Downloaded by guest on September 24, 2021 a G fragmentation geometry of natural the and cube Plato’s www.pnas.org/cgi/doi/10.1073/pnas.2001037117 G and 2D for compute link, We this fragmentation. field. establishing 3D stress generator pattern formative universal the the to averages. linked cuboid that genetically patterns exotic are of more produce the ubiquity fracture toward binary from the mosaics Deviations all explaining drives attractor, breakup Platonic cuboid. binary frag- generic is natural Simulations that forms. fragments mosaics, Plato’s show of convex shadows rock geometric of indeed natural are lens ments the of through viewed shape When average In Remark- the dominant: is hexagons. two ably, attractor has Platonic “Voronoi” the plates, (3D), and dimensions tectonic three quadrangles Earth’s “Platonic” to cracks attractors: to mud (2D) mosaics two-dimensional from convex natural with of fragments, of geometry however, theory average the littered, the that ubiqui- show apply is by We produced system fragmentation. ice and tous solar rock rarely of The shape polyhedra—shards 2020) distorted a nature. 17, cubes, January in review as for blocks (received found 2020 building 27, Earth’s May envisioned approved and Plato MA, Cambridge, University, Harvard Weitz, A. 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(Fig. cycles cracks (32) reopening–healing “Y” as during release wetting/drying into or energy rearrange (29–31) from maximize terminate bulk junctions either the that “T” to penetrate cracks and 28) (25, (26); secondary dry- junctions junctions of slow 3); “T” formation (Fig. at the (25–27) criss-cross junctions allows that “X” ing that cracks make (global) and tension primary sample strong the of produces following formation drying the the Fast drives where patterns corn-starch observed: systems, and been These fragmentation mud have (25). 2D of experiments rocks model in cracks, volcanic sometimes reproduced joints, of been columnar have solidification Alterna- as (24). in planes such form intersecting these patterns, random of quasi-2D of size tively, set and cube, a Plato’s shape resemble approaching The even or regular, 2). highly be (Fig. may outcrops polyhedra two-dimensional at by observer planes mosaics the (2D) (3D) to revealed three-dimensional often forms polyhedra, of masses rock in Jointing producing in 23) 22, (16, indi- composition distributions. than these models material more or matters and input dimensionality) self-similar, and energy (size are geometry geometri- (18–21) that mass of cate fragment shape zoo of and a distributions (13) (14–17) observed is fragmentation Nevertheless, there with 2). that associated (Fig. know polyhedra Platonic now permissible space-filling We only cally of the 12). form cube, (11, idealized a the solid is that blocks postulated building sys- Plato Earth’s solar 1). the (Fig. throughout building (6–10) rings literally tem and are surfaces that planetary materials for granular blocks Moreover, creates shells and planetary (4). 6) within pervasive applications 5, pro- is (1, industrial ice this and in but rock of 1), exploited fragmentation (Fig. also 3) is (1, cess catastrophic be may fragmentation S T-M opoyaisRsac ru,Bdps nvriyo ehooyadEoois 11Bdps,Hungary; Budapest, 1111 Economics, and Technology of University Budapest Group, Research Morphodynamics MTA-BME eateto hoeia hsc,Uiest fDbee,43 ercn ugr;and Hungary; Debrecen, 4032 Debrecen, of University Physics, Theoretical of Department omb ¨ brDomokos abor ´ rgetto ie h at’ ufc ihtltl mosaics. telltale with surface Earth’s the tiles Fragmentation ewrsproaetruhtemtra 1 ) alr by Failure 2). (1, material the crack through growing when