A Functional Morphospace, Based on Dimensionless Numbers, for a Circumferential, Calcite, Stabilizing Structure in Sand Dollars

A sand dollar, living on sand in fluctuating flow, is at risk of either sliding downstream or flipping over by rotating about its downstream edge. Stability on the sediment is enhanced by a belt of calcite located peripherally in the sand dollar. While the calcite belt stabilizes, in part, because of its contribution to the weight of the sand dollar, our paper focuses on how the calciteʹs peripheral distribution increases rotational inertia, which raises the energy and moment required for a destabilizing flow to cause catastrophic dislodgement. We define a dimensionless inertia number as rotational inertia per mass per squared radius of a sand dollar. Inertia per mass is a function of mass distribution, which we describe using two dimensionless numbers—the location number, which is the inner calcite belt radius divided by the sand dollar radius; and the areal density number, which is the relative mass per area of the peripheral and central regions. We plot a functional morphospace in which performance, quantified by the inertia number, is a function of the two morphological parameters, location number and areal density number. The functional morphospace has local maxima and can be interpreted as an adaptive landscape that predicts directions of morphological evolution under circumstances in which rotational inertia is a selected trait. Our model enables future, comparative, evolutionary tests of the function and adaptive importance of this peripheral calcite belt.