07-31-2020, 05:24 AM

Several of my games can alternatively be played on toruses. In general, any of your defined grids can be converted to a torus, by defining a unit cell of the tessellation to be the torus (any unit cell,not just the primary unit cell) You can restrict this unit cell to a parallelogram shape without diminishing the possibilities - but if the cells of the torus are only represented once, or only repeated at the edges some games may have a different preferred shap such as a limping board.

The problem is in representation. There are 4 general methods:

1) show each cell once, do not show the edge linking connections (implicit)

2) As in 1, but add connection lines from opposite edges (messy)

3) As in 1, but repeat cells on one of each pair of opposite edges (doubled edges)

4) Continuous tessellation showing a multiple of unit cells worth of duplicated pieces, with a play region within it highlighted (and reference numbered)

Other topological boards such as mobius strips, Kline bottles, polyhedra (gridded spheres) may also be of interest. - But toruses are the most common and easiest to implement.

The problem is in representation. There are 4 general methods:

1) show each cell once, do not show the edge linking connections (implicit)

2) As in 1, but add connection lines from opposite edges (messy)

3) As in 1, but repeat cells on one of each pair of opposite edges (doubled edges)

4) Continuous tessellation showing a multiple of unit cells worth of duplicated pieces, with a play region within it highlighted (and reference numbered)

Other topological boards such as mobius strips, Kline bottles, polyhedra (gridded spheres) may also be of interest. - But toruses are the most common and easiest to implement.