toroidal go / maths

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.

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.