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I know there was some thought put into how Go looks or feels like if it's not plaey on a flat, rectangular surface, but the surface of a 3-d object like a sphere or a donut. I'm mostly interested in:

  • How is the grid mapped to the surface, so that every node has 4 neighbors (the number of neighbors is important for how hard it is to capture stones)?

  • What does the absence of a border and corners mean for the game?

  • What are variants that are actually taken seriously as a game and played, and not only seen as obscure applications of weird topologies?

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The typical way to get these topologies using a square surface is by joining edges up: en.m.wikipedia.org/wiki/Surface#Construction_from_polygons I don't have enough experience of go to tell you if that would work. –  tttppp Aug 18 '13 at 7:34
see also for the topological problem: math.stackexchange.com/questions/470352/… –  mart Aug 18 '13 at 12:39
Once you eliminate corners and sides, it's not Go anymore. –  Pieter Geerkens Aug 18 '13 at 14:47

4 Answers 4

up vote 9 down vote accepted

I started writing another comment, but it got too large, so I'm putting this as an answer instead.

Your other question suggests that you're envisaging playing go on the surface of an actual sphere/torus. In particular that question mentions that you want "all nodes more or less equidistant". Searching on the internet, I didn't find much evidence that this is how go is played on other topologies.

There are a few reasons why I doubt torus/sphere go has been played on actual spheres or tori. Firstly - who has a suitable torus lying around that they could use? Secondly - getting the stones to stay in place would be awkward, and examining the position would be even worse!

Instead I suggest that the usual* way to convert a flat board game into one on a sphere/torus/Klein bottle is by imagining that the edges are joined up. For some diagrams of how the edges of the board should be joined see here.

I found a few references to playing tic tac toe and gomoku (5-in-a-row) on different surfaces, and one to playing go. All of these assumed a standard square grid with the edges defined to be connected in some way. The main differences from standard go seem to be:

  • Fewer corners and edges: A cylindrical board has just two edges, and a torus or Klein bottle surface game has no corners or edges.
  • The lack of corners and edges makes it harder to create immortal groups.

I did also come across a game of go on a cylindrical surface which was not just made of a standard board with edges connected. Each point still has four liberties, but the regions between edges are not all square:

cylindrical board projected onto a square

*I'm not sure a 'usual' actually exists here, but this is at least the usual way to draw diagrams on these surfaces in mathematics.

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+1 - Well, I could envision a steel go board in a spherical or toroidal form, combined with magnetic "stones". Still tough to play and visualize, as you cannot see the entire "board". In fact, this seems a serious problem for such a variant. Worse, a Klein bottle board might be a bit difficult to create for sale. :) –  user3264 Aug 19 '13 at 3:12
If you play on a standard board, but consider all edge points to wrap around to the other side, you've got a torus. Imagine taking a sheet of paper and rolling it into a tube. Now, take the two ends and bend the tube to connect them. The paper would tear/crumple, but you've got the idea. –  Dane Aug 23 '13 at 16:18
@Dane That's right. However if you add a twist before joining the first pair of edges then you get a Klein bottle. Have a look at the wikipedia link to see the other surfaces you can make. –  tttppp Aug 24 '13 at 6:18
It's probably worth pointing out that you can't actually make a Klein bottle-it's a shape that needs four dimensional space. But in theory you would get a Klein bottle! –  tttppp Aug 24 '13 at 6:28

I found this program that a guy at university made to abstract Go to any topology, using any number of vertices as well, it is called 3D Tashoku Go. Other than as a curiosity I haven't seen anyone take play in this way as a serious variant of Go.

Just changing the size of the grid leads to games that feel very different from a standard 19x19 game.

You shouldn't let that really dissuade you from exploring weird variants of Go, the game was originally played on a 17x17 board and I've seen people play on up to 31x31 on KGS at times.

These games are vastly different strategically. At 23x23 and greater the rush for the corners and sides feel less important than the fight for the center. Life is easy to be had and the game feels fairly relaxed and open.

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Looks like he's not active anymore, but the kgs archives still have game records for user beerslayer, who played almost exclusively 37x37 games. –  Gregor Jun 9 '14 at 17:30
I tried to play a lot of 37 x 37 games on KGS. People would always resign or escape 100 moves in, so I never got a feel for it. –  coolpapa Jun 12 '14 at 1:05

The implications in the game are that board boundaries are used by players for defensive purposes (corner and side enclosures).

The game usually starts with corner approaching moves because turns spent in the corner are more point efficient as they protect more territory with fewer moves (with the help of the board boundary).

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Although not strictly a new topology, I often play where the regular bound board is present, but on top of that draw n lines diagonally through points, connecting them. Through having a bottom left to top right, and a top left to bottom right diagonal go through the centre point. Through having 3 to 5 of these pairs of lines, they significantly change the strategy, also making it more interesting to amateur players. Of course this has significant FTA(first turn advantage), which is another reason its useful with large skill differences.

Finally, with the few diagonal lines running along the board, connect these lines at the edges so that they are continuous, while the normal vertical/horizontal grid is bound.

NB:Alternative rule: Territory only needs to surround the regular vertical/horizontal lines, not the diagonals.

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So a number of nodes have 6 liberties? –  mart Oct 19 '14 at 19:32
It is like a regular Goban, except with an "X" cross through the middle points, and this X is fractal, with multiple diagonals. Points that are crossed by no diagonal have 4, points crossed by 1 diagonal have 6, and points at the intersection of the two diagonals(like the centre of the X) have 8 liberties. –  Brayton Oct 20 '14 at 6:03

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