Narrow Results

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.

To construct symmetrical patterns on the unit sphere from the planar iterative function systems (IFSs), we present a method of constructing IFSs with $D_3$ symmetry which is composed of three-fold rotational symmetries together with reflections. An algorithm is developed to generate strange attractors with $D_3$ symmetry on a triangular face and then project it onto the surface of the unit sphere to form aesthetics patterns with spherical symmetry. As an illustrative example, we consider the regular inscribed icosahedron in the unit sphere which contains 20 triangular faces. This method is valid to randomly generate aesthetic spherical patterns using planar IFSs.