A Peano-based space-filling surface of fractal dimension three

Although space-filling curves are well known, and have many applications in parallel computing and data mapping, there is a need for a space-filling surface that is a continuous mapping from two-dimensional domain onto the unit cube. This would allow efficient implementation of a 2D problem on parallel processors which are interconnected into a 3D grid. Such a surface is presented in this paper, which uses Hilbert’s geometric approach to generate a mapping from a unit square to a triangular prism. Using two such mappings we can create a mapping from a rectangle to a unit cube. To culminate, we use the mapping to produce a continuous omnichromatic picture, that is, one for which the colors change continuously, and under sufficient resolution, contains every possible RGB value. •We construct a continuous mapping from a rectangle onto a cube.•Construction is Peano-like, defined recursively using square partitions.•Mapping is easily implemented with a computer program.•Mapping can be used to create an omnichromatic picture, containing all colors.