Analysis of Turing Patterns on a Spherical Surface Using Polyhedron Approximation

We considered Turing patterns on a spherical surface from the viewpoint of polyhedron geometry. We restrict our consideration to a set of parameters that produces a pattern of spots. We obtained numerical solutions for the Turing system on a spherical surface and approximated the solutions to convex polyhedrons. The polyhedron structure was dependent on both the radius of the sphere R and the initial condition. The number n of faces of the polyhedron increased with an increase in R. For small values of R, highly ordered structures were observed. With an increase of the value of R, a variety of structures were observed for each n, and the symmetry property of the spots, which determined the regularity of the polyhedron structure, gradually disappeared. We classified the numerical results according to their symmetrical properties of the approximated polyhedrons. The results revealed that the obtained Turing patterns lost symmetrical properties and varied the structures within same number of spots.