Wilson's proof of a finite upper bound on c using strips going in a 22.5 degree direction... will also show there is a finite upper bound on c for the problem using ANY rotation angle and ANY translative offset, to define the grid G2 from G1. The strips need to go in the angular direction exactly midway between the x-directions for G1 and for G2. All strips same width. The strip width is chosen so that a G1 gridline parallel to the y axis intersects the strip at a line segment of length=1, open on one end, closed on other. Result, c<=3 in all these problem variants. But wait, there is more. You can also pose Wilson's problem in dimensions 3,4, etc. I claim we can induct on dimension to prove a finite upper bound on c, for any fixed dimension, any and grid G2 defined as G1 with any given translative offset, and any given rotation, where G1 allowed to be any nondegenerate point lattice. To explain this in 3 dimensions where G1=Z^3 simple cubic lattice: Divide the world into slabs using parallel planes. All slabs same width. The planes need to be oriented exactly midway between the xy planes for grid G1, and the xy planes for grid G2, i.e. using the square root of the rotation matrix. The width needs to be chosen so each G1 gridline parallel to z-axis cross slab at a line segment of length=1, closed at one end, open at other. Great. Now WITHIN each slab, we have a 2-dimensional Wilson-esque problem, up to (a) bounded-distance perturbations of each grid point away from its unperturbed grid location, and (b) the two grids G1 and G2 have been distorted by an affine, they are no longer square grids. But they still are geometrically identical lattices, just rotated with respect to one another, and that is all we really needed. QED. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)