It is quite easy to tell what positions are drawn, and in return what appropriate moves should be dealt with.

Simply let's figure this out, for different types of pieces and boards (later also with boundaries of course). Well, actually the meaning of radial/angular is valid on a plain board (standard poly=4 grid is ok), but already on a torus (that is to say there is some identifying modulo in the graph) an overall angular rotation makes no real sense probably. We better simply think of a relative movement in the two directions separately (and doing them alternatingly or in parallel delivers some sort of angular and radial modes, so you can see it either way). In case of the 3x4-torus some mode is suppressed because there is not enough room for it to happen.

The generators don't commute, or do they?! Actio = Reactio is involved probably. After all we want a group, and that's why once one piece is reflected (it avoids being captured), the other mustn't continue as if nothing had happened. And again we'll have some flags indicating a reflected state, so the situation can be distinguished from that achieved by simple translation. Well, the modes are probably not what we will dump out directly as permutations, but (as in the single piece case) we work locally and assign labels using as much information as is necessary to be faithful. Actually (at the moment) we think that the _intended_ locations (of both pieces) at times t=0,1 are needed (same as for the case of a single piece), of course normalized to where the "a"s are pointing to. It's not necessarily the actual locations arising in due course, but (actually similar to the tricks (low-high flipping) used for the case when two boundary lines are neighboured) one ply before a central collision the piece not at turn still intends to go straight forward, this is important because when the piece at turn changes its mind and rotates (does "b" instead of "a"), this original direction will be realized. Additionally the reflection states are also taken into account, for each piece separately (whereby it doesn't make a difference whether the reflection takes place at a boundary or by interaction with another piece). Because we don't use single piece generators but only combined stuff like (a,a), we must set this doubledly (a one piece ply apart) to reach all situations in general (parity).

For the record we also investigated the approach to use higher orders (that is the final real locations at times t=0,1,2) for the labels, thereby obviously the evolution in time can be modelled as well, but putting the solutions into relation to each other (say this one is (a,a), then what is (b,a)) didn't work out. And of course just the _actual_ locations at t=0,1 aren't enough. since we can't be sure that we are in the middle of a collision when our labelling procedure is beginning, whereas we must distinguish between a half-reflected and a simple translation situation. We hope that now everything is clearly explained :-)

It turned out that it is a good strategy to simply ensure that the 'other' piece (that is the piece not having as the first one to change direction to avoid suicide, it only changes direction as a reactio effect) should bounce against a collision place when going the further development backwards in time. It is not necessarily the original collision place, but in case the two locations of the pieces are neighboured over multiple faces then one might decide to use another collision place for the bouncing, deciding on criteria like the conservation of angular momentum rather than aesthetic grounds like avoiding an abrupt back-reflection into itself whereby the latter nonetheless often serves as a good hint signalling the aforementioned problem.

And we applied some rotations (before the collision is taking place to the piece not to move next, or in other words to the other piece after the collision) in order to see if such a position would result in a bad intended move by the piece to move next (thus invoking a procedure similar to the bouncing treatment), because neighboured repairings make it necessary to do the same for other situations as well in order to avoid 'disjoint-and-duplicate-free' conflicts.

On zonotope_5 and deltoid (the odd poly clearly showed the bouncing idea necessity) we ended up with using two iterations to arrive at the final new directions for the two pieces, that is to say the high/low flip occured twice, since after the first exchange of momenta and subsequent tweaking around we still found the piece to move in a suicidal position. We don't know yet if this is only a curiosity, perhaps even unnecessarily blowing up our groups, or if there are subtle interaction rules behind it. First of all we were already satisfied with one working group for the moment fulfilling our purposes.

Now some general words of wisdom about our method. It's already implicit that the players move alternatingly, and that they do not risk a loss by bad play.

One can state that we can work locally (we need not look around globally in order to decide how the orbits should evolve). Probably one can formulate the procedure within a difference equation framework, and it should be possible to prove that by this artificial restriction to obey some additional rules one doesn't lose any solution. In contrast, it will enable us to follow an approach to solve the stuff systematically instead of searching around.

In general the generators (a,a), (b,a) and (a,b) do not generate the full group. Thereby (b,a) means first of all the piece with number 1 moves in 'b' direction, then the other does an 'a'. For example on the trapezohedron graph we also need (c,a), (a,c). And one might think that on a torus (b,b) could be the same as (b,a)*(a,b)*(a,a)^-1, but this is not the case (only for some labels). But nevertheless adding (b,b) to the generators doesn't make a difference for this particular group, in general we better use some more additional generators like (c,a), (b,b) and (a,c) (but perhaps the results below are still only subgroups). The general situation will be investigated when the time is ready.

Well, of course this way to present the result is not very informative. One would like to see how the overall movement (2-dim) can be separated from the other structures (movement relative against each other (in 2 dimensions), and links). When changing the board, what changes, and what stays the same? The cases where we weren't able to figure out the degree (that is the number of labels) of the permutation representation indicate that there is a certain need for improvement of the current strategy, for example constructing and storing the orbits for every possible starting position again and again is of course a waste of space and time. Much work to be done.

We also tried out some strategy to not care about angular momentum but simply exchanged the directions of the pieces when colliding. For example on the torus_3x4 we then got a doubled number of labels 432 and a group something like lossy C2³³ × C3⁹ × A(9). Well, for whatever it's good, but what should we make out of it? Let's keep in mind our final goal, that is the simplest group possible while covering all necessary moves of Chess pieces.

Cautious readers pointed out that boards with boundaries could lead to even higher levels of insight. Guess what, it is already on our agenda:-)

When there are boundaries around, we anticipate the following issues. First of all there is the additional topic called stalemate, which simply means one party might decide to do a Null-move, which mathematically still fits perfectly into the framework of group theory.

It is still completely trivial to decide what positions are drawn or not. Imagine an exotical board with a corridor attached to an area with enough space for a piece to move around and feel free a little bit. Place one piece in the middle of the corridor, and the other into the wide area. Then (because whereas a pass in general is not considered a valid move, in case of a stalemate it is) the (trapped) piece might dare to move further down into the corridor, it doesn't depend on how far away from the entry the other piece is.

But another issue troubling us happens when the two pieces as an ensemble approach a boundary, let us say on a board with faces having four sides (poly=4). The first piece to move reaches the boundary and gets reflected, ready to move on in a reflected mood. But then the other piece follows up immediately, thus squeezing the unlucky guy. What shall he do? He might go to the left or to the right, probably depending on his (high/low) former reflection status, and the other piece will simply be reflected back. Ok, but now all labels are taken, and we have other situations arriving here. The first piece might have gotten trapped by another piece coming not from the back but from the right or left, leading to the same result (the other possible further developments are taken already by other input positions). Perhaps one can say it has to do with the sort of crossing explicitely noticed there (as there are the unknotted, the left-over right and the right-over-left crossing). We think that we must triple these labels. This somehow resembles the trick we used in the single piece case when doubling the labels because of the reflection (and yes, three left-right crossings in a row and we are back at the original label set, why not). But it may well be that in the next week we will think differently about it.

To cut a long story short, you will understand that the boundaries have to wait until the next release.