Todd Rowland
Wolfram Research
Maryland
Registered: Oct 2003
Posts: 115 
The rule number for this 2D CA is Sum[2^(7 n), {n, 0, Quotient[512, 7]}] (154 digits).
Alternatively, one can use
CellularAutomaton[{1  Sign[Mod[Flatten[# ].coef, 7]] &, {},{1, 1}}, init, k]
after defining the coefficients
coef7={4,2,1,4,2,1,4,2,1}
(i.e. from Flatten[Table[Mod[2^(3ij),7],{i,3},{j,3}]]).
Unlike, the results in the last post, when one starts from an initial black cell on an unbounded grid, one gets a neat nested pattern. From the animation it is a bit hard to tell, but by looking at the slices it is apparent that it is a 2D version of rule 129. Also in the case of the 2D grid with the edges wrapped around as in a torus, the slices are the same as rule 129.
The interesting effect in fbinard's example comes from his boundary condition. The boundary cells are forced to be white cells on a finite grid. The reason this differs from the unbounded case is that the neighborhood of all white cells causes its center cell to become a black cell. So on an unbounded grid all the white cells turn black, and all black stays black. But here the boundary cells are always white, and that is a somewhat forced condition.
It appears to be quite formidable to attempt to predict the effects, so this is a type of NKS phenomenon, and even though there is forcing on the system from the outside, the specification is still simple.
For instance, look at the slice of this forced evolution compared to the evolution of rule 129 with its boundary also forced to be white cells. The case with rule 129 looks much more behaved.
Looking at the alternative rules of this type, one gets interesting behavior from a single black cell, even on the usual unbounded grid. Included in the attachment is the case for Mod[i,7]==1, where I substituted Flatten[# ].coef1 in the code above.
The other pictures in the zip show this CA on an unbounded grid, and also with the boundary forced to be zero, in a similar manner to the way some 2D CA's are displayed in NKS, e.g. step by step on page 171 and looking at slices on page 175.
Here is the code I used to get the forced boundary picture in the case of rule 129.
NestList[CellularAutomaton[129, {#, 0}, 1, {All, {0, 114}}][[1]] &, ReplacePart[Table[0, {115}], 1, 58], 119]
Attachment: mod7ca.zip
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