title: Optimal and stable random pattern formations.
abstract: We develop a graph model to describe the vast range of patterns observed in biological structures. For any given number of spotty patterns, a finite number of structures (optimal graphs) is precisely described. The construction of the optimal graphs is based on the minimization of the diffusion dissipation energy. The notion of geometrical stability of structures is introduced. It is demonstrated that the hexagonal array is stable and the square array is not. This explains the reason why the hexagonal array appears more frequently in practice than the square one.
authors.affiliation: Department of Computer Science and Computational Methods, Pedagogical University, Krakow, Poland. Electronic address: http://www.mityu.up.krakow.pl.
authors.forename: Vladimir
authors.initials: V
authors.lastname: Mityushev
mesh_terms: D008954:Models, Biological
keywords: Biological structure; Discrete energy; Optimal graph; Pattern formation; Random graph; Voronoi tessellation