Plato's Cube and the Natural Geometry of Fragmentation

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 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¨ | statistical physics | fracture mechanics | pattern formation 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; 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 process 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 First published July 17, 2020.

18178–18185 | PNAS | August 4, 2020 | vol. 117 | no. 31 www.pnas.org/cgi/doi/10.1073/pnas.2001037117 Downloaded at University of Debrecen, Hungary on August 12, 2020 Downloaded at University of Debrecen, Hungary on August 12, 2020 edfietetidparameter third call the We (42): define node a We at mosaic meet that convex tices param- a three of and by polytopes, cells characterized the statistically Frag- be as 41). eters. may regarded (40, that which interfaces be model (39), fracture then a of can choose texture ments we of local loss 1.1), the Without section ignores solids. Appendix, of (SI disintegration generality the by form fragments ( Institute. Arizona. Lipman. NASA/JPL-Caltech/SETI of Survey/Peter under NASA/Goddard/University credit: licensed credit: Image Image is (E Bennu. shell. Earth. asteroid which the planetary on of fragments cracked Surface forming (D) showing NASA/JHUAPL/SwRI. processes Europa credit: Image moon Pluto. Jupiter’s on cracks (B) Institute. Science Caltech/Space 1. Fig. ooo tal. et Domokos introduce also we mosaics, 3D For the n reua oac shaving correspond- cells, as other mosaics of of irregular junctions (3D) “T” and to faces 2D or with in (2D) ing junctions edges “Y” corre- along and lie vertices, “X” other with to respectively. coincide 2D only in vertices sponding cell which in regularity eldegree Cell rgetto cospaesadscales. and planets across Fragmentation N ftemosaic. the of I oa degree nodal stenme firglrndsweevertices where nodes irregular of number the is ( Inset v ¯ steaeaenme fvrie fthe of vertices of number average the is ) ocnceutosta rdc yolsi os omn rci eois(Inset deposits breccia forming flows, pyroclastic produce that eruptions Volcanic ( G) 4.0. BY CC .(H (http://www.sandatlas.org/). Sepp Siim credit: image D C B A N ( n n ¯ R steaeaenme fver- of number average the is ) 2 = stenme of number the is 0 p ≤ cbr avn.Iaecei:Asrla naci iiinInPilp.(F Phillips. Division/Ian Antarctic Australian credit: Image calving. Iceberg ) .W en regular define We 2B). (Fig. 1 = f ¯ p steaeaenme of number average the as [¯ ≡ n and ¯ , N A–D v R ] p /(N the aunsrnscmoe fie(Inset ice of composed rings Saturn’s (A) rings. and surfaces planetary show 0 = R yblcplane. symbolic respectively. , regular n + = N I n 3, and 4 ) ≤ nodes 1 as iaybekpdie n nta oactwr uodaverages. cuboid by toward mosaic fragmentation initial any secondary drives breakup how averages binary show field. cuboid stress simulations generic that most Geometric the show under fracture primary (DEM) mechanics from emerge method fracture element of Discrete frag- simulations averages. attractor”: cuboid “Platonic with the most ments to the corresponds Remarkably, chart. cluster global that significant the find within and clusters fragments form rock 3D they and mosaics fracture 2D natural possible ones. geometrically formed human-made all including mosaics include mosaics, on charts in global here mosaics these focus fracture, 3D we by and Although 2D plane. a symbolic admissible provides the geometrically theory This of 3). chart (Fig. global may values fractures parameter periodic identical or have random deterministic from from made stochastic networks delineate mosaics; (24), not networks does fracture framework of descriptions our other to contrast In faces. iebatn.Iaecei:SrlaBdr(photographer). Bodor Sarolta credit: Image blasting. Mine ) nti ae,w esr h emtyo ievreyof variety wide a of geometry the measure we paper, this In G H E F PNAS | uut4 2020 4, August okfls erdcdfo e.58, ref. from Reproduced falls. Rock ) .Iaecei:U Geological US credit: Image ). | o.117 vol. .Iaecei:NASA/JPL- credit: Image ). E | –H o 31 no. hwexample show C Polygonal ) | 18179

