Abstract
By constructing invariant mappings associated with wallpaper groups, this paper presents a simple and efficient method to generate colorful wallpaper patterns. Although the constructed mappings have simple form and only two parameters, combined with the color scheme of orbit trap algorithm, such mappings can create a great variety of aesthetic wallpaper patterns. The resulting wallpaper patterns are further projected by central projection onto the sphere. This creates the interesting spherical patterns that possess infinite symmetries in a finite space.
Highlights
Wallpaper groups are mathematical classification of two-dimensional repetitive patterns
By constructing invariant mappings associated with wallpaper groups, this paper presents a simple and efficient method to generate colorful wallpaper patterns
Patterns with wallpaper symmetry can be widely found in architecture and decorative art [3,4,5]
Summary
Wallpaper groups (or plane crystallographic groups) are mathematical classification of two-dimensional repetitive patterns. In [7], Field and Golubitsky first proposed the conception of equivariant mappings They constructed equivariant mapping to generated chaotic cyclic, dihedral, and wallpaper attractors. We present a simple invariant method to create wallpaper patterns. It has independent mapping form and only two parameters. Escher’s Circle Limits I–IV are unusual and visually attractive because they realized infinity in a finite unit disc Inspired by his arts, we use central projection to project wallpaper patterns onto the finite sphere. We use central projection to project wallpaper patterns onto the finite sphere This obtains the aesthetic patterns of infinite symmetry structure in the finite sphere space.
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