percolate point breaking networks their to stressed are olids oc ¨ | ttsia physics statistical a,b oga .Jerolmack J. Douglas , | rcuemechanics fracture | atr formation pattern d c,d,1 ehnclEgneigadApidMcais nvriyo enyvna hldlha A19104; PA Philadelphia, Pennsylvania, of University Mechanics, Applied and Engineering Mechanical eecKun Ferenc , e n J and , Geometric doi:10.1073/pnas.2001037117/-/DCSupplemental at online information supporting contains article This ( GitHub from the download 1 simulations—i.e., to for geometric free Center the the models—is at for break including available code and fragments, freely All rock are cut 3D datasets, ( https://osf.io/h2ezc/). experimental measured Science large all Open and for small data the mass both and Shape deposition: Data the under Published Submission.y Direct PNAS a is article This interests.y competing and paper. y no F.K., the declare D.J.J., wrote authors J.T. G.D., The and F.K., tools; research; D.J.J., reagents/analytic G.D., performed and new J.T. data; contributed and analyzed J.T. J.T. F.K., and G.D., F.K., D.J.J., research; G.D., designed G.D. contributions: Author n htteesae utfilsaewtotgp,since gaps, without space fill must shapes these polyhedra; approxi- that 3D and well and 2A) polygons be Fig. (2D can (24) shape polytopes convex fragment by that mated principles: relies key approach This two problem. on fragmentation the to systems (39) 2D mosaics how and a if to unclear applied 3D. is to categories it map descriptive univer- Third, merely continuum. distinct are pattern represent or patterns classes know not sality fracture do inhibiting we different Second, fragments, scales. whether and and systems mosaics among fracture comparison different are describe use systems communities to different different metrics First, among however, skin-deep. the patterns determine, than to to fracture more difficult related in is similarities genetically It if is (38). field mosaics stress fracture formative of geometry to the supplied sediment of patterns size grain fracture 37). reorga- (36, initial Moreover, rivers the and 36). determine 34) rock (35, (33, in erosion patterns materials and landscape 3) these dissolution, nize (Fig. disintegrate flow, landscapes fluid further stressed focused that into for cut pathways mosaics form fracture the soil, owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom To ouiul a itntfamn atrst hi formative and their to fragmentation conditions. patterns stress in fragment cube distinct Plato’s map models uniquely of geometric to ubiquity and the mechanical explain the use to reproduce We fragments 3D cube. demon- natural aver-topological most Appropriately We correct: of together. essentially properties is packed aged idea tightly this source: cubes that be of ancient strate made may is and study, Earth they element unlikely this because the In an that proposed bodies. from who Plato, planetary inspiration other draw land- on we for assessing safely even to and probes resources, central Under-ing extracting is continents. and hazards to fragmentation natural glaciers taming to and falls standing rock from fragmentation, to of by-products nanoparticles the among and on live We Significance ee eitouetemteaia rmwr fconvex of framework mathematical the introduce we Here, that evidence anecdotal provide simulations and Experiments nsT anos ´ fragmentation f eateto hoeia hsc,Bdps nvriyof University Budapest Physics, Theoretical of Department or ¨ NSlicense.y PNAS ok ¨ a,f c ). eateto at n niomna Science, Environmental and Earth of Department y . y b eateto Mechanics, of Department https://www.pnas.org/lookup/suppl/ NSLts Articles Latest PNAS https://github.com/torokj/ | f8 of 1

