From: Allan Wechsler <acwacw@gmail.com> Date: 9/11/20, 4:05 PM
I would think that for big enough N = a^2 + b^2, the moderately obvious "Pythagorean" solution would *not* be optimal, and that you could do better with big patches of points arranged in a triangular lattice.
With this nearly-golden-ratio method, for N < 23, the nearly-square lattices hold their own matching 1/sqrt(n). As N increases, there appear patches of packings where the min distance is > 1/sqrt(N). These do tend to have nearly-equilateral triangular lattices; most of them are not Pythagorean so far. The first one appears at N=23. I only checked up to N=200 (easy to calculate but information overload for me). The best & second best so far are at N=146 and 151. min N distance 1/sqrt(N) ratio sum(sq) lattice type 2 0.707107 0.707107 1.000000 1,1 square 3 0.471405 < 0.577350 0.816497 4 0.500000 0.500000 1.000000 ~triangular 5 0.447214 0.447214 1.000000 1,2 ~square 6 0.372678 < 0.408248 0.912871 7 0.319438 < 0.377964 0.845154 ~triangular 8 0.353553 0.353553 1.000000 2,2 ~triangular 9 0.333333 0.333333 1.000000 10 0.316228 0.316228 1.000000 1,3 ~square 11 0.287480 < 0.301511 0.953463 12 0.250000 < 0.288675 0.866025 13 0.277350 0.277350 1.000000 2,3 ~square 14 0.225877 < 0.267261 0.845154 15 0.240370 < 0.258199 0.930949 16 0.225347 < 0.250000 0.901388 17 0.212091 < 0.242536 0.874475 1,4 18 0.235702 0.235702 1.000000 3,3 19 0.189766 < 0.229416 0.827170 20 0.223607 0.223607 1.000000 2,4 21 0.202031 < 0.218218 0.925820 22 0.203279 < 0.213201 0.953463 (Only those with min d > 1/sqrt(N) shown below.) 23 0.217391 >>> 0.208514 1.042572 ~diamonds 53 0.143694 >>> 0.137361 1.046107 2,7 ~triangular 56 0.135996 >>> 0.133631 1.017700 58 0.134659 >>> 0.131306 1.025536 3,7 61 0.132168 >>> 0.128037 1.032266 5,6 66 0.124943 >>> 0.123091 1.015038 71 0.120338 >>> 0.118678 1.013987 120 0.093169 >>> 0.091287 1.020621 133 0.090538 >>> 0.086711 1.044139 ~diamonds 138 0.087558 >>> 0.085126 1.028577 141 0.085401 >>> 0.084215 1.014085 143 0.084497 >>> 0.083624 1.010435 146 0.087714 >>> 0.082761 1.059853 5,11 ~triangular 151 0.086093 >>> 0.081379 1.057925 ~triangular 154 0.083157 >>> 0.080582 1.031957 156 0.083333 >>> 0.080064 1.040833 159 0.082723 >>> 0.079305 1.043097 161 0.080745 >>> 0.078811 1.024544 162 0.079051 >>> 0.078567 1.006154 9,9 164 0.079502 >>> 0.078087 1.018128 8,10 166 0.078313 >>> 0.077615 1.008996 167 0.079890 >>> 0.077382 1.032409 172 0.076471 >>> 0.076249 1.002903 175 0.077723 >>> 0.075593 1.028175 183 0.074325 >>> 0.073922 1.005450 --Steve