APPLIED PHYSICAL SCIENCES (a1) (a5)(a5) (f,v)=(6,8) (f,v)=(9,14)(f,v)=(9,14)

(a2) (f,v)=(6,8)

((a6)a6) (a3)(a33) (f,v)=(6,8)(f,v)=(6,8) (f,v)=(5,6)(f,vv)=(5, 6)

(a4)(a4) (a7)(a7) (f,v)=(6,7)(f,v)=(6,7) (f,v)=(7,9)7,9) (b1) (b2)

Fig. 2. Examples of fragments and fracture lines. (A) Natural fragments approximated by convex polyhedra. (B, 1) Granite wall showing global cracks. (B, 2) Approximation of fragmentation pattern by regular primitive mosaic (black lines) and its irregular version with secondary cracks (red lines).

The 2D Mosaics in Theory and in Nature. The geometric theory of We describe two important types of mosaics, which help to 2D convex mosaics is essentially complete (39) and is given by organize natural 2D patterns. First are primitive mosaics, pat- the formula (42) terns formed by binary dissection of domains. If the dissection is global, we have regular primitive mosaics (p = 1) composed 2¯n v¯ = , [1] entirely of straight lines, which, by deﬁnition, bisect the entire n¯ − p − 1 sample. These mosaics occupy the point [¯n,v ¯] = [4, 4] in the symbolic plane (39). In nature, the straight lines appear as pri- which delineates the admissible domain for convex mosaics mary, global fractures. Next, we consider the situation where the within the [¯n,v ¯] symbolic plane (Fig. 3)—i.e., the global chart. cells of a regular primary mosaic are sequentially bisected locally. Boundaries on the global chart are given by: 1) the p = 1 and Irregular (T-type) nodes are created resulting in a progressive p = 0 lines; and 2) the overall constraints that the minimal degree decrease p → 0 and concomitant decrease n¯ → 2 toward an irreg- of regular nodes and cells is three, while the minimal degree of ular primitive mosaic. The value v¯ = 4, however, is unchanged by irregular nodes is two. We constructed geometric simulations of this process (Fig. 3), so in the limit, we arrive at [¯n,v ¯] = [2, 4]. a range of stochastic and deterministic mosaics (SI Appendix, sec- In nature, these local bisections correspond to secondary frac- tion 2) to illustrate the continuum of patterns contained within turing (3, 38). Fragments produced from primary vs. secondary the global chart (Fig. 3) fracture are indistinguishable. Further, any initial mosaic subject

GEOMETRY (1) (11) (23) 6 NATURE 2D VORONOI MOSAICS (13) (14) (1) (8) (13) (14) (2) (9) (15)

(3) (10) (16) (15) (2) (4) (11) (17) THE 2D PLATONIC ATTRACTOR (5) (18) (12) (19) (20) (3)(10)(22) (16) (17) (6) 2D PRIMITIVE MOSAICS (6) (8) (9) (21) (7) (4) (18) (19) (7) 2D DELAUNAY MOSAICS (5)(12) MECHANICS 3 2 4 6 (22) (20) (21) (23)

Fig. 3. Mosaics in 2D. (Left) Symbolic plane [n¯, ¯v] with geometrically admissible domain (deﬁned in Eq. 1) shaded gray. Patterns 1 to 7 marked with black circles are deterministic periodic patterns. Patterns 8 to 12 are geometric simulations of random mosaics: 8, regular primitive; 9 and 10, advanced (irregular) primitive; 11, Poisson–Voronoi; and 12, Poisson–Delaunay. (Right) Red squares (13 to 21) correspond to analyzed images of natural 2D mosaics shown: 13, columnar joints, Giant’s Causeway (Image credit: Japan Society for the Promotion of Science–UK Research and Innovation Joint Research Project/Akio Nakahara); 14, mud cracks (Image credit: Charles E. Jones [University of Pittsburgh, Pittsburgh, PA]); 15, tectonic plates; 16, Martian surface (Image credit: NASA/JPL-Caltech/University of Arizona); 17, permafrost in Alaska (Image credit: Matthew L. Druckenmiller [University of Colorado Boulder, Boulder, CO]); 18, mud cracks (Image credit: Hannes Grobe [Alfred Wegener Institute, Bremerhaven, Germany]); 19, dolomite outcrop; 20, permafrost in Alaska (Image credit: Bretwood Higman [photographer]); and 21, granite rock surface. Patterns 22 and 23 are generated by generic DEM simulation (Materials and Methods): 22, general stress state with eigenvalues σ1 > σ2; and 23, isotropic stress state with σ1 = σ2.

