually possible?

Hard to say for sure. There are a number of points where it's barely updating any cells at all, [1] and it's not hard to find local branches, even large complex ones, that resolve to repeating patterns absent any interaction with the larger plot. [2] On the other hand, after 13,000 steps the number of cells updated at any given step doesn't seem to drop below a couple dozen.[3]

On the third hand, I found a CA that exhibits complex transient behavior for ~32,000 steps before settling down, [4] so there's a good chance that there's no way to know without running it.

1. DifferencePattern.png

2. DeadBranches.png

3. UpdatedCells.png

4. Rule 670738548

Meant to attach this: