Better explicating the strengths and shortcomings of these models will help refine projections and improve transparency in the years ahead.
Image credit: Witsawat.S.
Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved May 27, 2020 (received for review January 17, 2020)
Significance
We live on and among the by-products of fragmentation, from nanoparticles to rock falls to glaciers to continents. Understanding and taming fragmentation is central to assessing natural hazards and extracting resources, and even for landing probes safely on other planetary bodies. In this study, 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. We use mechanical and geometric models to explain the ubiquity of Plato’s cube in fragmentation and to uniquely map distinct fragment patterns to their formative stress conditions.
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.
Footnotes
- ↵1To whom correspondence may be addressed. Email: sediment{at}sas.upenn.edu.
Author contributions: G.D. designed research; G.D., F.K., and J.T. performed research; G.D., D.J.J., F.K., and J.T. contributed new reagents/analytic tools; G.D., D.J.J., F.K., and J.T. analyzed data; and G.D., D.J.J., F.K., and J.T. wrote the paper.
The authors declare no competing interests.
This article is a PNAS Direct Submission.
Data deposition: Shape and mass data for all measured 3D rock fragments, including both the small and large experimental datasets, are freely available at the Center for Open Science (https://osf.io/h2ezc/). All code for the geometric simulations—i.e., the cut and break models—is free to download from GitHub (https://github.com/torokj/Geometric_fragmentation).
This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.2001037117/-/DCSupplemental.
Data Availability.
Shape and mass data for all measured 3D rock fragments, including both the small and large experimental datasets, are freely available at https://osf.io/h2ezc/. All code for the geometric simulations—i.e., the cut and break models—is free to download from https://github.com/torokj/Geometric_fragmentation.
Published under the PNAS license.
Plato’s cube and the natural geometry of fragmentation
Article Classifications
- Physical Sciences
- Applied Physical Sciences
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