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APPLIED PHYSICAL SCIENCES and three faces, and their dual polyhedra which have trian- the intersection of 3D Voronoi mosaics with a surface is a 2D gular faces (lower; Fig. 4). Simple polyhedra arise as cells of Voronoi mosaic. mosaics in which the intersections are generic—i.e., at most three planes intersect at one point—and this does not allow for odd Connecting Primary Fracture Patterns to Mechanics with Simulations. values of v. We hypothesize that primary fracture patterns are genetically As in 2D, 3D regular primitive mosaics are created by linked to distinct stress fields, in order of most generic to most intersecting global planes. These mosaics occupy the point rare. In a 2D homogeneous stress field, we may describe the [¯n,v ¯] = [8, 8] on the 3D global chart (Fig. 4). Cells of regular stress tensor with eigenvalues |σ1| ≥ |σ2| and characterize the primitive mosaics have cuboid averages [f¯,v ¯] = [6, 8] (39, 42). stress state by the dimensionless parameters µ = σ2/σ1 and This is the Platonic attractor, marked by the v¯ = 8 line in the i = sgn(σ1), whose admissible domain is µ ∈ [−1, 1] (and it is global chart. The 3D Voronoi mosaics, similar to their 2D coun- double covered due to i = ±1). In 3D, this corresponds to eigen- terparts, are associated with the Voronoi tessellation defined by values |σ1| ≥ |σ2| ≥ |σ3|, stress state µ1 = σ2/σ1, µ2 = σ3/σ1, i = some random process. If the latter is a Poisson process, then we sgn(σ1), and domain µ1, µ2 ∈ [−1, 1], |µ1| ≥ |µ2| (and it is dou- obtain (39) [¯n,v ¯] = [4, 27.07], [f¯,v ¯] = [15.51, 27.07] (Fig. 4). ble covered due to i = ±1) (Fig. 5). There is a unique map Prismatic mosaics are created by regarding the 2D pattern from these stress-field parameters to the location of the resul- as a base that is extended in the normal direction. The pris- tant fracture mosaics in the global chart; we call this map the matic mosaic constructed from a 2D primitive mosaic has cuboid mechanical pattern generator. [Results may be equivalently cast averages and is therefore statistically equivalent to a 3D prim- on the Flinn diagram (51) commonly used in structural geology: itive mosaic. The prismatic mosaic created from a 2D Voronoi SI Appendix, Fig. S10 ]. The 2D pattern generator is described base is what we call a columnar mosaic, and it has distinct sta- by the single scalar function v¯(µ, i) (and n¯ can be computed tistical properties: [¯n,v ¯] = [6, 12], [f¯,v ¯] = [8, 12]. Thus, the three from Eq. 1), while the 3D pattern generator is characterized main natural extensions of the two dominant 2D patterns are 3D by scalar functions n¯(µ1, µ2, i),v ¯(µ1, µ2, i), f¯(µ1, µ2, i). Com- primitive, 3D Voronoi, and columnar mosaics. puting the full pattern generator is beyond the scope of the Regular primitive mosaics appear to be the dominant 3D pat- current paper, even in 2D. Instead, we perform generic DEM tern resulting from primary fracture of brittle materials (46). simulations (52, 53) of a range of scenarios to interpret the Moreover, dynamic brittle fracture produces binary breakup important primary mosaic patterns described above (Materials in secondary fragmentation (23, 47), driving the 3D averages and Methods). [f¯,v ¯] toward the Platonic attractor. The most common exam- In 2D, we find that pure shear produces regular primitive ple in nature is fractured rock (Figs. 2 and 4). The other two mosaics (Fig. 3), implying v¯(−1, ±1) ≈ 4. This corresponds to 3D mosaics are more exotic and seem to require more spe- the Platonic attractor. In contrast, hydrostatic tension creates cialized conditions to form in nature. Columnar joints like the regular Voronoi mosaics (Fig. 3 and SI Appendix, section 4), such celebrated Giants Causeway, formed by the cooling of large that v¯(1, 1) ≈ 6—the Voronoi attractor. Both are in agreement basaltic rock masses (25, 31, 48) (Fig. 5B), appear to corre- with our expectations. spond to columnar mosaics. In these systems, the hexagonal In 3D, we first conducted DEM simulations of hard materials arrangement and downward (normal) penetration of cracks arise at [µ1, µ2] locations corresponding to shear, uniform 2D ten- as a consequence of maximizing energy release (29–31). The sion and uniform 3D tension ([−0.5, −0.25], [1, −0.2] and [1, 1], only potential examples of 3D Voronoi mosaics that we know respectively) (Materials and Methods and SI Appendix, section 4). of are septarian nodules, such as the famous Moeraki Boul- The resulting mosaics displayed the expected fracture patterns ders (49) (Fig. 4). These enigmatic concretions have complex for brittle materials: primitive, columnar, and Voronoi, respec- growth and compaction histories and contain internal cracks tively (Fig. 5). To obtain a global, albeit approximate, picture of that intersect the surface (50). Similar to primitive mosaics, the 3D pattern generator, we ran additional DEM simulations