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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. Geometric simulations show how secondary fragmentation by For 3D mosaics, we also introduce f¯ as the average number of binary breakup drives any initial mosaic toward cuboid averages.

2 of 8 | www.pnas.org/cgi/doi/10.1073/pnas.2001037117 Domokos et al. Downloaded by guest on September 24, 2021 Downloaded by guest on September 24, 2021 oilsrt h otnu fpten otie within contained patterns 3) of (Fig. continuum chart global the the illustrate sec- to Appendix, 2) (SI mosaics of tion of deterministic degree and simulations stochastic minimal geometric of constructed the range a We while two. three, is is nodes cells irregular and nodes regular of onaiso h lblcataegvnb:1 the 1) by: given are mosaics chart convex global p the for on domain Boundaries admissible the the within delineates which by given is and (39) complete (42) essentially formula the is Nature. mosaics in and convex Theory 2D in lines). Mosaics (red 2D cracks The secondary with version irregular its and lines) (black mosaic primitive regular by pattern fragmentation of Approximation 2) i.3. Fig. 2. Fig. :22, al. Methods): et and Domokos (Materials simulation DEM generic credit: by (Image generated Alaska are in 23 credit: permafrost and 20, (Image eigenvalues 22 outcrop; surface with Patterns dolomite Martian surface. state 19, rock stress 16, Germany]); granite general Project/Akio plates; 18, Bremerhaven, 21, CO]); Research tectonic Institute, Boulder, and Boulder, Wegener Joint 15, [photographer]); Colorado [Alfred PA]); of Higman Innovation Grobe [University Bretwood Pittsburgh, and Druckenmiller Hannes Pittsburgh, L. Research credit: Matthew of (Image Science–UK credit: [University cracks (Image of Alaska mud Jones Promotion in E. permafrost the (irregular) 17, Charles advanced for Arizona); credit: of 10, Society NASA/JPL-Caltech/University (Image and Japan cracks 9 credit: primitive; mud regular (Image 14, 8, Causeway Nakahara); mosaics: (Right Giant’s random Poisson–Delaunay. joints, of simulations columnar 12, geometric 13, and are 12 Poisson–Voronoi; to 11, 8 Patterns primitive; patterns. periodic deterministic are circles 0 = ie;ad2 h vrl osrit httemnmldegree minimal the that constraints overall the 2) and lines; aua rget prxmtdb ovxplhda (B, polyhedra. convex by approximated fragments Natural (A) lines. fracture and fragments of Examples oac n2.(Left 2D. in Mosaics [¯ n ¯ , v ] yblcpae(i.3—.. h lblchart. global the 3)—i.e., (Fig. plane symbolic yblcplane Symbolic ) (f,v)=(6,7)( (a4) (a (f,v)=(5,6)( (a3) (a3 (f,v)=(6,8) (a2) (f,v)=(6,8) (a1) v ¯ f f , ,v 4 3) v v = MECHANICS ) ) ) GEOMETRY = =

(6,7) 6) (5 (3) (1) (7) (6) (5) (4) (2) n ¯ , 6) − σ 2¯

(22) (23) 1 p n σ > − (8) (12) (9) (10) (11) 2 1 6 n 3 stoi tessaewith state stress isotropic 23, and ; 3 [ , n, ¯ h emti hoyof theory geometric The 2 (f,v)=(7,9) (a7) ( (f,v)=(6,8)( ( (a6) (f,v)=(9,14)( (a5) ( (6) ¯ v a a a f f ,v ,v ] (18) 5) 7 6 ihgoerclyamsil oan(endi Eq. in (defined domain admissible geometrically with ) ) (20) ) ) = = (8)

(16) (13) (1) (6 ( 9,1 7,9 (15) , (19) (11) (9) 8 ) ) 4

(17) (7) ) (14) (23) 2D VORONOIMOSAICS (2) (21) p THE 2DPLATONIC ATTRACTOR 1 = e qae 1 o2)crepn oaaye mgso aua Dmsisshown: mosaics 2D natural of images analyzed to correspond 21) to (13 squares Red ) 4 (3) 2D DELAUNAYMOSAICS 2D PRIMITIVE MOSAICS (10) and [1] (22) (4) h value progressive The mosaic. a primitive in ular resulting created are decrease nodes locally. bisected (T-type) sequentially are pri- the Irregular mosaic where as primary regular situation appear a the of lines consider cells we straight Next, the fractures. nature, global entire mary, point In the (39). the bisect plane occupy definition, symbolic mosaics by These which, lines, sample. dissection straight the of If have entirely domains. we of global, dissection is binary are by First formed patterns. terns 2D natural organize rcueaeidsigihbe ute,ayiiilmsi subject secondary mosaic initial vs. any primary frac- Further, from indistinguishable. secondary are produced to fracture at Fragments arrive correspond 38). we bisections (3, limit, local turing the these in nature, so In 3), (Fig. process this σ 1 b)(b2) (b1) edsrb w motn ye fmsis hc epto help which mosaics, of types important two describe We = σ 6 2 (5) . (12) p → NATURE 0 (13) (18) (15) (20) (16) n ocmtn decrease concomitant and eua rmtv mosaics primitive regular hddga.Pten o7mre ihblack with marked 7 to 1 Patterns gray. shaded 1) rnt alsoiggoa rcs (B, cracks. global showing wall Granite 1) v ¯ 4 = (19) (21) (17) (14) oee,i nhne by unchanged is however, , NSLts Articles Latest PNAS pat- mosaics, primitive n ¯ [¯ n → ¯ , (p v 2 ] 1 = oadan toward = [¯ n composed ) ¯ , 4 4] [4, v ] = | nthe in 2 4] [2, irreg- f8 of 3 .

