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.