In the past 50 years, L-systems have been successfully used to model the development of filamentous and branching forms in biology. Their success is largely due to the fact that they rely on the mathematical notion of discrete rewriting systems, that in essence simply reflects the idea that the evolution of a structure results from the evolution of its individual components. This core property is reminiscent of how biological organisms develop and happens to be critical to model their growth. The formalism of L-systems has been developed to model the growth of forms in Euclidean 1-D, 2-D, or 3-D spaces. These spaces have the property to be flat and show no curvature anywhere. However, the growth of various forms or processes in biology takes place in curved spaces. This is for example the case of vein networks growing within curved leaf blades, of unicellular tubes, such as pollen tubes, growing on curved surfaces to fertilize 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 growing forms in curved spaces, we thus developed further the theory of L-systems. 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 space, while being intrinsically curved. We suggest that this abstract extension can be used to provide a new approach for effective modeling of tropism phenomena and illustrate this idea on a few conceptual examples.
To play the different examples of the paper online using Binder, press the following link :
This installation is based on Conda. Conda is a package manager that can be installed on Linux, Windows, and Mac. If you have not yet installed conda on your computer, follow these instructions:
Conda Installation. Follow instructions for Miniconda.
Conda Download. Use the Python 3.10 based installation.
Then, after install of conda, run the following command to create an environment with openalea.lpy and its notebooks extension.
conda env create -f binder/environment.yml
Activate the environment using
conda activate riemaniannlsystem
Run the notebook with the following command
jupyter notebook notebooks/PaperExamples.ipynb