APPLIED PHYSICAL SCIENCES to secondary splitting of cells will, in the limit, produce fragments analysis doesn’t solve the surface/mantle question, the geometry with v¯ = 4 (SI Appendix, section 1). Thus, we expect primitive of the Tectonic Mosaic is compatible with either 1) an evolu- mosaics associated with the line v¯ = 4 in the global chart to be tion consisting of episodes of brittle fracture and healing or 2) an attractor in 2D fragmentation, as noted by ref. 43, and we cracking via thermal expansion. expect the average angle to be a rectangle (26) (Fig. 2). We The rest of our observed natural 2D mosaics plot between call this the Platonic attractor. As a useful aside, a planar sec- the Platonic and Voronoi attractors (Fig. 3). We suspect that tion of a 3D primitive mosaic (e.g., a rock outcrop) is itself a these landscape patterns, which include mud cracks and per- 2D primitive mosaic (Fig. 2). The second important pattern is mafrost, either initially formed as regular primitive mosaics and Voronoi mosaics, which are, in the averaged sense, hexagonal are in various stages of evolution toward the Voronoi attractor; tilings [¯n,v ¯] = [3, 6]. They occupy the peak of the 2D global or were Voronoi mosaics that are evolving via secondary fracture chart (Fig. 3). toward the Platonic attractor. For the case of mosaics in per- We measured a variety of natural 2D mosaics (SI Appendix, mafrost, however, we acknowledge that mechanisms other than section 2) and found, encouragingly, that they all lie within fracture—such as convection—could also be at play. the global chart permitted by Eq. 1. Mosaics close to the Pla- tonic (v¯ = 4) line include patterns known or suspected to arise Extension to 3D Mosaics. There is no formula for 3D convex under primary and/or secondary fracture: jointed rock outcrops, mosaics analogous to the p = 1 line of Eq. 1 that defines the mud cracks, and polygonal frozen ground. Mosaics close to global chart. There exists a conjecture, however, with a strong Voronoi include mud cracks and, most intriguingly, Earth’s tec- mathematical basis (42); at present, this conjecture extends tonic plates. Hexagonal mosaics are known to arise in the limit only to regular mosaics. We define the harmonic degree as h¯ = for systems subject to repeated cycles of fracturing and healing n¯v¯/(¯n +v ¯). The conjecture is that d < h¯ ≤ 2d−1, where d is sys- (25) (Fig. 3). We thus consider Voronoi mosaics to be a second tem dimension. For 2D mosaics, we obtain the known result important attractor in 2D. Horizontal sections of columnar joints (39, 42) h¯ = 2, consistent with the p = 1 substitution in Eq. 1. also belong to this geometric class; however, their evolution is In 3D, the conjecture is equivalent to 3 < h¯ ≤ 4, predicting that inherently 3D, as we discuss below. all regular 3D convex mosaics live within a narrow band in the It is known that Earth’s tectonic plates meet almost exclu- symbolic [¯n,v ¯] plane (42) (Fig. 4). Plotting a variety of well- sively at “Y” junctions; there is debate, however, about whether studied periodic and random 3D mosaics (SI Appendix, section this “Tectonic Mosaic” formed entirely from surface fragmenta- 3), we confirm that all of them are indeed confined to the tion or contains a signature of the structure of mantle dynamics predicted 3D global chart (Fig. 4). Unlike the 2D case, we can- underneath (5, 44, 45). We examine the tectonic plate configura- not directly measure n¯ in most natural 3D systems. We can, tion (45) as a 2D convex mosaic, treating the Earth’s crust as a however, measure the polyhedral cells (the fragments): the aver- thin shell. We find [¯n,v ¯] = [3.0, 5.8], numbers that are remark- age numbers f¯,v ¯ of faces and vertices, respectively. Values for ably close to a Voronoi mosaic. Indeed, the slight deviation from [f¯,v ¯] may be plotted in what we call the Euler plane, where [¯n,v ¯] = [3, 6] is because the Earth’s surface is a spherical man- the lines bounding the permissible domain correspond to sim- ifold, rather than planar (SI Appendix, section 2.3). While this ple polyhedra (upper) where vertices are adjacent to three edges

