I keep an eye on the Go/Baduk subreddit and answer random questions when it feels like I have something to contribute.
In one question "If aliens exist and they play Go, which board size do they mainly play on?", a lot of the usual arguments were totted out, the best balance between the 3rd and 4th line (see my previous posts on this subject), a slightly tongue-in-cheek 19x19x19, the odd, rectangular board size.
I posted something simple with no real justification, probably 19x19, but I went a bit in depth into the idea that you can play go on any grid, any graph, beyond the one we play on with four edges for every vertex (with the exception of the edges that have 3 to 1 and the corners with 2 to 1). I made some broad statements about the effects of changing the graph to have a larger edge to vertex ratio but with no justification.
A gentleman named Keenan Crane over at Carnegie Mellon created a program called 3d Tashoku Go back in 2003. There he generalized the rules of Go to any topology. I remember playing with it a bit but found that novel as those topologies are, they are unsatisfying.
Broadly, and with little thought, I would think that the finite topologies would break down into the same three categories that you could map a topology into based upon it's curvature. It's either
- Zero, or flat
- Positive, or closed like a sphere
- Negative, like a saddle or a hyperbolic space
We're already familiar with flat spaces for go, i.e. the 19x19 grid we play on. So I don't think I need to speak much on this. Other than to note that what makes these topologies interesting is that they have edges and corners. On a standard Go board/grid/graph/topology (oh man, the mathematicians are going to come after me with pitchforks for conflating all of those) Joseki work mostly because of those corners and edges. They bound the problem in a way that allows us to fully explore the problem space.
A closed topology where the edges disappear would cause joseki to naturally disappear. This would be similar if you played Go on an infinitely large flat board/grid/graph. How does a 3-3 joseki work if there is no edge? It doesn't, the same with any approaches, pincers, anything. Your stones are unmoored, adrift in a see of possibilities. Always able to be chased but never caught.
This too is why I feel playing on a very large board doesn't feel satisfying. The closer you get to that infinite board the closer you are to just being able to make a line of stones as long as you want and easily make life anywhere you want.
For a closed topology I suppose the main idea would be to use the wrapping around effect to catch your opponent. This would be similar to some strategies that already exist in Go. I believe that this would be highly sensitive to the size of the board as well, just as the concept of territory and influence is on a flat topology.
While I have experimented with closed topologies I have never even thought to explore a game of Go with a negative curvature. What would it be like to find that your stones further away from the center have more liberties than the ones in the middle? Imagine, the center has four liberties. Those four have six, those six have eight liberties each! You could never kill anything! You'd never be able to play enough stones!
I feel like I have exhausted some of my thoughts about this subject. I would like to see how modern AI could adapt to expanded graphs to play Go on. Maybe a higher ratio of edges to vertexes would be more interesting!