A Peano-based space-filling surface of fractal dimension three. (March 2023)

• Record Type: Journal Article
• Authors: Paulsen, William
• Is Part Of: Chaos, solitons and fractals. Volume 168(2023)
• Publication Date: 2023-03
• Subjects: 28A80; Fractals -- Hilbert curve -- Peano curve -- Space filling curves -- Space filling surfaces -- Omnichromatic picture; Chaotic behavior in systems -- Periodicals; Solitons -- Periodicals; Fractals -- Periodicals; Chaotic behavior in systems; Fractals; Solitons; Periodicals; 003.7
• DOI: 10.1016/j.chaos.2023.113130
• Languages: English
• ISSNs: 0960-0779

Abstract

Abstract: 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. Highlights: 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.