You can get a pretty nice packing on a square torus using the golden ratio. For the unit square, pick one of these spacings (Yes, they are > 1.) h = floor(N phi) / N or h = ceil(N phi) / N, put the points at (k h (mod 1), k / N) for k = 1 to N, and pick the better of the two packings. They are uniform, which makes it < O(N) to measure the closest spacing.

But what would seem to be the best arrangements for 1 <= N <= 10 on T^2 and T^3 ???

Here are my numbers. The sequence is not monotonic, which means you can sometimes do better by using the next spacing and leaving out a point! Using the best of `floor()` or `ceil()` (on a unit square): N min distance 2 0.7071067811865476 3 0.4714045207910316 4 0.5 5 0.44721359549995765 6 0.3726779962499649 7 0.3194382824999699 8 0.3535533905932738 9 0.3333333333333333 10 0.3162277660168369 11 0.2874797872880344 12 0.25 13 0.27735009811261385 14 0.22587697572631218 I don't know an analogous method for the cube torus. But you *could* "pack" points on a unit interval: k phi (mod 1) for k = 1 to N. ...Why would you want to do that!? you ask. Because you can keep adding points to the interval incrementally, and the neighbor distances (mod 1) are always within some constant factor of 1/N. I have searched but not found a similarly simple way to add points incrementally to the square torus that gets remotely good packing. (I call this the peppering problem.) --Steve

From: Dan Asimov <dasimov@earthlink.net> Date: 9/10/20, 6:26 PM

The Tammes problem is the well-known problem to find the arrangement of N points on the unit sphere S^2 that maximizes the minimum distance between any pair of them.

What about the same question for N points on the square torus

T^2 = R^2/Z^2

or the cubical torus

T^3 = R^3/Z^3

???