Fig. 5. Illustration of the 3D pattern generator. (Left) The i = 1 leaf of the [µ1, µ2] plane. Colors (symbols) indicate patterns observed in 74 DEM simulations; marked lattice points are associated with multiple simulations. Blue plus, primitive; green square, columnar; and red circle, Voronoi. Overlapping colors (symbols) indicate intermediate mosaic patterns. (Right) Simulation and ﬁeld examples. (A) A 3D Voronoi-type mosaic at µ1 = µ2 = i = 1; observe surface patterns on all planar sections agreeing with 2D Voronoi-type tessellations, and also with surface patterns of the shown septarian nodule. Image credit: FossilEra/Matt Heaton. (B) Columnar mosaic at µ1 = 1, µ2 = 0, i = 1; surface pattern on horizontal section is a 2D Voronoi-type tessellation, while parallel vertical lines extend normal to the surface, both matching observations on the illustrated basalt columnar joints. Image credit: Japan Society for the Promotion of Science–UK Research and Innovation Joint Research Project/Akio Nakahara. (C) A 3D primitive mosaic at µ1 = −0.5, µ2 = −0.25, and i = 1; note surface patterns on all sections agree with 2D primitive mosaics, and also correspond to fracture patterns on the illustrated rock.

18182 | www.pnas.org/cgi/doi/10.1073/pnas.2001037117 Domokos et al. 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[6, = ] ae nteptengener- pattern the on Based eewti 2 fthe of 12% within were DE N f (D . f 9 vlto ftefc n etxnme averages number vertex and face the of Evolution ) and × and and h u oe hw with shown model cut The (B) Inset. 9 IAppendix, SI (∆ v v E lsmass plus , rbblt itiuin of distributions Probability ) eecen- were µ 0 = and .25) SI itiuinadtoacutn o netit nexperimen- Methods mass in and 5 descrip- the uncertainty (Materials in shape for cutoff protocols the accounting the tal to for two one models and parameters: distribution these three of using data both fragmentation tor fit secondary We from resulting processes. the mosaics with while primitive cube 6), initial ular (Fig. an intersecting planes by global patterns primi- fracture irregular primary and as regular The of mosaics. simulations tive geometric used we the in variability significant data. recognizing natural also while hypothesis, htaevr ls otoeo aua rget ma values (mean fragments natural of those to [ close distributions very shape are topological that produced mosaic, primitive ular trcosi hsgoa hr,aiigfo h ehnc of mechanics the because from nature in arising prevails are chart, attractor Platonic There global The chart. fragmentation. frag- this global the in geometric mosaics—and a attractors fracture mosaics produce—into all convex they organize of ments to theory lens the a of provides extension and application The are Implications and to fragments Discussion model more as cut attractor the 6). Platonic (Fig. produced used the we toward asymp- converge totically Finally, mosaics fracture primitive S12). breakup 3D how binary demonstrate Fig. good by formed very Appendix, and predominantly found cut (SI are the We fragments to 3D ). 5 ral fit section to (R Appendix, used agreement (SI be could models values 21) break Although (19, descriptors (19). shape conditions diversity formative a for containing and particles) materials (3,728 of dataset collected ously f ¯ .Tebs-tmdl hc orsod oamdrtl irreg- moderately a to corresponds which model, best-fit The ). ¯ , obte nesadtefl itiuin ffamn shapes, fragment of distributions full the understand better To v v 8 8. [6.58, = ] and armashat ´ f f and N eentrpre,maue ausfrclassical for values measured reported, not were = 2 v .W loaaye uhlre,previ- larger, much a analyzed also We 74]). ,poiigfrhreiec htnatu- that evidence further