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Currently I have a plane with square tiling. It pretty trivial to get point on plane mapped to vertex of square tiling: plane point (x; y) -> vertex of square tiling (x div A; y div A). How to get same, but for Triangular tiling?

This is important! As you can see, there may be many (up to 4) closest points in square tiling, but by using quotient I map center of square just to 1 point, not four.

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