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 th curve is , i.e., the length grows exponentially with , even though each curve is contained in a square with area .
^D. Hilbert: Über die stetige Abbildung einer Linie auf ein Flächenstück. Mathematische Annalen 38 (1891), 459–460.
^G.Peano: Sur une courbe, qui remplit toute une aire plane. Mathematische Annalen 36 (1890), 157–160.
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and Zhang, Tianrong. "On the Fractal Structure of the Boundary of Dragon Curve". Archived from the original on 14 June 2011. Retrieved 9 February 2019...
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