providing >0.95), rey ugr,fo hc esmldadmeasured and sampled we which from Hungary, arhegy, 0itretn lnsadeape fdgtlfragments digital of examples and planes intersecting 50 ´ epciey o aua ooiefamnsadfisof fits and fragments dolomite natural for respectively, , PNAS u model cut B N=50 INTERSECTING PLANES 600.000 DIGITAL FRAGMENTS PRODUCED BY ¯ f | and uut4 2020 4, August ¯ v iuae eua rmtv mosaics primitive regular simulated hwn ovrec oadtecuboid the toward convergence showing , ra model break and | o.117 vol. section Appendix, SI iuae irreg- simulated | o 31 no. | 18183

APPLIED PHYSICAL SCIENCES binary breakup is the most generic fragmentation mechanism, acted as a homogeneous drag to the particles, which ensured a producing averages corresponding to quadrangle cells in 2D and homogeneous stress field in the system. cuboid cells in 3D. Remarkably, a geometric model of random For any given shear rate, the fragment size is controlled by intersecting planes can accurately reproduce the full shape dis- the particle space viscosity and the Tresca criterion limit. We set tribution of natural rock fragments. Our findings illustrate the values that produced reasonable-sized fragments relative to our remarkable prescience of Plato’s cubic Earth model. One cannot, computational domain, allowing us to characterize the mosaics. however, directly “see” Plato’s cubes; rather, their shadows are Another advantage of the periodic system was that we could seen in the statistical averages of many fragments. The relative avoid any wall effect that would distort the stress field. We note rarity of other mosaic patterns in nature make them exceptions here that it is possible to slowly add a shear component to the that prove the rule. Voronoi mosaics are a second important isotropic tensile shear test and obtain a structure which has aver- attractor in 2D systems such as mud cracks, where hydrostatic age values of n¯ and v¯ that are between the primitive mosaics tension or healing of fractures forms hexagonal cells. Such con- and the Voronoi case. Details of mechanical simulations are ditions are rare in natural 3D systems. Accordingly, columnar discussed in SI Appendix, section 4. mosaics arise only under specific stress fields that are consistent with iconic basalt columns experiencing contraction under Materials and Methods directional cooling. The 3D Voronoi mosaics require very spe- Methods for Fitting Geometric Model Results to Field Data. In the simula- cial stress conditions, 3D hydrostatic tension, and may describe tion, we first computed a regular primitive mosaic by dissecting the unit rare and poorly understood concretions known as septarian cube with 50 randomly chosen planes, resulting in 6 × 105 fragments. nodules. We refer to this simulation as the cut model. Subsequently, we further We have shown that Earth’s Tectonic Mosaic has a geometry evolved the mosaic by breaking individual fragments. We implemented that is consistent with what is known about fragmentation related a standard model of binary breakup (14, 19) to evolve the cube by sec- to plate tectonics (5) (Fig. 3). This opens the possibility of con- ondary fragmentation: At each step of the sequence, fragments either break with a probability p into two pieces or keep their current size until straining stress history from observed fracture mosaics. Space b the end of the process with a probability 1 − pb. The cutting plane was missions are accumulating an ever-growing catalog of 2D and placed in a stochastic manner by taking into account that it is easier to 3D fracture mosaics from diverse planetary bodies that challenge break a fragment in the middle perpendicular to its largest linear extent.

