BlinkerSpawn wrote:Kiran wrote:If this is caused, as I suspect, by waves propagating cyclically around the boundaries, why should the period be 256 and not 512?
I would like to say that this is caused by the fact that B4678/S35678 is self-similar, so that the trough of the wave is congruent to the peak of the wave, leaving only 256 "degrees of freedom", so to speak.
However, the top boundary doesn't appear to have the sort of translational symmetry that this explanation hinges on.
At first it didn't look like there was anything traveling around the boundaries. Was looking for something moving left or right, and there wasn't any obvious directionality. But the light bulb kind of goes off as soon as you think about XOR effects, along the lines of the B1/S bounded grid example
here.
Notice that each of the isolated high points on each boundary sends a new live cell left and right on each tick. Those two ripples travel at lightspeed ... and that means they're going to meet and cancel each other out in 256 ticks, not 512.
Assuming that the XOR effect is managing to propagate on the angled sections, where it's a lot harder to see than on the flat sections, it's a little easier to see why the period can't be 512 with a minimal example:
Code: Select all
x = 512, y = 9, rule = B4678/S35678:T512,16
512o$512o$512o$512o$512o$512o$512o$512o$256bo!
It seems like a good sign for this theory, that each boundary only stabilized to period 256 just after it evolved to the point where its variation from horizontal never exceeded 45 degrees. You're not going to get lightspeed east/west ripples if there's a two-cell-high vertical cliff anywhere.
-- Drat these bounded grids anyway. By rights Golly should be totally running away with a pattern like this!
As the
A268754 sequence shows (explained a little in the link above) the period could get a lot more complicated at non-power-of-two widths. Would advise starting with artificially simple examples if experimenting with other widths. These three aren't so bad --
Code: Select all
#C period 510 oscillator
x = 510, y = 9, rule = B4678/S35678:T510,16
510o$510o$510o$510o$510o$510o$510o$510o$254bo!
Code: Select all
#C period 511 oscillator
x = 511, y = 9, rule = B4678/S35678:T511,16
511o$511o$511o$511o$511o$511o$511o$511o$255bo!
Code: Select all
#C also a period 511 oscillator
x = 513, y = 9, rule = B4678/S35678:T513,16
513o$513o$513o$513o$513o$513o$513o$513o$255b3o!
-- but it seems like there might be some surprises lurking at other widths.