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.
Highlights
Plato envisioned Earth’s building blocks as cubes, a shape rarely found in nature
We draw inspiration from an unlikely and ancient source: Plato, who proposed that the element Earth is made of cubes because they may be tightly packed together. We demonstrate that this idea is essentially correct: Appropriately averaged properties of most natural 3D fragments reproduce the topological cube
The Platonic attractor prevails in nature because
Summary
Fragments form by the disintegration of solids. Without loss of generality (SI Appendix, section 1.1), we choose a model that ignores the local texture of fracture interfaces [40, 41]. While our DEM simulations do not model secondary fracture, we remind the reader that binary breakup drives any initial mosaic toward an irregular primitive mosaic with cuboid averages (SI Appendix, section 1)—emphasizing the strength of the Platonic attractor. The cut model simulated regular primitive mosaics as primary fracture patterns by intersecting an initial cube with global planes (Fig. 6), while the break model simulated irregular primitive mosaics resulting from secondary fragmentation processes. We fit both of these models to the shape descriptor data using three parameters: one for the cutoff in the mass distribution and two accounting for uncertainty in experimental protocols (Materials and Methods and SI Appendix, section 5). We used the cut model to demonstrate how 3D primitive fracture mosaics converge asymptotically toward the Platonic attractor as more fragments are produced (Fig. 6)
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call