  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 2D
fragments, from mud cracks to Earth's tectonic plates, has two attractors:
"Platonic" quadrangles and "Voronoi" hexagons. In 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.
