Growth and form, Lie algebras and special functions

The formation of a biological organism, or an organ within it, can often be regarded as the unfolding of successive equilibria of a mechanical system. In a mathematical model, these changes of equilibria may be considered to be responses of mechanically constrained systems to a change of a reference configuration and of a reference metric, which are in turn driven by genes and their expression. This paper brings together three major threads of research. These are: Lie-type symmetries of equations; models as well as data on growth and pattern formation; and the relation between Lie algebras (and groups) and special functions associated with them. We show that symmetry methods can be generalized to map between solutions to models with different reference metrics. In the case in which we attempt to obtain such equations, they seem too complicated to be of any immediate service to the community of researchers on cortical growth. However, models and data on growth may be used to obtain generators of these Lie algebras empirically and numerically. These generators result in new classes of special functions. The paper is an invitation to develop what we may call empirical Lie algebras and associated functions. The hypothesis that remains to be tested is whether the confluence of ideas described in the paper, namely the Lie algebraic-related consequences of pattern formation and growth, prove useful for deepened understanding of biological growth patterns.