understanding (Fig. 1). Geometric analysis of surface mosaics Inspired by similar computational models (19), we used pb dependent on may inform debates on planetary dynamics, such as whether axis ratios (SI Appendix, section 5). This computation, which we call the Pluto’s polygonal surface (Fig. 1C) is a result of brittle frac- break model, provides an approximation to an irregular primitive mosaic; ture or vigorous convection (7). Another potential application is this secondary fragmentation process influenced the nodal degree n¯, but ¯ using 2D outcrop exposures to estimate the 3D statistics of joint not [¯v, f]. networks in rock masses, which may enhance prediction of rock- In order to compare numerical results with the experimental data obtained by manual measurements, we have to take into account several fall hazards and fluid flow (55). While the present work focused sampling biases. First, there is always a lower cutoff in size for the exper- on the shapes of fragments, the theory of convex mosaics (39) imental samples. We implemented this in simulations by selecting only is also capable of predicting particle-size distributions resulting fragments with m > m0, m0 being the cutoff threshold. Second, there is from fragmentation, which may find application in a wide range experimental uncertainty when determining shape descriptors—especially of geophysical problems. marginally stable or unstable equilibria for the larger dataset (SI Appendix, The life cycle of sedimentary particles is a remarkable expres- section 5). We implemented this in the computations by letting the location sion of geometry in nature. Born by fragmentation (19) as of the center of mass be a random variable with variation σ0 chosen to be statistical shadows of an invisible cube, and rounded during small with respect to the smallest diameter of the fragment. We kept only transport along a universal trajectory (56), pebble shapes appear those equilibria which were found in 95% of the cases. Third, there is experimental uncertainty in finding very small faces. We implemented this into to evolve toward the likewise invisible gomb¨ oc—albeit¨ with- the computations by assuming that faces smaller than A0P will not be found out reaching that target (57). The mathematical connections by experimenters, where P denotes the smallest projected area of the frag- among these idealized shapes, and their reflections in the natu- ment. Using the above three parameters, we fitted the seven computational ral world, are both satisfying and mysterious. Further scrutiny of histograms to the seven experimental ones by minimizing the largest devi- 2 these connections may yet unlock other surprising insights into ation, and we achieved matches with Rmax ≥ 0.95 from all histograms (see nature’s shapes. SI Appendix, section 5 for details). Results for the small dataset are shown in Fig. 6. Methods of Mechanical Simulations Initial samples were randomized cubic assemblies of spheres Data Availability. Shape and mass data for all measured 3D rock fragments, including both the small and large experimental datasets, are freely avail- glued together, with periodic boundary conditions in all direc- able at https://osf.io/h2ezc/. All code for the geometric simulations—i.e., the tions. The glued contact was realized by a flat, elastic cylinder cut and break models—is free to download from https://github.com/torokj/ connecting the two particles, which was subject to deforma- Geometric fragmentation. tion from the relative motion of the glued particles. Forces and torques on the particles were calculated based on the deforma- ACKNOWLEDGMENTS. We thank Zsolt Langi´ for mathematical comments; Krisztian´ Halmos for invaluable help with field data measurements; Andrew tion of the gluing cylinder. The connecting cylinder broke perma- Gunn for careful review of the figures; and David J. Furbish and Mikael nently if the stress acting upon it exceeded the Tresca criterion Attal, whose comments helped to improve the manuscript substantially. This (53). The stress field was implemented by slowly deforming the research was supported by Hungarian National Research, Development, and underlying space. In order to avoid that there is only one perco- Innovation Office Grants K134199 (to G.D.), K116036 (to J.T.), and K119967 (to F.K.); Hungarian National Research, Development, and Innovation Office lating crack, we set a strong viscous friction between the particles TKP2020 IE Grant BME Water Sciences & Disaster Prevention (to G.D. and and the underlying space. The seeding for the evolving cracks was J.T.); US Army Research Office Contract W911NF-20-1-0113 (to D.J.J.); and provided by the randomized initial geometry of the spheres. This US NSF National Robotics Initiative INT Award 1734355 (to D.J.J.).

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