AN EQUILIBRIUM THEORY OF ECHINOID PLATE GEOMETRY

The skeleton of an echinoid can be considered as a tessellation of a regular closed body surface by a collection of polygonal tiles or plates. Overall growth of such a modeled skeleton can be accommodated by the growth of the individual plates and by the insertion (and growth) of additional plates. We demonstrate, via a 3-D computer-based growth simulation, that coronal plate growth is optimized in echinoids, such that the plates can be closely approximated by Voronoi polygons. Plates represented in this manner have the property that every point within the plate is closer to the growth-center of the associated plate than to any other growth center. In a dual sense, the mesh created by connecting plate-growth centers meets the criteria for a spherical Delaunay triangulation. This property constrains relationships between accommodation, insertion, and plate growth to maintain an equilibrium definable in terms of hypothetical developmental processes. Results of the growth simulations explain several characteristics of the echinoid skeleton, including zigzag suture patterns, spiral-like plate insertion, and ambulacral plate compounding. The same growth simulation procedures have been applied successfully to several groups of extinct, somewhat bizarre, Paleozoic echinoids and other pelmatozoan echinoderms. Insights derived from these models can be used to refine current theories of skeletal homology in the echinoderms. For example, models indicate that a common underlying mechanism can describe plate insertion and growth in the echinoid corona and eocrinoid theca. This is counter to the current theory of skeletal homology in which these two skeletal regions are considered to be completely unrelated.