Plato’s cube and the natural geometry of fragmentation

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)v¯=2n¯n¯−p−1,[1]which delineates the admissible domain for convex mosaics within the [n¯,v¯] symbolic plane (Fig. 3)—i.e., the global chart. Boundaries on the global chart are given by: 1) the p=1 and p=0 lines; and 2) the overall constraints that the minimal degree of regular nodes and cells is three, while the minimal degree of irregular nodes is two. We constructed geometric simulations of a range of stochastic and deterministic mosaics (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 (p=1) composed entirely of straight lines, which, by definition, bisect the entire sample. These mosaics occupy the point [n¯,v¯] = [4,4] in the symbolic plane (39). In nature, the straight lines appear as primary, global fractures. Next, we consider the situation where the cells of a regular primary mosaic are sequentially bisected locally. Irregular (T-type) nodes are created resulting in a progressive decrease p→0 and concomitant decrease n¯→2 toward an irregular primitive mosaic. The value v¯=4, however, is unchanged by this process (Fig. 3), so in the limit, we arrive at [n¯,v¯] = [2,4]. In nature, these local bisections correspond to secondary fracturing (3, 38). Fragments produced from primary vs. secondary fracture are indistinguishable. Further, any initial mosaic subject to secondary splitting of cells will, in the limit, produce fragments with v¯=4 (SI Appendix, section 1). Thus, we expect primitive mosaics associated with the line v¯=4 in the global chart to be an attractor in 2D fragmentation, as noted by ref. 43, and we expect the average angle to be a rectangle (26) (Fig. 2). We call this the Platonic attractor. As a useful aside, a planar section of a 3D primitive mosaic (e.g., a rock outcrop) is itself a 2D primitive mosaic (Fig. 2). The second important pattern is Voronoi mosaics, which are, in the averaged sense, hexagonal tilings [n¯,v¯] = [3,6]. They occupy the peak of the 2D global chart (Fig. 3).

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 (v¯=4) line include patterns known or suspected to arise under primary and/or secondary fracture: jointed rock outcrops, mud cracks, and polygonal frozen ground. Mosaics close to Voronoi include mud cracks and, most intriguingly, Earth’s tectonic plates. Hexagonal mosaics are known to arise in the limit for systems subject to repeated cycles of fracturing and healing (25) (Fig. 3). We thus consider Voronoi mosaics to be a second important attractor in 2D. Horizontal sections of columnar joints also belong to this geometric class; however, their evolution is inherently 3D, as we discuss below.

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 [n¯,v¯] = [3.0,5.8], numbers that are remarkably close to a Voronoi mosaic. Indeed, the slight deviation from [n¯,v¯] = [3,6] is because the Earth’s surface is a spherical manifold, rather than planar (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 p=1 line of Eq. 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 h¯=n¯v¯/(n¯+v¯). The conjecture is that d<h¯≤2d−1, where d is system dimension. For 2D mosaics, we obtain the known result (39, 42) h¯=2, consistent with the p=1 substitution in Eq. 1. In 3D, the conjecture is equivalent to 3<h¯≤4, predicting that all regular 3D convex mosaics live within a narrow band in the symbolic [n¯,v¯] plane (42) (Fig. 4). Plotting a variety of well-studied periodic and random 3D mosaics (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 n¯ in most natural 3D systems. We can, however, measure the polyhedral cells (the fragments): the average numbers f¯,v¯ of faces and vertices, respectively. Values for [f¯,v¯] may be plotted in what we call the 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.

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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).

As in 2D, 3D regular primitive mosaics are created by intersecting global planes. These mosaics occupy the point [n¯,v¯] = [8,8] on the 3D global chart (Fig. 4). Cells of regular primitive mosaics have cuboid averages [f¯,v¯]=[6,8] (39, 42). This is the Platonic attractor, marked by the v¯=8 line in the global chart. The 3D Voronoi mosaics, similar to their 2D counterparts, are associated with the Voronoi tessellation defined by some random process. If the latter is a Poisson process, then we obtain (39) [n¯,v¯]=[4,27.07],[f¯,v¯]=[15.51,27.07] (Fig. 4).

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: [n¯,v¯]=[6,12],[f¯,v¯]=[8,12]. Thus, the three main natural extensions of the two dominant 2D patterns are 3D primitive, 3D Voronoi, and columnar mosaics.

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 [f¯,v¯] toward the Platonic attractor. The most common example in nature is fractured rock (Figs. 2 and 4). The other two 3D mosaics are more exotic and seem to require more specialized conditions to form in nature. Columnar joints like the celebrated Giants Causeway, formed by the cooling of large basaltic rock masses (25, 31, 48) (Fig. 5B), 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.

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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.

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 |σ1|≥|σ2| and characterize the stress state by the dimensionless parameters μ=σ2/σ1 and i=sgn(σ1), whose admissible domain is μ∈[−1,1] (and it is double covered due to i=±1). In 3D, this corresponds to eigenvalues |σ1|≥|σ2|≥|σ3|, stress state μ1=σ2/σ1,μ2=σ3/σ1,i=sgn(σ1), and domain μ1,μ2∈[−1,1],|μ1|≥|μ2| (and it is double covered due to i=±1) (Fig. 5). There is a unique map from these stress-field parameters to the location of the resultant fracture mosaics in the global chart; we call this map the mechanical 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 v¯(μ,i) (and n¯ can be computed from Eq. 1), while the 3D pattern generator is characterized by scalar functions n¯(μ1,μ2,i),v¯(μ1,μ2,i),f¯(μ1,μ2,i). Computing the full pattern generator is beyond the scope of the current paper, even in 2D. Instead, we perform generic DEM simulations (52, 53) of a range of scenarios to interpret the important primary mosaic patterns described above (Materials and Methods).

In 2D, we find that pure shear produces regular primitive mosaics (Fig. 3), implying v¯(−1,±1)≈4. This corresponds to the Platonic attractor. In contrast, hydrostatic tension creates regular Voronoi mosaics (Fig. 3 and SI Appendix, section 4), such that v¯(1,1)≈6—the Voronoi attractor. Both are in agreement with our expectations.

In 3D, we first conducted DEM simulations of hard materials at [μ1,μ2] locations corresponding to shear, uniform 2D tension and uniform 3D tension ([−0.5,−0.25],[1,−0.2] and [1,1], respectively) (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 9×9 (Δμ=0.25) grid, for both i=+1 and i=−1 (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, [f¯,v¯]=[6,8]. To test this, we collected 556 particles from the foot of a weathering dolomite rock outcrop (Fig. 6) and measured their values of f and v, plus mass and additional shape descriptors (Materials and Methods and SI Appendix, section 5) (54). We found striking agreement: The measured averages [f¯,v¯]=[6.63,8.93] were within 12% of the theoretical prediction, and distributions for f and v were centered around the theoretical values. Moreover, odd values for v were much less frequent than even values, illustrating that natural fragments are well approximated by simple polyhedra (Fig. 6). We regard these results as direct confirmation of the hypothesis, while also recognizing significant variability in the natural data.

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Fig. 6.

Natural rock fragments and geometric modeling. (A) Dolomite rock outcrop at Hármashatárhegy, 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).

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 [f¯,v¯]=[6.58,8.74]). We also analyzed a much larger, previously collected dataset (3,728 particles) containing a diversity of materials and formative conditions (19). Although values for v and f were not reported, measured values for classical shape descriptors (19, 21) could be used to fit to the cut and break models (SI Appendix, section 5). We found very good agreement (R2>0.95), providing further evidence that natural 3D fragments are predominantly formed by binary breakup (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).