  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.
