Peano (1890) constructed the first space-filling curve — a continuous surjection from [0,1] onto the unit square. At each order, the curve visits every 3ⁿ × 3ⁿ grid cell exactly once.
3
50
At order 3: the curve has 729 segments through a 27×27 grid.
Hausdorff dimension = 2 (same as the plane).