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Hilbert curve

Hilbert curve

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First 8 steps toward building the 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,[1] as a variant of the space-filling 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 of course 2 in any definition of dimension; its graph is a compact set homeomorphic to the closed unit interval, with Hausdorff dimension 2).

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

For multidimensional databases, Hilbert order has been proposed to be used instead of Z order because it has better locality-preserving behavior.