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Posted by Chris Sunami on 07-20-2004 12:27 AM:

Binary Numbers & the Replicator CA

Hi,

I first encountered CAs about a decade ago, and discovered the shapes discussed in the link below. Recently, after reading NKS, I decided to take another look at this particular CA, and to see if anything was known about it. I recently found out that this is a well-known CA called the "Replicator", and that it's a Game-of-Life type 2d totalistic replicator with the ruleset 1357/1357

But as far as I know, no one has made the same (rather striking) connection between the shapes generated by this ruleset and binary numbers that I have.

However, I'm not really in contact with anyone working in the field, or up on the latest innovations.

Does anyone have any insight to offer?

You can find my results here: http://philosophy.kitoba.com/fractal/bicarp.html

Thanks,
Chris

Posted by Kovas Boguta on 07-20-2004 01:10 AM:

how does this work?

"If you take the first pixel and number it "0" and the next shape as "1" then each successive shape will correspond to a binary number. "

I dont understand this proceedure for associating a number to the picture.

my uninformed guess is that this rule is couting in binary in some elaborate way.

The binary digits of the integers form a nested sequence - for instance see http://www.wolframscience.com/nksonline/page-117

By the way, all non-degenerative additive rules form nested patterns.

There are a few different mechanisms we know about that achieve nesting. (see http://www.wolframscience.com/nksonline/page-357 )

among cellular automata a common form is caused by the rule literally emulating itself in larger blocks -- see for instance http://www.wolframscience.com/nksonline/page-270

a subtle point is that trying to say all forms of nesting are caused by mechanism X usually just ends up being a tautology, because 'mechanism x' is typically just a restatement of the property of nesting itself.

i think your statement that "it may even be the case that every CA that produces fractal shapes could be considered a self-simulator in this sense" is a true statement, but i could be wrong.

By the way, i know Richard discovered a very interesting 2d rule that gave elaborate nesting patterns.... perhaps we can dig it up and compare it to this one.

__________________
Everything is an expression.

Posted by Chris Sunami on 07-20-2004 02:07 AM:

Re: how does this work?

I dont understand this proceedure for associating a number to the picture.

my uninformed guess is that this rule is couting in binary is some elaborate way.

No, I'm just numbering each iteration in binary... nothing elaborate about it at all. You could think of it as the number of steps elapsed since the inital conditions.

By the way, all non-degenerative additive rules form nested patterns.

Why is that?

among cellular automata a common form is caused by the rule literally emulating itself in larger blocks -- see for instance http://www.wolframscience.com/nksonline/page-270

Yes, I don't consciously remember seeing that particular page, but I think I'm saying exactly the same thing as Wolfram is, at least with regards to the self-emulation.

Interestingly enough, both those patterns (rule 90 and rule 150) can be generated using my CA. The rule 90 pattern is identical to what you get if you just look at the 1d horizontal strip passing through the original start pixel. And the rule 150 pattern is what you get if you always look at the 1d strip corresponding to the bottommost edge of the each iteration.

By the way, i know Richard discovered a very interesting 2d rule that gave elaborate nesting patterns.... perhaps we can dig it up and compare it to this one.

Yes, I'd appreciate that.

By the way, the interesting thing about these patterns is not that they are nesting or fractal per se, but that they have a strong geometric relationship with the binary number of the step in the iteration.

EDIT: Thank you again. Your comments reminded me of some results I had found but not written up. I've added them at the end of the original file.