Many roads to success: alternative routes to building an economic shell in land snails.

Land snails exhibit an extraordinary variety of shell shapes. The way shells are constructed underlies biological and mechanical constraints that vary across gastropod clades. Here, we quantify shell geometry of the two largest groups, Stylommatophora and Cyclophoroidea, to assess the potential causes for variation in shell shape and its relative frequency. Based on micro-computed tomography scans, we estimate material efficiency through 2D and 3D generalizations of the isoperimetric ratio, quantifying the ratios between area and perimeter of whorl cross-sections (2D) and shell volume and surface (3D), respectively. We find that stylommatophorans optimize material usage through whorl overlap, which may have promoted the diversification of flat-shelled species. Cyclophoroids are bound to a circular cross-section because of their operculum; flat shells are comparatively rare. Both groups show similar solutions for tall shells, where local geometry has a smaller effect because of the double overlap between previous and current whorls. Our results suggest that material efficiency is a driving factor in the selection of shell geometry. Essentially, the evolutionary success of Stylommatophora likely roots in their higher flexibility to produce an economic shell.

Author Correction: Shape evolution of ooids: a geometric model.

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Shape evolution of ooids: a geometric model.

Striking shapes in nature have been documented to result from chemical precipitation - such as terraced hot springs and stromatolites - which often proceeds via surface-normal growth. Another studied class of objects is those whose shape evolves by physical abrasion - the primary example being river and beach pebbles - which results in shape-dependent surface erosion. While shapes may evolve in a self-similar manner, in neither growth nor erosion can a surface remain invariant. Here we investigate a rare and beautiful geophysical problem that combines both of these processes; the shape evolution of carbonate particles known as ooids. We hypothesize that mineral precipitation, and erosion due to wave-current transport, compete to give rise to novel and invariant geometric forms. We show that a planar (2D) mathematical model built on this premise predicts time-invariant (equilibrium) shapes that result from a balance between precipitation and abrasion. These model results produce nontrivial shapes that are consistent with mature ooids found in nature.

How river rocks round: resolving the shape-size paradox.

River-bed sediments display two universal downstream trends: fining, in which particle size decreases; and rounding, where pebble shapes evolve toward ellipsoids. Rounding is known to result from transport-induced abrasion; however many researchers argue that the contribution of abrasion to downstream fining is negligible. This presents a paradox: downstream shape change indicates substantial abrasion, while size change apparently rules it out. Here we use laboratory experiments and numerical modeling to show quantitatively that pebble abrasion is a curvature-driven flow problem. As a consequence, abrasion occurs in two well-separated phases: first, pebble edges rapidly round without any change in axis dimensions until the shape becomes entirely convex; and second, axis dimensions are then slowly reduced while the particle remains convex. Explicit study of pebble shape evolution helps resolve the shape-size paradox by reconciling discrepancies between laboratory and field studies, and enhances our ability to decipher the transport history of a river rock.