Visualization of Developmental Processes by Extrusion in Space-time

Developmental processes in nature may involve complex changes in the topology, shape, and patterns of growing structures. Processes taking place in one or two dimensions can be visualized as objects in three-dimensional space, obtained by extruding the growing structures along a line or curve representing the progress of time. In this paper, we extend the notion of L-systems with turtle interpretation to facilitate the construction of such objects. This extension is based on the interpretation of the entire derivation graph generated by an L-system, as opposed to the interpretation of individual words. We illustrate the proposed method by applying it to visualize the development of compound leaves, a sea shell with a pigmentation pattern, and a filamentous bacteria. In addition to serving as visualization examples, these models are of interest on their own. The sea shell model uses an L-system to express a reaction-diffusion process, thus relating these two models of morphogenesis. The model of bacteria, which is also of the reaction-diffusion type, sheds new light on one of the basic problems of morphogenesis, the formation of equally spaced organs in a developing medium.