Riemannian L-systems: modelling growing forms in curved spaces

In the past 50 years, the formalism of L-systems has been successfully used and developed to model the growth of filamentous and branching biological forms. These simulations take place in classical 2-D or 3-D Euclidean spaces. However, various biological forms actually grow in curved, non-Euclidean, spaces. This is, for example, the case of vein networks growing within curved leaf blades, of unicellular filaments, such as pollen tubes, growing on curved surfaces to fertilise distant ovules, of teeth patterns growing on folded epithelia of animals, of diffusion of chemical or mechanical signals at the surface of plant or animal tissues, etc. To model these forms growing in curved spaces, we thus extended the formalism of L-systems to non-Euclidean spaces. In a first step, we show that this extension can be carried out by integrating concepts of differential geometry in the notion of turtle geometry. We then illustrate how this extension can be applied to model and program the development of both mathematical and biological forms on curved surfaces embedded in our Euclidean space. We provide various examples applied to plant development. We finally show that this approach can be extended to more abstract spaces, called abstract Riemannian spaces, that are not embedded into any higher-dimensional space, while being intrinsically curved. We suggest that this abstract extension can be used to provide a new approach for effective modelling of growth of branching systems within non-uniform substrates and illustrate this idea on a few conceptual examples.