Pattern formations and optimal packing

TL;DR

This paper links pattern formations in reaction-diffusion systems to optimal random packing of disks, using a discrete network approximation and energy minimization to classify and construct optimal packings.

Contribution

It introduces a novel approach connecting reaction-diffusion pattern formation to geometric packing problems via discrete network models and graph isomorphism classes.

Findings

Reaction-diffusion patterns correspond to classes of optimal random packings.

Discrete energy minimization identifies unique optimal packings within each class.

Finite disk count leads to a finite number of optimal packing solutions.

Abstract

Patterns of different symmetries may arise after solution to reaction-diffusion equations. Hexagonal arrays, layers and their perturbations are observed in different models after numerical solution to the corresponding initial-boundary value problems. We demonstrate an intimate connection between pattern formations and optimal random packing on the plane. The main study is based on the following two points. First, the diffusive flux in reaction-diffusion systems is approximated by piecewise linear functions in the framework of structural approximations. This leads to a discrete network approximation of the considered continuous problem. Second, the discrete energy minimization yields optimal random packing of the domains (disks) in the representative cell. Therefore, the general problem of pattern formations based on the reaction-diffusion equations is reduced to the geometric problem of random packing. It is demonstrated that all random packings can be divided onto classes associated with classes of isomorphic graphs obtained form the Delaunay triangulation. The unique optimal solution is constructed in each class of the random packings. If the number of disks per representative cell is finite, the number of classes of isomorphic graphs, hence, the number of optimal packings is also finite.