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# Plato’s cube and the natural geometry of fragmentation

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.

Solids are stressed to their breaking point when growing crack networks percolate through the material (1, 2). Failure by fragmentation may be catastrophic (1, 3) (Fig. 1), but this process is also exploited in industrial applications (4). Moreover, fragmentation of rock and ice is pervasive within planetary shells (1, 5, 6) and creates granular materials that are literally building blocks for planetary surfaces and rings throughout the solar system (6⇓⇓⇓–10) (Fig. 1). Plato postulated that the idealized form of Earth’s building blocks is a cube, the only space-filling Platonic solid (11, 12). We now know that there is a zoo of geometrically permissible polyhedra associated with fragmentation (13) (Fig. 2). Nevertheless, observed distributions of fragment mass (14⇓⇓–17) and shape (18⇓⇓–21) are self-similar, and models indicate that geometry (size and dimensionality) matters more than energy input or material composition (16, 22, 23) in producing these distributions.

Fragmentation tiles the Earth’s surface with telltale mosaics. Jointing in rock masses forms three-dimensional (3D) mosaics of polyhedra, often revealed to the observer by two-dimensional (2D) planes at outcrops (Fig. 2). The shape and size of these polyhedra may be highly regular, even approaching Plato’s cube, or resemble a set of random intersecting planes (24). Alternatively, quasi-2D patterns, such as columnar joints, sometimes form in solidification of volcanic rocks (25). These patterns have been reproduced in experiments of mud and corn-starch cracks, model 2D fragmentation systems, where the following have been observed: Fast drying produces strong tension that drives the formation of primary (global) cracks that criss-cross the sample and make “X” junctions (25⇓–27) (Fig. 3); slow drying allows the formation of secondary cracks that terminate at “T” junctions (26); and “T” junctions rearrange into “Y” junctions (25, 28) to either maximize energy release as cracks penetrate the bulk (29⇓–31) or during reopening–healing cycles from wetting/drying (32) (Fig. 3). Whether in rock, ice, or soil, the fracture mosaics cut into stressed landscapes (Fig. 3) form pathways for focused fluid flow, dissolution, and erosion that further disintegrate these materials (33, 34) and reorganize landscape patterns (35, 36). Moreover, fracture patterns in rock determine the initial grain size of sediment supplied to rivers (36, 37).

Experiments and simulations provide anecdotal evidence that the geometry of fracture mosaics is genetically related to the formative stress field (38). It is difficult to determine, however, if similarities in fracture patterns among different systems are more than skin-deep. First, different communities use different metrics to describe fracture mosaics and fragments, inhibiting comparison among systems and scales. Second, we do not know whether different fracture patterns represent distinct universality classes or are merely descriptive categories applied to a pattern continuum. Third, it is unclear if and how 2D systems map to 3D.

Here, we introduce the mathematical framework of convex mosaics (39) to the fragmentation problem. This approach relies on two key principles: that fragment shape can be well approximated by convex polytopes (24) (2D polygons and 3D polyhedra; Fig. 2*A*) and that these shapes must fill space without gaps, since 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). Fragments can then be regarded as the cells of a convex mosaic (39), which may be statistically characterized by three parameters. *Cell degree* (*nodal degree* (*symbolic plane*. We define the third parameter *regularity* of the mosaic. *regular* nodes in which cell vertices only coincide with other vertices, corresponding in 2D to “X” and “Y” junctions with *B*). We define regular and irregular mosaics as having

In this paper, we measure the geometry of a wide variety of natural 2D fracture mosaics and 3D rock fragments and find that they form clusters within the global chart. Remarkably, the most significant cluster corresponds to the “Platonic attractor”: fragments with cuboid averages. Discrete element method (DEM) simulations of fracture mechanics show that cuboid averages emerge from primary fracture under the most generic stress field. Geometric simulations show how secondary fragmentation by binary breakup drives any initial mosaic toward cuboid averages.

