Optimal and stable random pattern formations

• Published in: Journal of theoretical biology Vol. 422; pp. 12 - 17
• Main Author: Mityushev, Vladimir
• Format: Journal Article
• Language: English
• Published: England Elsevier Ltd 07-06-2017
• Subjects: Biological structure; Discrete energy; Models, Biological; Optimal graph; Pattern formation; Random graph; Voronoi tessellation; Pattern formation; Biological structure; 92C15; Voronoi tessellation; Discrete energy; 52C15; Optimal graph; 68R10; Random graph
• ISSN: 0022-5193; 1095-8541
• DOI: 10.1016/j.jtbi.2017.04.010

•Graph model to describe the vast range of patterns is developed.•Optimal graphs which determine biological structures are described.•It is explained why nature prefers the hexagonal array over the square one.•Why the leopard skin has rather hexagonal structure than square one.
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.