Plato’s cube and the natural geometry of fragmentation Gabor´ Domokosa,b, Douglas J. Jerolmackc,d,1 , Ferenc Kune , and Janos´ Tor¨ ok¨ a,f aMTA-BME Morphodynamics Research Group, Budapest University of Technology and Economics, 1111 Budapest, Hungary; bDepartment of Mechanics, Materials and Structure, Budapest University of Technology and Economics, 1111 Budapest, Hungary; cDepartment of Earth and Environmental Science, University of Pennsylvania, Philadelphia, PA 19104; dMechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104; eDepartment of Theoretical Physics, University of Debrecen, 4032 Debrecen, Hungary; and fDepartment of Theoretical Physics, Budapest University of Technology and Economics, 1111 Budapest, Hungary Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved May 27, 2020 (received for review January 17, 2020) Plato envisioned Earth’s building blocks as cubes, a shape rarely soil, the fracture mosaics cut into stressed landscapes (Fig. 3) found in nature. The solar system is littered, however, with form pathways for focused fluid flow, dissolution, and erosion distorted polyhedra—shards of rock and ice produced by ubiqui- that further disintegrate these materials (33, 34) and reorga- tous fragmentation. We apply the theory of convex mosaics to nize landscape patterns (35, 36). Moreover, fracture patterns show that the average geometry of natural two-dimensional (2D) in rock determine the initial grain size of sediment supplied to fragments, from mud cracks to Earth’s tectonic plates, has two rivers (36, 37). attractors: “Platonic” quadrangles and “Voronoi” hexagons. In Experiments and simulations provide anecdotal evidence that three dimensions (3D), the Platonic attractor is dominant: Remark- the geometry of fracture mosaics is genetically related to the ably, the average shape of natural rock fragments is cuboid. formative stress field (38). It is difficult to determine, however, When viewed through the lens of convex mosaics, natural frag- if similarities in fracture patterns among different systems are ments are indeed geometric shadows of Plato’s forms. Simulations more than skin-deep. First, different communities use different show that generic binary breakup drives all mosaics toward the metrics to describe fracture mosaics and fragments, inhibiting Platonic attractor, explaining the ubiquity of cuboid averages. comparison among systems and scales. Second, we do not know Deviations from binary fracture produce more exotic patterns that whether different fracture patterns represent distinct univer- are genetically linked to the formative stress field. We compute sality classes or are merely descriptive categories applied to a APPLIED PHYSICAL SCIENCES the universal pattern generator establishing this link, for 2D and pattern continuum. Third, it is unclear if and how 2D systems 3D fragmentation. map to 3D. Here, we introduce the mathematical framework of convex Gomb¨ oc¨ j statistical physics j fracture mechanics j pattern formation mosaics (39) to the fragmentation problem. This approach relies on two key principles: that fragment shape can be well approxi- mated by convex polytopes (24) (2D polygons and 3D polyhedra; olids are stressed to their breaking point when growing crack Fig. 2A) and that these shapes must fill space without gaps, since Snetworks percolate through the material (1, 2). Failure by fragmentation may be catastrophic (1, 3) (Fig. 1), but this pro- cess is also exploited in industrial applications (4). Moreover, Significance fragmentation of rock and ice is pervasive within planetary shells (1, 5, 6) and creates granular materials that are literally building We live on and among the by-products of fragmentation, from blocks for planetary surfaces and rings throughout the solar sys- nanoparticles to rock falls to glaciers to continents. Under- tem (6–10) (Fig. 1). Plato postulated that the idealized form of standing and taming fragmentation is central to assessing Earth’s building blocks is a cube, the only space-filling Platonic natural hazards and extracting resources, and even for land- solid (11, 12). We now know that there is a zoo of geometri- ing probes safely on other planetary bodies. In this study, cally permissible polyhedra associated with fragmentation (13) we draw inspiration from an unlikely and ancient source: (Fig. 2). Nevertheless, observed distributions of fragment mass Plato, who proposed that the element Earth is made of cubes (14–17) and shape (18–21) are self-similar, and models indi- because they may be tightly packed together. We demon- cate that geometry (size and dimensionality) matters more than strate that this idea is essentially correct: Appropriately aver- energy input or material composition (16, 22, 23) in producing aged properties of most natural 3D fragments reproduce the these distributions. topological cube. We use mechanical and geometric models