# Hilbert's Curve

See Wikipedia and Kerry Mitchell's explanation.

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 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 $n$th curve is ${2}^{n}-\frac{1}{{2}^{n}}$ i.e., the length grows exponentially with $n$, even though each curve is contained in a square with area $1$

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