The Geometry of Macroevolution: Phenotypic Evolution on Non-Euclidean Manifolds

AbstractPhylogenetic comparative methods typically rely on an often unstated and potentially unrealistic assumption: that phenotypes evolve within a flat Euclidean space. We advocate for explicitly considering the "geometry of macroevolution," proposing that complex developmental and genetic constraints may cause phenotypes to evolve on curved, non-Euclidean manifolds. On such manifolds, the shortest path between two forms (the geodesic) is not a straight line. We demonstrate how measuring evolutionary divergence on a curved manifold with an inappropriate Euclidean metric can systematically underestimate true path lengths, creating analytical artifacts. Specifically, this geometric distortion can produce the appearance of declining evolutionary rates over time, offering a novel complementary explanation for widely observed patterns like age-rate scaling. Characterizing this geometry can be approached through a priori theoretical models or empirically through data-driven manifold learning. The convergence of large-scale phenomic datasets and machine learning is making it increasingly feasible to infer this geometric structure directly from data. Our goal is to encourage the field to move from implicitly assuming a geometry to deliberately characterizing it, ensuring that inferred macroevolutionary patterns reflect biological reality rather than the constraints of our analytical framework.