A space-filling curve of the given order. Rotation turns the square; the path always visits every cell once.
David Hilbert, 1891
A space-filling curve: a 1D index folded until it visits every cell of a 2ⁿ×2ⁿ grid. Order is how many times the U is folded; rotation turns the square. The path is always a single stroke.
gridfractal
1D foldThe same index, first as a line, then folded into the plane. Order 2 visits every cell of a 4×4.