### The 2D Mosaics in Theory and in Nature.

The geometric theory of 2D convex mosaics is essentially complete (39) and is given by the formula (42)*SI Appendix*, section 2) to illustrate the continuum of patterns contained within the global chart (Fig. 3)

We describe two important types of mosaics, which help to organize natural 2D patterns. First are *primitive mosaics*, patterns formed by binary dissection of domains. If the dissection is global, we have *regular primitive mosaics* (*irregular primitive mosaic*. The value *SI Appendix*, section 1). Thus, we expect primitive mosaics associated with the line *Voronoi mosaics*, which are, in the averaged sense, hexagonal tilings

We measured a variety of natural 2D mosaics (*SI Appendix*, section 2) and found, encouragingly, that they all lie within the global chart permitted by Eq. **1**. Mosaics close to the Platonic (

It is known that Earth’s tectonic plates meet almost exclusively at “Y” junctions; there is debate, however, about whether this “Tectonic Mosaic” formed entirely from surface fragmentation or contains a signature of the structure of mantle dynamics underneath (5, 44, 45). We examine the tectonic plate configuration (45) as a 2D convex mosaic, treating the Earth’s crust as a thin shell. We find *SI Appendix*, section 2.3). While this analysis doesn’t solve the surface/mantle question, the geometry of the Tectonic Mosaic is compatible with either 1) an evolution consisting of episodes of brittle fracture and healing or 2) cracking via thermal expansion.

The rest of our observed natural 2D mosaics plot between the Platonic and Voronoi attractors (Fig. 3). We suspect that these landscape patterns, which include mud cracks and permafrost, either initially formed as regular primitive mosaics and are in various stages of evolution toward the Voronoi attractor; or were Voronoi mosaics that are evolving via secondary fracture toward the Platonic attractor. For the case of mosaics in permafrost, however, we acknowledge that mechanisms other than fracture—such as convection—could also be at play.

### Extension to 3D Mosaics.

There is no formula for 3D convex mosaics analogous to the **1** that defines the global chart. There exists a conjecture, however, with a strong mathematical basis (42); at present, this conjecture extends only to regular mosaics. We define the *harmonic degree* as **1**. In 3D, the conjecture is equivalent to *SI Appendix*, section 3), we confirm that all of them are indeed confined to the predicted 3D global chart (Fig. 4). Unlike the 2D case, we cannot directly measure *Euler plane*, where the lines bounding the permissible domain correspond to simple polyhedra (upper) where vertices are adjacent to three edges and three faces, and their dual polyhedra which have triangular faces (lower; Fig. 4). Simple polyhedra arise as cells of mosaics in which the intersections are generic—i.e., at most three planes intersect at one point—and this does not allow for odd values of v.

As in 2D, 3D regular primitive mosaics are created by intersecting global planes. These mosaics occupy the point

Prismatic mosaics are created by regarding the 2D pattern as a base that is extended in the normal direction. The prismatic mosaic constructed from a 2D primitive mosaic has cuboid averages and is therefore statistically equivalent to a 3D primitive mosaic. The prismatic mosaic created from a 2D Voronoi base is what we call a *columnar mosaic*, and it has distinct statistical properties:

Regular primitive mosaics appear to be the dominant 3D pattern resulting from primary fracture of brittle materials (46). Moreover, dynamic brittle fracture produces binary breakup in secondary fragmentation (23, 47), driving the 3D averages *B*), appear to correspond to columnar mosaics. In these systems, the hexagonal arrangement and downward (normal) penetration of cracks arise as a consequence of maximizing energy release (29⇓–31). The only potential examples of 3D Voronoi mosaics that we know of are septarian nodules, such as the famous Moeraki Boulders (49) (Fig. 4). These enigmatic concretions have complex growth and compaction histories and contain internal cracks that intersect the surface (50). Similar to primitive mosaics, the intersection of 3D Voronoi mosaics with a surface is a 2D Voronoi mosaic.

