Peano Curve

The Peano curve was the first continuous curve ever shown to pass through every point of a filled square — a genuinely startling result when Giuseppe Peano published it in 1890, since it seemed to contradict the intuitive idea that a 1-dimensional curve could not possibly have the same "size" as a 2-dimensional area. It is built recursively by dividing a square into a 3×3 grid of subsquares and visiting them in a back-and-forth (boustrophedon) order, with each subsquare in turn recursively subdivided the same way, mirrored where needed so the path connects smoothly from one subsquare to the next. Its discovery directly inspired David Hilbert to publish his own, visually tidier space-filling curve (see the Hilbert Curve page on this site) just a year later, based on subdividing into a 2×2 grid instead.