# Toroidal Go

Keywords: Variant

# Description

Toroidal Go (henceforth referred to as “t-Go”) is a Go variant which is played on a graph that can be seen as a normal board where the edges have been “removed” as follows: the edges “wrap around” horizontally and vertically; they link up with each other so all points have 4 neighbours; each point on the far left is linked to the point on the same horizontal line on the far right; similarly for linking up points at the top and the bottom.

The name Toroidal Go comes from the fact that this board fits naturally onto a toroid / doughnut three-dimensional surface.

T-go is just one example of a family of go variants called edgeless Go.

# Strategy

• In 11x11 toroidal Go, aiming to make eyes isn’t always easy, nor is it a good primary goal to aim at. One strategy can be to try to split the opponent into two eyeless groups and then aim at killing one of them. (from a blog post)

In the fuseki as Black I tend to aim for several things at once. I try to develop territorial frameworks (I choose this term instead of “moyo” because on an 11x11 board frameworks have to be fairly small). I try to make them large enough in scale so that White has to make thin shape in order to prevent them from becoming territory. I look out for opportunities to split White into separate groups with thin shape. I also try not to let my shape become too thin.

With White I attempt to limit Black’s territorial development without making overly thin shape; if neither player has large territories at the end there is more chance that the komi will swing the balance in White’s favour.

In n x n t-Go, when the board is represented as an n x n grid, a point on the edge of the grid is linked to the point on the opposite edge, which can be hard to visualise. There is a natural way of expanding the flat n x n grid by adding extra lines around it in order to make visualisation easier. Typically three or four lines are added; the number of lines added could be a matter of personal preference. The extra space added is called 'wraparound' in this article. Each point in the wraparound is just an additional representation of a point in the original grid, which we call the 'main area'. It’s not really necessary to explain more formally, when there are examples readily available (see the next section).

Ideally when wraparound is included, there will also be some way of easily seeing which part of the board is the wraparound area, and which part is the main area.

## Examples of wraparound

• On LittleGolem, three extra lines are added to the 11x11 t-Go boards, and the background colour of the wraparound area is a different colour to the background of the main area.

Move #30 from a game on LittleGolem

Move #72 from a game on LittleGolem

• On a standard 19x19 board, 11x11 t-Go games can easily be played and reviewed by using the first four lines closest to the edges as wraparound; the main area of the board then consists of the 11x11 square whose corners are the four 5-5 points.
• A blog (also discussed below). In the games discussed on this blog, four lines of wraparound are added to 11x11 t-Go; signs are added to the points of wraparound that are adjacent to the main area (the "edge" of the wraparound) in order to distinguish between the wraparound and the main area.

A tsumego - White to play and win - from a blog post - after move #69

• The online viewer/editor for t-Go leaves the choice of the number of wraparound lines up to the person using the page; it also provides a choice of how to display the edge of the wraparound (no marks / box symbols / coordinates).

# Transformations

All points are equivalent, in a toroidal graphs. In formal maths, they satisfy a homogeneity property called being “ vertex transitive” – actually, they satisfy an even stronger homegeneity property which is discussed here.

## Rotations and reflections

No special description is needed here; there are 3 rotations and 4 reflections that can be applied.

## Panning or translation (shifting the board)

The term “panning” from film refers to moving the camera. This is more or less the same as the mathematical term “translation”: in geometry, a translation is a transformation that shifts all points in a certain direction. In t-Go, arbitrary translations can be applied without changing the internal structure of the board. This online viewer/editor for t-Go implements panning.

## Usage; normalisation

The above transformations could be useful for helping to visualise the structure of the board, and also for “normalising” games. “Normalising” here means taking a collection of t-Go games and applying transformations to each game to ensure the first move of each game is at the same point, and that as many moves as possible are in the same triangular region. Formally, for each game there is an N > 1 and a combination of transformations s.t. the first N moves are in the triangle {y>=0; x>=y; x,y < m/2 }, for (m x m) t-Go; normalising the game means applying such transformations for a maximal N.

# Online servers and tournaments

• On the turn-based server littlegolem there is an ongoing, open-ended tournament of t-Go (11x11; 4.5 komi) with 24 players participating as of Oct 2017: link
• One-off tournament on littleGolem with 7 participants: link
• The Go Without Borders project http://www.gowithoutborders.com/ is an online server where players can meet to play 9x9, 13x13, or 19x19 games. Alternatively, users can download the free client application to play two-player games on their own Windows or Mac PCs.
• Torigo server Torigo.io. The edgeless game of Go with "drag & drop" and arrows to navigate easily through the goban. Mobile friendly with 9x9, 13x13 and 19x19 sizes. Challenging AlphaGo for a rematch vs humans.

# Other links and resources

• An online tool for playing, editing and reviewing games (for existing games, you can copy/paste the SGF into the viewer): link
• The code implementing t-Go for the item above is open source: link.
• T-go is called Daoqi / 道棋 by some Chinese people. There is a SL page about it (link). On the Daoqi site there is an SGF editor (with an interface in Chinese) (based on Maxigos) that has been adapted for t-Go ( link).
• The SGF editor q5go supports toroidal boards as of version 0.4. It can save and load t-Go games in a slightly nonstandard format, and you can modify a normal SGF file: find "GM[1]" and replace it with "GMq5go-1?", and add a "TO[3]" property to the root.

## Malcolm's blog

Toroidal Go last edited by 90.91.135.20 on December 11, 2020 - 15:18