Plato’s cube and the natural geometry of fragmentation

Abstract:

Plato envisioned Earth’s building blocks as cubes, a shape rarely found in nature. The solar system is littered, however, with distorted polyhedra—shards of rock and ice produced by ubiquitous fragmentation. We apply the theory of convex mosaics to show that the average geometry of natural two-dimensional (2D) fragments, from mud cracks to Earth’s tectonic plates, has two attractors: “Platonic” quadrangles and “Voronoi” hexagons. In three dimensions (3D), the Platonic attractor is dominant: Remarkably, the average shape of natural rock fragments is cuboid. When viewed through the lens of convex mosaics, natural fragments are indeed geometric shadows of Plato’s forms. Simulations show that generic binary breakup drives all mosaics toward the Platonic attractor, explaining the ubiquity of cuboid averages. Deviations from binary fracture produce more exotic patterns that are genetically linked to the formative stress field. We compute the universal pattern generator establishing this link, for 2D and 3D fragmentation. Our findings explain the relative abundance of distinct fracture patterns on Earth, and open the possibility of inferring stress conditions on other planetary bodies from images of surface fracture patterns. More personally, this project has been an incredible and unexpected journey following the science into the philosophical realm: It turns out that Plato’s conception about the element earth being made up of cubes is, literally, the statistical average model for real earth. And that is just mind-blowing.