![]() ![]() Next I'm relaxing the restriction: I'll let the vertices fall also on the middle points of the sides of paper squares. So, the sheet of paper in the top-left corner of the figure collects my best results of this family. In fact, I didn't really find head-to-head confrontations whose "winner" depended on the scoring details, and even if I did I would have happily declared a tie and kept all the alternatives. Anyway I've found substantial agreement among different strategies. Moreover I tested different size factors to take into account the fact that getting better shape approximations with bigger figures is somewhat obvious and also less useful for sketching. ![]() I put no requirements about the center should it happen to fall on or close to a crossing, even better. I've explored different ways to quantitatively assess the "precision" of an approximation, starting from length uniformity of sides (satisfactory only for triangles) later combined with uniformity of angles, to the mean squared distance of vertices from the ideal ones not lying on the grid. Apart from the square, it is not possible to get regular polygons with this restriction, so the intent is to obtain the closest approximations keeping the figures "small". The rule is simple: the vertices must lie at the grid crossings, that is their coordinates must be expressible as integer numbers. I'm attaching a picture of my best findings, intentionally left handmade to show the kind of result I'm pursuing (following an answer I introduced a couple of digital adjustments). ![]() In particular I'm thinking of regular polygons and circles. So I began looking for ways to sketch geometric figures as precisely as possible without using compass and/or ruler. I find myself often fooling around with pen and paper, preferably squared paper. ![]()
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