Hilbert Curve

A Hilbert curve (or Hilbert space-filling curve) is a continuous fractal curve first described by the German mathematician David Hilbert in 1891, as a variant of the space-filling curves discovered a year earlier by Giuseppe Peano. It is built recursively: at each iteration, every straight segment of the previous curve is replaced by a smaller, rotated copy of the whole curve, so the path folds back on itself more and more finely while always covering the square it lives in without ever crossing itself. Because it is space-filling, its image is the entire unit square (a 2-dimensional shape) even though it is traced by a single 1-dimensional path — giving it a Hausdorff dimension of exactly 2. Its locality-preserving property (points close together along the curve tend to be close together in the plane, and vice versa) makes it useful well beyond pure mathematics, in areas like image compression, database indexing, and network routing.