Hilbert curve

The Hilbert curve (also known as the Hilbert space-filling curve) is a continuous fractal space-filling curve first described by the German mathematician David Hilbert in 1891,^[1] as a variant of the space-filling Peano curves discovered by Giuseppe Peano in 1890.^[2]

Because it is space-filling, its Hausdorff dimension is 2 (precisely, its image is the unit square, whose dimension is 2 in any definition of dimension; its graph is a compact set homeomorphic to the closed unit interval, with Hausdorff dimension 2).

The Hilbert curve is constructed as a limit of piecewise linear curves. The length of the ${\displaystyle n}$th curve is ${\displaystyle \textstyle 2^{n}-{1 \over 2^{n}}}$, i.e., the length grows exponentially with ${\displaystyle n}$, even though each curve is contained in a square with area ${\displaystyle 1}$.