codeholic wrote:There are 3 ways of building (13,1)c/31 glider-based track:

That's pretty cool, because the 1st one moves the track horizontally by 1 cell, while delaying the track period by 1, the 2nd advances the track period by 3 and the 3rd one by 2, while keeping the track position intact. This set is complete in the sense, that if there are two distant enough glider tracks, you can bring them to any relative position and period!

Sorry to keep digging up flaws in your reasoning but I did some math and I think that the set of SE g -> SE g rephasings isn't actually complete (I only considered the three in the quoted post and not the NE g -> SE g reaction because I doubt it will find use). My reasoning is as follows:

**EDIT: YOU WERE RIGHT AND I WAS WRONG. Specifically, my formalism lacked the constraint that real gliders must have x=y mod 4.**
Abstractify the glider into an object that moves (1,1) in each generation, where cells are a quarter the size of the CGOL grid. This gets rid of phases, which just complicate the math anyway.

We take positive to be in the glider's direction of motion, and coordinate pairs are (x,y). In this metric, the three rephasings above are (3,-1), (3,3) and (2,2).

We want to show that two streams can be brought to any relative displacement. Choose a glider in the first stream: we care about its position relative to the second stream, so there's some modular arithmetic. In our new metric the ship moves 4*(+1,-13) in the time it takes a glider to move 31*(+1,+1), the distance between gliders in a stream is the difference, (27,83).

Now it's just 2D modular arithmetic. Unfortunately, since the three generators and mod vector are all "color-preserving" where "color" here is x+y mod 2 (new-metric analog of color on the CGOL grid) , we cannot reach a relative offset that has a different "color".

Now, everything in the block+boat

copier gets lucky. Burning the block and boat creates streams at relative offset (73+a,65+a), where a is the number of generations waited to burn the block after the earliest possible opportunity. The rubbing reaction that drops block+boat can occur if the streams have relative offset (56,48), the same color.

The maker of the linked copier chose to make a=1 so that all that is needed is 6*(3,3) on the back stream. Note that a=0 and 6*(3,3)+1*(2,2) works as well (verify this by moving the B eating the topmost block forward one generation). Also note that since it is the back of the crawlers that interact to make the constellation, the pattern actually features 7*(3,3) on the back stream and 1*(3,3) on the front stream.

Ivan also got lucky extending the copier to a

doubler, because the boats burn to give a stream at (38+b,10+b) relative to the remaining glider stream (again b is set by the generation delay from earliest possible opportunity). This is once again the correct color.

If I am to categorize the whole possible interaction space there will be two color classes, and tracks cannot be rephased simply from one to the other. This also cuts in half the number of lane+phase combinations usable for any eventual slow salvo construction with p2n targets. There will probably be reasonable workarounds though.

Post edit: so since x=y mod 4, all possible glider values are reachable, and the set is complete after all. However much this notation tripped me up at first, I think it is the notation I will be sticking with because it does avoid phase confusion and can be interpreted easily from the position of the gliders. So if in the future I reference weird distances between gliders, this is how I mean it.

Physics: sophistication from simplicity.