spacefilling

Spacefilling curves are mathematical objects that map a one-dimensional parameter onto a higher-dimensional space in a continuous and surjective way. The prototypical case is a curve that fills a two-dimensional region such as the unit square. Formally, a spacefilling curve is a continuous surjection from the unit interval to the unit square, f: [0,1] → [0,1]^2. Although the domain is one-dimensional, the image has two-dimensional extent and positive area, and the mapping is highly non-smooth.

History and construction. The first explicit spacefilling curve was the Peano curve, constructed by Giuseppe Peano in 1890. David Hilbert later provided another classic construction in 1891. Both are obtained as limits of polygonal paths that traverse finer and finer grids; at each stage the path visits every small cell of the grid, and the limit as the grid size tends to zero yields a continuous surjection onto the target region.

Properties and applications. Spacefilling curves are continuous but not rectifiable: their length is infinite, and the mapping is not one-to-one, with many parameter values mapping to the same point. They are fractal-like and exhibit self-similarity in their construction. In practice, such curves are used for spatial indexing and data layout, for example Hilbert and Morton (Z-order) orders, to map two-dimensional data to one dimension while preserving locality; they also appear in image processing, computer graphics, and numerical simulations.