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, as a variant of the space-filling Peano curves discovered by Giuseppe Peano in 1890.

Because it is space-filling, its Harsdorf 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 Harsdorf dimension 2).

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

37 seconds / 1000 x 1000 mp4 / Loop / 30 fps #objkt4objkt