[Redundancy in ECA Rule 30 Emulation] - A New Kind of Science: The NKS Forum

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# Redundancy in ECA Rule 30 Emulation

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Posted by: Lawrence J. Thaden

I took my 6 year old granddaughter out for Chinese the other day. The restaurant had glass mirrors on opposing walls. When she saw her image reflected in the mirrors she said, “There are hundreds of me.”

Later I got to thinking about this. If her image is mirrored over the entire space, maybe something similar occurs in the mirroring of ECA rule 30 over three space.

I had thought, incorrectly, I might add, that it was restricted to 243 rules according to:

(FoldList[Plus, #, Table[9, {2}]]&/@FoldList[Plus, 1162281261, Table[243, {80}]]);

But why the restriction? So I started looking from the very beginning of three space rules one rule at a time. I soon found that the first image of ECA rule 30’s behavior is at rule 19794.

Then I looked backward from the last rule in three space. I found the last image of ECA rule 30’s behavior is at rule 7625593095564.

But how many in between? Is it like the mirrors, distributed over the entire space?

Since the range is so large, I settled on starting at rule 1157498292. This is the first rule before the three color rule equivalent of the logic expression for ECA 30. It has the advantage of being a ways away from the first rule. So it can be used to test something more than just the tail of three space.

Before long I found over 14 million three space rule numbers emulating ECA rule 30.

The last of these is rule number 95296267902. But there is no reason why more can’t be found. In fact rule number 95300677119 is the next one, with none intervening.

Here is the code to generate the 14348907 rule numbers (3^15).

FoldList[Plus, #, Table[3^2, {3^1 -1}]]&/@
FoldList[Plus, #, Table[3^5, {3^4 -1}]]&/@
FoldList[Plus, #, Table[3^11, {3^1 -1}]]&/@
FoldList[Plus, 1157498292, Table[3^14, {3^9 - 1}]];

This time I am being cautious: Although I tested each one of these 14 million rule numbers and found that each does emulate rule 30 behavior, I make no claim that there are not others that I missed.

This certainly raises questions:

(1) Why all the redundancy?
(2) Is this only true for rule 30?
(3) Why is the NKS Book silent on this issue?

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