Fig. 4. Mosaics in 3D. A total of 28 uniform honeycombs, their duals, Poisson–Voronoi, Poisson–Delaunay, and primitive random mosaics plotted on the parameter planes. (Left) The [n¯, ¯v] plane, where continuous black line corresponds to prismatic mosaics (42). The shaded gray area marks the predicted domain based on the conjecture d < h¯ ≤ 2d−1.(Right) The [¯f, ¯v] plane, where straight black lines correspond to simple polyhedra (top) and their duals (bottom); for the tetrahedron [¯f, ¯v] = [4, 4], these two are identical. Mosaics highlighted in Fig. 5 are marked by red circles on both panels: 3D Voronoi (A); columnar mosaics (B); and 3D primitive mosaic (C).

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Platonic the h Dptengnrtr ernadtoa E simulations DEM additional ran of we picture approximate, generator, albeit pattern global, 3D a respec- the obtain Voronoi, To and 5). patterns (Fig. columnar, tively fracture primitive, expected materials: the brittle for displayed mosaics resulting The Methods and ([−0.5, (Materials tension respectively) 3D uniform and sion 2 ] sgn( = ln.Clr smos niaepten bevdi 4DMsimulations; DEM 74 in observed patterns indicate (symbols) Colors plane. n2,w n htpr ha rdcsrglrprimitive regular produces shear pure that find we 2D, In n3,w rtcnutdDMsmltoso admaterials hard of simulations DEM conducted first we 3D, In [µ v ¯ 1 1 1 1) (1, , |σ n domain and ), µ σ 2 1 1 ] | ≥ | ,wiete3 atr eeao scharacterized is generator pattern 3D the while 1), hs disbedmi is domain admissible whose ), C oain orsodn oser nfr Dten- 2D uniform shear, to corresponding locations Dpiiiemsi at mosaic primitive 3D A ) ≈ Rslsmyb qiaetycast equivalently be may [Results generator. pattern σ teVrniatatr ohaei agreement in are Both attractor. Voronoi 6—the 2 | ≥ | n σ ¯ i 3 (µ µ = .Te2 atr eeao sdescribed is generator pattern 2D The ]. i tesstate stress |, 1 = 1 , Fg ) hr sauiu map unique a is There 5). (Fig. ±1) , µ .I D hscrepnst eigen- to corresponds this 3D, In ±1). µ 2 v ¯ 2 ∈ , (−1, i v ¯ ,1], [−1, ,¯ ), |σ (µ, v 1 ±1) (µ | ≥ | µ i µ ,such 4), section Appendix, SI ) 1 and 1 µ 5,[1, −0.25], 1 = , (and 1 |µ ≈ σ = µ = −0.5, 2 NSLts Articles Latest PNAS 2 1 hscrepnsto corresponds This 4. eto 4). section Appendix, SI σ | µ , | ≥ | i 2 2 µ n hrceiethe characterize and ), n /σ = ¯ ∈ µ f ¯ µ i 2 a ecomputed be can 1 = (µ ,1] [−1, 2 , = −0. | µ ;osresurface observe 1; µ 1 5 and −0.25, adi sdou- is it (and 2 , = µ = 2] 2 σ σ , and 2 adi is it (and i 3 /σ Com- ). /σ | 1 1 1] [1, 1 f8 of 5 i , and = i = 1; ,

