Given a finite square grid, I can rotate it and for any angle expressible as a Pythagorean triple that fits in to the grid, points on the rotated grid will align to the original.
For an infinite grid, I believe any angle
atan(a/b) can be obtained where a and b are integers.
This would imply that given integers
(a,b), there exists a Pythagorean triple
(r.a)^2 + (r.b)^2 = c^2 where c is an integer, and r is rational?
Q) Does this also occur on a hexagonal grid, or are the only alignments of a hexagonal grid those from rotational symmetry, at multiples of