Random wear shapes all pointy tips alike

Despite their ubiquity in nature, spikes or stingers rarely exhibit sharp tips. Instead, a closer look at their roughly conical tips reveals a striking similarity in shape: they follow a power-law profile, z~radius^n, where n ≈ 2. This conformity appears across a wide range of spatial scales and materials. In this work, we use model experiments and continuum theory to demonstrate that this universal tip morphology results from stochastic weathering processes. Specifically, we examine the evolution of the geometry of pencil tips, tracking the transformation of their initial, sharp, linear profile to the convergent universal contour, when subjected to random wear. This finding is particularly significant in light of recent observations of consistent tip profiles of biological stingers (Quan et al., 2024; Evans et al., 2021), the generality of tip contours of spindles formed by dissolution (Nakouzi et al., 2014), and the geomorphic evolution of natural pinnacles due to wet erosion (Huang et al., 2020) – all of which exhibit this universal profile.