|Argh! The pain of discovering that I too had forgotten about subtraction after my recent rant about accountants of my generation.|
Demonstrating Golly 2.0's Perl scripting to Melbourne Perl Mongers in March I used as one code example a script which made it very obvious that I had not properly analysed the problem, but still finished up with data that told me that outcomes of the interaction of a trailing Gobbler with a shifter repeated with period 42.
Even with all the data anyone could use, that and other phase combinatorics pertaining to significant interactions under Generations 345/3/6 continue to tax my residual mental powers. Finally this morning, I realised that part of the explanation for 42 lay in subtraction, but I need to work this through.
Gobblers are ugly enough that it is unsurprising that they gobble up block trails at 8/13 speed, that is 2 blocks every 13 iterations, significantly faster than the 1/2 speed for ships and track/trail laying engines. Period doubling and tripling sees Gobblers leave their own puffer-like trail of three blocks, two zeros and a p.8 every 48 cells, the phase difference between neighbouring p.8s being an effect of the underlying p.13 mechanism. (This trail spacing was added justification for using 48:1 compression in the iteration 200000 canvas prints, but that is another story.)
My first attempt to survey possible interaction between trailing Gobblers and shifters cross tabulated the 78 iterations of a Gobbler moving ahead 48 cells with the 10 iterations to move a shifter forward 4 cells as it shifts one block back by one cell in a 10 iteration cycle. My first mistake was immediately obvious with every outcome repeated one down and one across on the test grid. Moving both forward by the same number of cycles was always going to produce the same results.
But it was the second issue which has had me scratching my head ever since. Looking across any of the rows soon revealed that each outcome was repeated 42 columns across. Like Arthur Dent, I had no clue where 42 came from. It did not help in the least that my mind got stuck on (78 + 10)/2 = 44, nor that I often forgot whether I was actually looking for 42 or 43.
Making it even more curious, within the 42 cycle there are six runs each containing 4 identical outcomes one iteration apart, a unique outcome and two more one iteration apart, giving a total of 18 distinct interaction patterns. The most I can suggest about this is that looked at close up it seems tied up with the fact that both Gobblers and shifters move forward jumpily.
As with other classes of fully surveyed interactions, only two of the 18 fail to settled sown. I've linked a largish animation of one which has the bonus feature of sending a skimming ship forward in a phase that interacts with the same shifter, shifters never being directly disrupted by trailing Gobblers. Skimming ships are also produced by one particular alignment of a ship orthogonally interacting with a block trail so their also phase-dependent reactions with shifters are another recognised means of seeding chaotic growth at a distance downstream/trail.
This morning before getting out of bed, it finally occurred to me that 42 is actually the difference in iterations between how long it takes a shifter and a Gobbler to advance the 48 cell Gobbler trail period. In the usually even more clarifying shower, it was not quite so clear, but the arithmetic works so I'm going with it.
In a totally unrelated context, there is also a mechanism which produces ship streams with a separation of (42, 48) which is clearly driven by the 4/7 tail erosion speed of a two on gap tagalong, so that is a much clearer picture.