I'm in a hotel for one night for work and packed a few solo games. I haven't played TM:AE in a while so I decided to play it first. Wow, I forgot how brutal it can be. I was nowhere near completely terraforming Mars before the rounds ended.
... The valid positions are either all even coordinates or all odd coordinates.
So, your graph paper has a scale of 2 for each line. You can either be on an intersection (all even coordinates) or on a square center (all odd coordinates).
Either way, movement is intuitive. Either move 2 in an orthogonal direction, or (1,1,1) in a diagonal direction.
I'd say this all compares favorably with the better known rhombic dodecahedron tiling.