Fragmentation tiles the Earth’s surface with telltale mosaics. to explain the ubiquity of Plato’s cube in fragmentation and Jointing in rock masses forms three-dimensional (3D) mosaics to uniquely map distinct fragment patterns to their formative of polyhedra, often revealed to the observer by two-dimensional stress conditions. (2D) planes at outcrops (Fig. 2). The shape and size of these polyhedra may be highly regular, even approaching Plato’s cube, 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 or resemble a set of random intersecting planes (24). Alterna- J.T. analyzed data; and G.D., D.J.J., F.K., and J.T. wrote the paper.y tively, quasi-2D patterns, such as columnar joints, sometimes The authors declare no competing interests.y form in solidification of volcanic rocks (25). These patterns have been reproduced in experiments of mud and corn-starch This article is a PNAS Direct Submission.y cracks, model 2D fragmentation systems, where the following Published under the PNAS license.y have been observed: Fast drying produces strong tension that 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 drives the formation of primary (global) cracks that criss-cross Open Science (https://osf.io/h2ezc/). All code for the geometric simulations—i.e., the the sample and make “X” junctions (25–27) (Fig. 3); slow dry- cut and break models—is free to download from GitHub (https://github.com/torokj/ ing allows the formation of secondary cracks that terminate Geometric fragmentation).y at “T” junctions (26); and “T” junctions rearrange into “Y” 1 To whom correspondence may be addressed. Email: [email protected] junctions (25, 28) to either maximize energy release as cracks This article contains supporting information online at https://www.pnas.org/lookup/suppl/ penetrate the bulk (29–31) or during reopening–healing cycles doi:10.1073/pnas.2001037117/-/DCSupplemental.y from wetting/drying (32) (Fig. 3). Whether in rock, ice, or www.pnas.org/cgi/doi/10.1073/pnas.2001037117 PNAS Latest Articles j 1 of 8 Downloaded by guest on September 24, 2021 A E B F C G D H Fig. 1. Fragmentation across planets and scales. A–D show planetary surfaces and rings. (A) Saturn’s rings composed of ice (Inset). Image credit: NASA/JPL- Caltech/Space Science Institute. (B) Jupiter’s moon Europa showing cracked planetary shell. Image credit: NASA/JPL-Caltech/SETI Institute. (C) Polygonal cracks on Pluto. Image credit: NASA/JHUAPL/SwRI. (D) Surface of the asteroid Bennu. Image credit: NASA/Goddard/University of Arizona. E–H show example processes forming fragments on Earth. (E) Iceberg calving. Image credit: Australian Antarctic Division/Ian Phillips. (F) Rock falls. Reproduced from ref. 58, which is licensed under CC BY 4.0.(G) Volcanic eruptions that produce pyroclastic flows, forming breccia deposits (Inset). Image credit: US Geological Survey/Peter Lipman. Inset image credit: Siim Sepp (http://www.sandatlas.org/). (H) Mine blasting. Image credit: Sarolta Bodor (photographer). fragments form by the disintegration of solids. Without loss of faces. In contrast to other descriptions of fracture networks (24), generality (SI Appendix, section 1.1), we choose a model that our framework does not delineate stochastic from deterministic ignores the local texture of fracture interfaces (40, 41). Frag- mosaics; networks made from random or periodic fractures may ments can then be regarded as the cells of a convex mosaic have identical parameter values (Fig. 3). This theory provides a (39), which may be statistically characterized by three param- global chart of geometrically admissible 2D and 3D mosaics in eters. Cell degree (v¯) is the average number of vertices of the the symbolic plane. Although we focus here on mosaics formed polytopes, and nodal degree (n¯) is the average number of ver- by fracture, these global charts include all geometrically possible tices that meet at a node (42): We call [¯n,v ¯] the symbolic plane. mosaics, including human-made ones. We define the third parameter 0 ≤ p ≡ NR=(NR + NI ) ≤ 1 as In this paper, we measure the geometry of a wide variety of the regularity of the mosaic. NR is the number of regular nodes natural 2D fracture mosaics and 3D rock fragments and find that in which cell vertices only coincide with other vertices, corre- they form clusters within the global chart. Remarkably, the most sponding in 2D to “X” and “Y” junctions with n= 4 and 3, significant cluster corresponds to the “Platonic attractor”: frag- respectively. NI is the number of irregular nodes where vertices ments with cuboid averages. Discrete element method (DEM) lie along edges (2D) or faces (3D) of other cells, correspond- simulations of fracture mechanics show that cuboid averages ing in 2D to “T” junctions of n = 2 (Fig. 2B). We define regular emerge from primary fracture under the most generic stress field. and irregular mosaics as having p = 1 and p = 0, respectively.