Hilbert curve

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

Because it is space-filling, its Hausdorff dimension is ${\displaystyle 2}$ (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).

${\displaystyle H_{n}}$ is the ${\displaystyle n}$th approximation to the limiting curve. The Euclidean length of ${\displaystyle H_{n}}$ is ${\displaystyle \textstyle 2^{n}-{1 \over 2^{n}}}$, i.e., it grows exponentially with ${\displaystyle n}$, while at the same time always being bounded by a square with a finite area.

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