Growth of a Tessellation: Geometric rules for the Development of Stingray Skeletal Patterns.

Abstract

The skeletons of sharks and rays, fashioned from cartilage, and armored by a veneer of mineralized tiles (tesserae) present a mathematical challenge: How can the continuous covering be maintained as the skeleton expands? This study, using microCT and custom visual data analyses of growing stingray skeletons, systematically examines tessellation patterns and morphologies of the many thousand interacting tesserae covering the hyomandibula (a skeletal element critical to feeding), over a two-fold developmental change in hyomandibula length. The number of tesserae remains surprisingly constant, even as the hyomandibula expands isometrically, with all hyomandibulae displaying self-similar distributions of tesserae shapes/sizes. Although the distribution of tesserae geometries largely agrees with the rules for polyhedra tiling of complex surfaces-dominated by hexagons and a minor fraction of pentagons and heptagons, but very few other polygons-the agreement with Euler's classic mathematical laws is not perfect. Contrary to the assumed uniform growth rate (which is shown would create geometric incompatibilities), larger tesserae grow faster to accommodate skeletal expansion. It is hypothesized that this local regulation of global system complexity is driven by tension (from cartilaginous core expansion) in the fibers connecting tesserae, with strain-responsive cells orchestrating local mineral apposition.