### Connecting Primary Fracture Patterns to Mechanics with Simulations.

We hypothesize that primary fracture patterns are genetically linked to distinct stress fields, in order of most generic to most rare. In a 2D homogeneous stress field, we may describe the stress tensor with eigenvalues *pattern generator*. [Results may be equivalently cast on the Flinn diagram (51) commonly used in structural geology: *SI Appendix*, Fig. S10 ]. The 2D pattern generator is described by the single scalar function **1**), while the 3D pattern generator is characterized by scalar functions *Materials and Methods*).

In 2D, we find that pure shear produces regular primitive mosaics (Fig. 3), implying *SI Appendix*, section 4), such that

In 3D, we first conducted DEM simulations of hard materials at *Materials and Methods* and *SI Appendix*, section 4). The resulting mosaics displayed the expected fracture patterns for brittle materials: primitive, columnar, and Voronoi, respectively (Fig. 5). To obtain a global, albeit approximate, picture of the 3D pattern generator, we ran additional DEM simulations that uniformly sampled the stress space on a *SI Appendix*, section 4). The constructed pattern generator demarcated the boundaries in stress-state space that separate the three primary fracture patterns (Fig. 5). The vast proportion of this space is occupied by primitive mosaics, which are also the only pattern generated under negative volumetric stress. Such compressive stress conditions are pervasive in natural rocks. Columnar mosaics are a distant second in terms of frequency of occurrence; they occupy a narrow stripe in the stress space. Most rare are Voronoi mosaics, which only occur in a single corner of the stress space (Fig. 5). Boundaries separating the three patterns shifted somewhat for simulations that used softer materials (*SI Appendix*, Fig. S9), but the ranking did not. These primary fracture mosaics serve as initial conditions for secondary fracture. 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.

### Geometry of Natural 3D Fragments.

Based on the pattern generator (Fig. 5), we expect that natural 3D fragments should have cuboid properties on average, *Materials and Methods* and *SI Appendix*, section 5) (54). We found striking agreement: The measured averages

To better understand the full distributions of fragment shapes, we used geometric simulations of regular and irregular primitive mosaics. 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). The best-fit model, which corresponds to a moderately irregular primitive mosaic, produced topological shape distributions that are very close to those of natural fragments (mean values *SI Appendix*, section 5). We found very good agreement (*SI Appendix*, Fig. S12). Finally, 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).

## Discussion and Implications

The application and extension of the theory of convex mosaics provides a lens to organize all fracture mosaics—and the fragments they produce—into a geometric global chart. There are attractors in this global chart, arising from the mechanics of fragmentation. The Platonic attractor prevails in nature because binary breakup is the most generic fragmentation mechanism, producing averages corresponding to quadrangle cells in 2D and cuboid cells in 3D. Remarkably, a geometric model of random intersecting planes can accurately reproduce the full shape distribution of natural rock fragments. Our findings illustrate the remarkable prescience of Plato’s cubic Earth model. One cannot, however, directly “see” Plato’s cubes; rather, their shadows are seen in the statistical averages of many fragments. The relative rarity of other mosaic patterns in nature make them exceptions that prove the rule. Voronoi mosaics are a second important attractor in 2D systems such as mud cracks, where hydrostatic tension or healing of fractures forms hexagonal cells. Such conditions are rare in natural 3D systems. Accordingly, columnar mosaics arise only under specific stress fields that are consistent with iconic basalt columns experiencing contraction under directional cooling. The 3D Voronoi mosaics require very special stress conditions, 3D hydrostatic tension, and may describe rare and poorly understood concretions known as septarian nodules.

