I came up with the idea of non-euclidean chess(the chess board I'm working on will have 2D elliptical geometry) but I came across a problem. The question: What kind of grid do I put on it(what shapes will tesselate on it, how many are placed around a vertex, etc.) ? The grid on globes only cause parallel lines to intersect at only two points, the poles, which doesn't normally happen on this geometry. If I plot two points, from which I draw two parallel lines, they should intersect after a certain amount of distance, no matter where I place them on the sphere, aslong as the two points remain the same distance from each other, and the lines drawn from them are "parallel". I tried to do this, but I usually failed in one way or another. The goal is to find the right shape(or shapes) that will correctly tesselate on a sphere that will give me that property. I have a list of other desired properties, but it is most likely I won't get them all, and it's listed from most important to least:
- point A and B, are x distance apart, with two parallel lines drawn through each one, and the two lines will intersect after d distance from point C placed equidistant between point A and B. No matter where they are placed on the sphere.
- The tessellated shape used are quadrilaterals(with these shapes, making a pawn go forward is easy. One option. But with odd sided shapes, it gives them a choice, which breaks the idea of elliptical geometry, and could even be hyperbolic, in a way.)
- Rotationally symmetrical
- One type of shape used
- The angles of the type of shape used each have the same amount of degrees(Simply because I want to. The others have logical reasons. except for maybe the one before this one. It's technically not needed, it just has might some weird, unwanted effects on the board if this goal is not met)