SPINES, SPLINES, AND SINES: MODELING THE GROWTH OF LIVING AND FOSSIL ECHINOIDS

The simulated growth of an echinoid skeleton is implemented in an object-oriented programming language (C++) and run on a Windows-based PC. The echinoid skeleton is modeled as a tessellation over the surface of a sphere. The model incorporates various types of plates, including the genital and ocular plates, oral and anal plates, and ambulacral and interambulacral coronal plates. The basic geometric elements are (1) a set of plate-growth center points, and (2) a set of plate nodes representing the juncture of three adjacent plates. Any two connected nodes form the edge of two plates, and the collection of edges associated with a center point form a polygon representing a plate. The geometric relationship between the plates, i.e., the plate adjacency, is modeled using a Delaunay triangulation of plate-growth center points, and the results are maintained in two interrelated data arrays. One array is composed of the center points (plates) associated with each node, and the other array is composed of the nodes connected to each node. Each array is highly structured and indexed by node number, and all other geometric relationships are derived from them. Individual plate growth (in the simplest case) is defined by a linear growth function dependent on plate perimeter. Plate insertion is controlled by a non-linear diffusion function calculating a hypothetical morphogen concentration derived from adjacent plates and a threshold criterion for nucleation. Plate interactions, including migration of plates away from the apical system towards the mouth during growth, are calculated along the geodesics (great circles) of the sphere, and all plate perimeter and area calculations are made over corresponding spherical patches. A characteristic echinoid shape (for display purposes) is calculated by applying an affine deformation to the sphere. The resulting models can incorporate more than a thousand individual plates and realistically simulate growth in simple regular echinoids, including some Paleozoic forms.