CONSTRUCTION OF SPHERICAL PATTERNS FROM PLANAR DYNAMIC SYSTEMS

We investigated the generation of spherical continuous-tilings of the chaotic attractors or the filled-in Julia sets from the plane mappings. We build three plane mappings, which can be used to construct the continuous patterns on the surfaces of the hexahedron and the unit sphere. We discuss the coordinate transformation for a spatial point between the different coordinate systems and further discuss how to project a spherical point onto a surface of the inscribed hexahedron. We present a method of constructing a spherical pattern with the pattern of a square on the inscribed hexahedron from an arbitrary projective angle, and generate spherical patterns from the three plane mappings. The results show that we can construct a great number of spherical patterns of the chaotic attractors and the filled-in Julia sets from the plane mappings, which are based on the square lattice and meet the requirements of the continuity and the four rotation symmetries on the lattice's boundaries.