In 1891 David Hilbert showed that a single continuous line can come arbitrarily close to every point of a square if you follow a simple, recursive process: Draw a U through four squares. Replace each turn with a smaller U, rotated to join its neighbors, and repeat.

This print shows a curve built on these rules at several levels of recursion.

Computers use space-filling curves like this one to store maps, images and databases because this process keeps neighboring things side by side.