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Hilbert curve

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 th curve is , i.e., the length grows exponentially with , even though each curve is contained in a square with area . File:Hilbert curve 1.svg|Hilbert curve, first order File:Hilbert curve 2.svg|Hilbert curves, first and second orders File:Hilbert curve 3.svg|Hilbert curves, first to third orders File:Hilbert curve production rules!.svg|Production rules File:Hilbert.png|Hilbert curve, construction color-coded File:Hilbert3d-step3.png|A 3-D Hilbert curve with color showing progression File:Courbe de Hilbert.jpg|Variant, first three iterationsBourges, Pascale. "[http://pascale.et.vincent.bourges.pagesperso-orange.fr/fractales%20et%20chaos1/Chapitre%201.htm Chapitre 1: fractales]", ''Fractales et chaos''. Accessed: 9 February 2019. Both the true Hilbert curve and its discrete approximations are useful because they give a mapping between 1D and 2D space that preserves locality fairly well. This means that two data points which are close to each other in one-dimensional space are also close to each other after folding. The converse cannot always be true. Because of this locality property, the Hilbert curve is widely used in computer science. For example, the range of IP addresses used by computers can be mapped into a picture using the Hilbert curve. Code to generate the image would map from 2D to 1D to find the color of each pixel, and the Hilbert curve is sometimes used because it keeps nearby IP addresses close to each other in the picture.

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Geohash
Geohash is a public domain geocode system invented in 2008 by Gustavo Niemeyer which encodes a geographic location into a short string of letters and digits. Similar ideas were introduced by G.M. Morton in 1966. It is a hierarchical spatial data structure which subdivides space into buckets of grid shape, which is one of the many applications of what is known as a Z-order curve, and generally space-filling curves. Geohashes offer properties like arbitrary precision and the possibility of gradually removing characters from the end of the code to reduce its size (and gradually lose precision).
Space-filling curve
In mathematical analysis, a space-filling curve is a curve whose range reaches every point in a higher dimensional region, typically the unit square (or more generally an n-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, space-filling curves in the 2-dimensional plane are sometimes called Peano curves, but that phrase also refers to the Peano curve, the specific example of a space-filling curve found by Peano.
Fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called affine self-similar.
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