APPLIED PHYSICAL SCIENCES that uniformly sampled the stress space on a 9 × 9 (∆µ = 0.25) hypothesis, while also recognizing significant variability in the grid, for both i = +1 and i = −1 (SI Appendix, section 4). The natural data. constructed pattern generator demarcated the boundaries in To better understand the full distributions of fragment shapes, stress-state space that separate the three primary fracture pat- we used geometric simulations of regular and irregular primi- terns (Fig. 5). The vast proportion of this space is occupied tive mosaics. The cut model simulated regular primitive mosaics by primitive mosaics, which are also the only pattern gener- as primary fracture patterns by intersecting an initial cube with ated under negative volumetric stress. Such compressive stress global planes (Fig. 6), while the break model simulated irreg- conditions are pervasive in natural rocks. Columnar mosaics ular primitive mosaics resulting from secondary fragmentation are a distant second in terms of frequency of occurrence; they processes. We fit both of these models to the shape descrip- occupy a narrow stripe in the stress space. Most rare are Voronoi tor data using three parameters: one for the cutoff in the mass mosaics, which only occur in a single corner of the stress space distribution and two accounting for uncertainty in experimen- (Fig. 5). Boundaries separating the three patterns shifted some- tal protocols (Materials and Methods and SI Appendix, section what for simulations that used softer materials (SI Appendix, 5). The best-fit model, which corresponds to a moderately irreg- Fig. S9), but the ranking did not. These primary fracture ular primitive mosaic, produced topological shape distributions mosaics serve as initial conditions for secondary fracture. While that are very close to those of natural fragments (mean values our DEM simulations do not model secondary fracture, we [f¯,v ¯] = [6.58, 8.74]). We also analyzed a much larger, previ- remind the reader that binary breakup drives any initial mosaic ously collected dataset (3,728 particles) containing a diversity toward an irregular primitive mosaic with cuboid averages (SI of materials and formative conditions (19). Although values Appendix, section 1)—emphasizing the strength of the Platonic for v and f were not reported, measured values for classical attractor. shape descriptors (19, 21) could be used to fit to the cut and break models (SI Appendix, section 5). We found very good Geometry of Natural 3D Fragments. Based on the pattern gener- agreement (R2>0.95), providing further evidence that natu- ator (Fig. 5), we expect that natural 3D fragments should have ral 3D fragments are predominantly formed by binary breakup cuboid properties on average, [f¯,v ¯] = [6, 8]. To test this, we col- (SI Appendix, Fig. S12). Finally, we used the cut model to lected 556 particles from the foot of a weathering dolomite rock demonstrate how 3D primitive fracture mosaics converge asymp- outcrop (Fig. 6) and measured their values of f and v, plus mass totically toward the Platonic attractor as more fragments are and additional shape descriptors (Materials and Methods and SI produced (Fig. 6). Appendix, section 5) (54). We found striking agreement: The measured averages [f¯,v ¯] = [6.63, 8.93] were within 12% of the Discussion and Implications theoretical prediction, and distributions for f and v were cen- The application and extension of the theory of convex mosaics tered around the theoretical values. Moreover, odd values for provides a lens to organize all fracture mosaics—and the frag- v were much less frequent than even values, illustrating that ments they produce—into a geometric global chart. There are natural fragments are well approximated by simple polyhedra attractors in this global chart, arising from the mechanics of (Fig. 6). We regard these results as direct confirmation of the fragmentation. The Platonic attractor prevails in nature because

A B

556 DOLOMITE FRAGMENTS COLLECTED 600.000 DIGITAL FRAGMENTS PRODUCED BY AND MEASURED N=50 INTERSECTING PLANES C DE

Fig. 6. Natural rock fragments and geometric modeling. (A) Dolomite rock outcrop at Harmashat´ arhegy,´ Hungary, from which we sampled and measured natural fragments accumulating at its base, highlighted in Inset.(B) The cut model shown with N = 50 intersecting planes and examples of digital fragments (Inset) drawn from the 600,000 fragments produced. (C) Evolution of the face and vertex number averages ¯f and ¯v, showing convergence toward the cuboid values of eight and six, respectively, with increasing N.(D and E) Probability distributions of f and v, respectively, for natural dolomite fragments and fits of the cut and break models (for details on the latter, see main text).

