| > The equivalent loose approximation on a hex grid would seem to be long direction minus 1/4 short direction, which doesn't seem any harder. As promised in a sibling comment I would work it out on paper. By my path, using the 7/4 approximation for sqrt(3), the ratio between X and Y directions is 12:21 which is much harder to do in your head than the 3:2 that a square grid gives you. However, a simpler approximation of (5:3) with acceptably low error came out. Math (see diagram below) Normalizing your 1.75 approximation to 7 "units" then the sides of the hexagon are of length 7, the distance from center to corner is length 7, and the distance from center to side is 6. This gives a distance from A to other points as: B: 12 (Center -> Side -> Center) C: 12 (Center -> Side -> Center) D: 21 (Center -> Corner -> Corner -> Center) Given that the actual value of A->D is ~20.784 that suggests approximating A->D = 20 alowing us to reduce as follows: B: 3 C: 3 D: 5 This has a ~3.7% error in the horizontal direction. ______ ______
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/ B \______/ \___
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---- C ----
/ \ / \
/ A \______/ D \___
\ / \ /
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---- ----
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/ \______/ \___
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Well, yes, 4:7 is 12:21.
But I think 4:7 is a lot easier to do in your head than 12:21 (3:5 is a little easier, at the expense of being less accurate); not sure why you did all that work to make it more complicated when you started with a fraction that gave you the ratio in its simplest form.