3D->2D projection of a cube world?

[Hai-Etlik]

I probably should have mentioned that this is going to be a computer game, not an "actual" board game. I'm not planning on having the entire cube visible to any player at any time; at most, one hemisphere. Players will be able to pan the center of the projection to different points on the cube, so the area in focus will be least distorted at any given time. I've been doing more research, and I think an orthographic projection is going to be the easiest to interpret from the player's seat.

No, orthographic wouldn't work as you wouldn't be able to cover the full sphere, just one hemisphere of it. That would put you in about the same position as the "martian chess" board. In fact you don't need the sphere, you can do an orthographic projection of a sphere directly, it's just an ordinary drawing of a cube without perspective.

The polar aspect of the equidistant azimuthal projection is fairly simple to work with. If you can transform between Cartesian and Spherical coordinate systems, that's pretty much all you need to do.