6 of 8 | www.pnas.org/cgi/doi/10.1073/pnas.2001037117 Domokos et al. Downloaded by guest on September 24, 2021 Downloaded by guest on September 24, 2021 ooo tal. et Domokos g invisible likewise during as the rounded (19) toward and appear shapes fragmentation evolve cube, pebble (56), by to trajectory invisible universal Born a an along nature. transport of in shadows geometry statistical of sion range wide a in problems. application (39) geophysical resulting find of mosaics distributions may convex which particle-size of fragmentation, predicting from theory of the capable focused fragments, also work present is of the shapes While the (55). rock- flow on of fluid prediction and joint enhance hazards of may frac- fall which statistics brittle masses, 3D rock the of in estimate networks to result is exposures application outcrop a whether potential 2D Another is as using (7). convection 1C) such vigorous (Fig. or dynamics, ture surface planetary mosaics polygonal on surface Pluto’s of debates analysis inform Geometric may and 1). 2D challenge (Fig. that of bodies understanding planetary catalog diverse Space from ever-growing mosaics mosaics. fracture an con- 3D fracture of accumulating observed possibility are the from missions opens history This 3). stress (Fig. straining (5) related tectonics fragmentation about plate known to is what with septarian consistent is that as known concretions understood describe may nodules. poorly and spe- and tension, very hydrostatic rare require 3D conditions, mosaics consis- stress Voronoi under are cial 3D contraction that The experiencing fields cooling. columns directional stress basalt columnar iconic specific Accordingly, with under systems. tent 3D only natural con- arise Such in hydrostatic mosaics cells. rare where hexagonal important are forms cracks, second fractures mud ditions of a as healing such are or systems tension mosaics 2D Voronoi exceptions in them rule. attractor make the relative nature in prove The patterns fragments. that mosaic many other of are of averages shadows rarity statistical their the rather, in cubes; seen Plato’s the “see” cannot, illustrate directly One model. findings however, Earth cubic Our Plato’s dis- of fragments. prescience shape rock remarkable random full natural the of of reproduce model tribution accurately geometric can a planes Remarkably, intersecting 3D. and in 2D mechanism, in cells fragmentation cells cuboid quadrangle generic to corresponding most averages the producing is breakup binary rvddb h admzdiiilgoer fteshrs This spheres. the of geometry initial was randomized cracks the evolving the by for provided particles seeding the The perco- between space. one friction underlying only viscous the and strong is a there set the that we deforming avoid crack, lating slowly to criterion order by Tresca In implemented the space. was exceeded underlying field it stress upon The acting (53). stress perma- the deforma- broke if cylinder the connecting nently on The and cylinder. based deforma- gluing Forces calculated the to particles. of were tion glued particles subject the the of cylinder was on motion elastic torques which relative flat, the particles, a from by tion two direc- realized the all was spheres in contact connecting conditions glued of boundary The assemblies periodic tions. cubic with together, randomized glued were samples Initial Simulations Mechanical of Methods into insights surprising other shapes. unlock of nature’s yet scrutiny may Further natu- mysterious. connections the and connections these in satisfying both reflections mathematical are their world, The and ral shapes, (57). idealized target these among that reaching out .P .Alr .F Thovert, of physics F. J. Statistical Chakrabarti, Adler, K. B. M. P. Biswas, S. 3. Kato, N. Hatano, T. Kawamura, H. 2. Turcotte, L. D. 1. h iecceo eietr atce sarmral expres- remarkable a is particles sedimentary of cycle life The geometry a has Mosaic Tectonic Earth’s that shown have We ehrad,1999). Netherlands, earthquakes. and friction, fracture, 1997). UK, Cambridge, Press, rcasadCasi elg n Geophysics and Geology in Chaos and Fractals rcue n rcueNetworks Fracture and Fractures e.Md Phys. Mod. Rev. 3–8 (2012). 