Pointless optima in Nature: How stochastic mechanical processes shape universal geometries

John Sebastian, PhD Student, Department of Physics – FLUIDS Section, Technical University of Denmark (DTU)

Sharp, spine-like structures, or stingers, are widespread in nature, appearing in organisms as diverse as tiny zooplankton, elephants, honeybees, and narwhals. These pointed shapes are an example of convergent evolution, where different species independently develop similar solutions, in this case, a simple yet effective defence mechanism. Remarkably, although stingers vary in size, they exhibit the same tip geometry - a power-law profile, z ~ radius^n, where n ≈ 2. 

This cross-species similarity has recently been attributed to evolutionary selection pressures that favour efficient penetration into soft substrates.

However, we show that this characteristic geometry can arise from a more elementary physical mechanism: stochastic mechanical wear.

Using model experiments and continuum theory, we demonstrate that stochastic weathering alone is sufficient to account for this universal shape, regardless of material properties or initial geometry. This convergent form isn’t unique to biological stingers; the same shape appears in natural structures like icicles, sugar spindles (popsicles) and stone pinnacles formed by erosion. To reproduce the effect in the lab, we subjected sharpened pencil tips - a model stinger - to prolonged random collisions and vibrations and tracked their geometric evolution over time. The tips consistently evolved toward the same universal profile. Moreover, by modelling the evolution of the geometric envelope as a free-boundary problem, we find that this framework also explains the persistence of self-similar shapes in non-conical structures such as teeth and horns.

The emergence of a universal geometry as an inevitable mechanical consequence of use, which coincidentally takes the guise of an optimised shape, offers a cautionary tale in biomimetics and suggests that temporal stochastic processes may play a more fundamental role in shaping form in Nature than adaptive narratives alone would imply.