Self-similarity in plants: Integrating mathematical and biological perspectives

Self-similarity is a conspicuous feature of many plants. Geometric self-similarity is commonly expressed in terms of affine transformations that map a structure into its components. Here we introduce topological self-similarity, which deals with the configurations and neighborhood relations between these components instead. The topological self-similarity of linear and branching structures is characterized in terms of recurrence systems defined within the theory of L-systems. We first review previous results, relating recurrence systems to the patterns of development that can be described using deterministic context-free L-systems. We then show that topologically self-similar structures may become geometrically self-similar if additional geometric constraints are met. This establishes a correspondence between recurrence systems and iterated function systems, which is of interest as a mathematical link between L-systems and fractals. The distinction between geometric and topological self-similarity is useful in biological applications, where topological self-similarity is more prevalent then geometric self-similarity.