839–884 84, omb ¨ cabi with- oc—albeit ¨ CmrdeUniversity (Cambridge Srne,Dordrecht, (Springer, Geometric Kriszti ACKNOWLEDGMENTS. from download to free avail- models—is break freely and are cut datasets, experimental large at and able small the both including Availability. Data 6. Fig. in 5 section devi- with Appendix, largest matches SI the achieved minimizing we by and ones ation, experimental computational seven seven the the fitted to we histograms parameters, three above the Using ment. into this where than implemented experimenters, smaller We by faces that faces. assuming exper- small by is computations very there only the Third, finding kept cases. in We the of fragment. uncertainty 95% the imental in of found were diameter variation which smallest with equilibria the variable those to random respect a with be location the mass small letting of by center computations Appendix, the the (SI of in dataset this implemented larger We the 5). for section equilibria only unstable descriptors—especially or selecting shape stable marginally determining by exper- when simulations the uncertainty for in experimental size this in with cutoff implemented lower fragments several We a account always samples. into is take imental there to First, have biases. we sampling measurements, manual by obtained the to component aver- has shear of which a structure values a add obtain age slowly and note test could to We shear field. possible we tensile stress isotropic is that the it distort was that would that system here effect periodic wall any the avoid mosaics. of the advantage characterize to Another our us to allowing relative domain, set fragments We computational limit. reasonable-sized criterion produced Tresca that the values and viscosity space particle the a ensured which system. particles, the the in to field stress drag homogeneous homogeneous a as acted SNFNtoa ooisIiitv N wr 745 t D.J.J.). and (to and D.J.J.); 1734355 Award G.D. (to INT (to W911NF-20-1-0113 Initiative Prevention Robotics Contract National Disaster Office NSF & US Research Sciences Army Water US BME J.T.); Office Grant K119967 Innovation and IE and J.T.), Development, TKP2020 (to Research, Mikael K116036 National G.D.), Hungarian and F.K.); (to (to Furbish K134199 and Grants J. Development, Office Research, David National Innovation Hungarian This and substantially. by manuscript supported figures; the was improve research the to helped of comments whose review Attal, careful for Gunn degree the nodal mosaic; call the primitive influenced we irregular not process which an fragmentation computation, to secondary approximation This this an 5). provides to section model, extent. easier used Appendix, break linear we is (SI largest (19), it its ratios models that to axis computational account perpendicular similar middle into by the taking Inspired in 1 by fragment probability manner a a stochastic break with either a sec- process in fragments by the placed sequence, cube of the the end evolve of the to probability step a 19) with each implemented (14, break At We breakup binary fragmentation: fragments. of ondary individual model 6 breaking standard the by in a as mosaic resulting unit simulation the the planes, this evolved dissecting chosen to by randomly refer mosaic We primitive 50 Data. regular with Field a to cube computed Results first Model we Geometric tion, Fitting for Methods are Methods and simulations Materials mechanical of Details case. in discussed Voronoi the and .A .MEe,Tdlroinainadtefatrn fJptrsmo Europa. moon Jupiter’s of fracturing the and reorientation Tidal McEwen, M S. A. D. R. 6. Seton, M. Coltice, N. Mallard, C. 5. Prasher, L. C. 4. nodrt opr ueia eut ihteeprmna data experimental the with results numerical compare to order In by controlled is size fragment the rate, shear given any For Nature plates. tectonic Earth’s of fragmentation and distribution 1987). NY, [ ¯ v , nHlo o naubehl ihfil aamaueet;Andrew measurements; data field with help invaluable for Halmos an ´ ¯ f l oefrtegoercsmltosie,the simulations—i.e., geometric the for code All https://osf.io/h2ezc/. ]. 95 (1986). 49–51 321, fragmentation. rsigadGidn rcs Handbook Process Grinding and Crushing eto 4. section Appendix, SI m n ¯ hp n asdt o l esrd3 okfragments, rock 3D measured all for data mass and Shape > and m o eal) eut o h ml aae r shown are dataset small the for Results details). for 0 etakZotL Zsolt thank We , p v P ¯ m b eoe h mletpoetdae ftefrag- the of area projected smallest the denotes 0 htaebtentepiiiemosaics primitive the between are that notopee rke hi urn ieuntil size current their keep or pieces two into en h uoftrsod eod hr is there Second, threshold. cutoff the being usqety efurther we Subsequently, model. cut le,P .Tcly udcincnrl the controls Subduction Tackley, J. P. uller, ¨ R max 2 nifrmteaia comments; mathematical for angi ´ ≥ 5fo l itgas(see histograms all from 0.95 − NSLts Articles Latest PNAS p b https://github.com/torokj/ Jh ie os e York, New Sons, & Wiley (John h utn ln was plane cutting The . Nature A 0 P × p 4–4 (2016). 140–143 535, ilntb found be not will b σ 10 0 eedn on dependent ntesimula- the In 5 hsnt be to chosen fragments. | but n, ¯ f8 of 7

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