Registered: Jan 2004
Posts: 350

Each of the following 243 three color rule numbers exactly reproduces elementary two color rule number 30 behavior.

Every group of three rule numbers is 243 rules distant from the next group of three.
Within each group the rule numbers are 9 rule numbers apart.

Adding And[(-1 -p), (-1 -q), r] modulo 3 to each rule within a group gives the next rule number.

Adding And[(-1 -p), (q + 1), r] modulo 3 to the first rule within a group gives the first rule number of the next group.

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

This shows the distribution of the rule numbers: ListPlot[Flatten[rulenumbers]];

This generates a table of the graphs. It uses Richard Philips NiceCaptionedRaster code.

graphs = Table[NiceCaptionedRaster[CellularAutomaton[{rulenumbers[[i, j]], 3, {{-1}, {0}, {1}}}, {{1}, 0}, 128], 5, rulenumbers [[i, j]], "PixelsPerCell"->1], {i, 81},{j, 3}];


This displays the graphs in 81 rows, three graphs to a row: Show[GraphicsArray[graphs]];


I am conjecturing that each of the two color elementary rule numbers has exactly 243 corresponding three color rule numbers.

Although, admittedly, in the case of the two color rule numbers that have the same behavior as other two color rule numbers, sometimes it will be difficult to identify the corresponding three color rule numbers.

For instance, elementary rule number 2 has the same behavior as elementary rules 10, 34, 42, 66, 74, 98, 106, 130, 138, 162, 170, 194, 202, 226, and 234.

In this instance two color rule number 2 has three color rule number 3 as first in the list of 243 emulating rule numbers and rule number 6501 as the last. And there is no problem identifying these.

However, the next elementary rule number is 10, and its first corresponding three color rule number is 6564 while its last is 13062. Again, no apparent problem.

Then elementary rule number 34 has its first emulating three color rule number at 3486961551 and its last at 3486968049. There is a wide gap between the previous rule, 13062.

One wonders what might be found in this gap? And this is where the difficulty come in.

For when the range for three color rule numbers corresponding to elementary rule 10 is extended beyond 13062, we immediately find 243 rule numbers starting at 13125 and ending with 19623 that appear to emulate elementary rule 10 behavior.

The question this raises is: does rule 13125 belong to the set of rules emulating elementary rule 10 or is its behavior a replica of that of three color rule 3, just as elementary rule 10’s is of elementary rule 2?

Because three color rule numbers will also have replicas with the same behavior, there will be a problem identifying boundaries when isolating which three color rules are emulating the two color rules and which are replicas of the three color rules.

However, this said, there is no difficulty with elementary rule 30, nor with the other elementary rules that have unique behavior.

__________________
L. J. Thaden

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