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PeanoCurve

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Line through all cells of the 3^n x 3^n grid in Peano order (integer grid; DataRange maps to reals)

PeanoCurve[n]
Graphics[PeanoCurve[2]] → -Graphics-Table[RegionMeasure[PeanoCurve[n]], {n, 4}] → {8, 80, 728, 6560}Graphics[{Thickness[Large], PeanoCurve[3] /. Line -> BSplineCurve}] → -Graphics-
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