Dunno whether this is a good problem for MO anyway. Well, here it goes: On a screen you have only little squares, so a natural question is which n-omino approximates a circle best. The problem is less finding a solution but more to find a valid measure for the circle-ness of a n-omino. Boundary length is obviously no good. Boundary length of the convex hull seems better - only that the hull tends to go across Eric-Half-A-Pixel and I'd prefer a measure that could be defined in taxicab geometry. Suggestions?

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computational metrology. E.g., "Shape based circularity measures of planar point sets"; "Measure of circularity for parts of digital boundaries and its fast computation"; "Discrete circularity measure"; etc. – Joseph O'Rourke Jul 3 '11 at 13:53