We have shown that Earth’s Tectonic Mosaic has a geometry that is consistent with what is known about fragmentation related to plate tectonics (5) (Fig. 3). This opens the possibility of constraining stress history from observed fracture mosaics. Space missions are accumulating an ever-growing catalog of 2D and 3D fracture mosaics from diverse planetary bodies that challenge understanding (Fig. 1). Geometric analysis of surface mosaics may inform debates on planetary dynamics, such as whether Pluto’s polygonal surface (Fig. 1*C*) is a result of brittle fracture or vigorous convection (7). Another potential application is using 2D outcrop exposures to estimate the 3D statistics of joint networks in rock masses, which may enhance prediction of rock-fall hazards and fluid flow (55). While the present work focused on the shapes of fragments, the theory of convex mosaics (39) is also capable of predicting particle-size distributions resulting from fragmentation, which may find application in a wide range of geophysical problems.

The life cycle of sedimentary particles is a remarkable expression of geometry in nature. Born by fragmentation (19) as statistical shadows of an invisible cube, and rounded during transport along a universal trajectory (56), pebble shapes appear to evolve toward the likewise invisible gömböc—albeit without reaching that target (57). The mathematical connections among these idealized shapes, and their reflections in the natural world, are both satisfying and mysterious. Further scrutiny of these connections may yet unlock other surprising insights into nature’s shapes.

## Methods of Mechanical Simulations

Initial samples were randomized cubic assemblies of spheres glued together, with periodic boundary conditions in all directions. The glued contact was realized by a flat, elastic cylinder connecting the two particles, which was subject to deformation from the relative motion of the glued particles. Forces and torques on the particles were calculated based on the deformation of the gluing cylinder. The connecting cylinder broke permanently if the stress acting upon it exceeded the Tresca criterion (53). The stress field was implemented by slowly deforming the underlying space. In order to avoid that there is only one percolating crack, we set a strong viscous friction between the particles and the underlying space. The seeding for the evolving cracks was provided by the randomized initial geometry of the spheres. This acted as a homogeneous drag to the particles, which ensured a homogeneous stress field in the system.

For any given shear rate, the fragment size is controlled by the particle space viscosity and the Tresca criterion limit. We set values that produced reasonable-sized fragments relative to our computational domain, allowing us to characterize the mosaics. Another advantage of the periodic system was that we could avoid any wall effect that would distort the stress field. We note here that it is possible to slowly add a shear component to the isotropic tensile shear test and obtain a structure which has average values of *SI Appendix*, section 4.

## Materials and Methods

### Methods for Fitting Geometric Model Results to Field Data.

In the simulation, we first computed a regular primitive mosaic by dissecting the unit cube with 50 randomly chosen planes, resulting in *cut model*. Subsequently, we further evolved the mosaic by breaking individual fragments. We implemented a standard model of binary breakup (14, 19) to evolve the cube by secondary fragmentation: At each step of the sequence, fragments either break with a probability *SI Appendix*, section 5). This computation, which we call the *break model*, provides an approximation to an irregular primitive mosaic; this secondary fragmentation process influenced the nodal degree

In order to compare numerical results with the experimental data obtained by manual measurements, we have to take into account several sampling biases. First, there is always a lower cutoff in size for the experimental samples. We implemented this in simulations by selecting only fragments with *SI Appendix*, section 5). We implemented this in the computations by letting the location of the center of mass be a random variable with variation *SI Appendix*, section 5 for details). Results for the small dataset are shown in Fig. 6.

## 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.

## Acknowledgments

We thank Zsolt Lángi for mathematical comments; Krisztián Halmos for invaluable help with field data measurements; Andrew Gunn for careful review of the figures; and David J. Furbish and Mikael Attal, whose comments helped to improve the manuscript substantially. This research was supported by Hungarian National Research, Development, and Innovation Office Grants K134199 (to G.D.), K116036 (to J.T.), and K119967 (to F.K.); Hungarian National Research, Development, and Innovation Office TKP2020 IE Grant BME Water Sciences & Disaster Prevention (to G.D. and J.T.); US Army Research Office Contract W911NF-20-1-0113 (to D.J.J.); and US NSF National Robotics Initiative INT Award 1734355 (to D.J.J.).

## Footnotes

- ↵
^{1}To 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.

Published under the PNAS license.

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