Broken Sheets—On the Numbers and Areas of Tectonic Plates

Bruce H. Wilkinson, Department of Earth Sciences, Syracuse University, Syracuse, New York, 13244, USA, [email protected]; Brandon J. McElroy, Department of Geology & Geophysics, University of Wyoming, Laramie, Wyoming, USA; and Carl N. Dummond, Department of Physics, Purdue University Fort Wayne, Fort Wayne, Indiana 46805-1499, USA

ABSTRACT unpredictable, thus reflecting the inher- comprise a single continuous power law The sizes and numbers of tectonic ently complicated nature of processes distribution of size frequency, but with the plates are thought to record the impor- associated with their formation. proviso that a finite Earth surface area tance of plate division, amalgamation, and imposes an upper limit on larger plate INTRODUCTION destruction at divergent and convergent areas. Like Bird (2003), Morra et al. (2013) margins. Changes in slope apparent on log The outer brittle layer of the Earth con- concluded that plate sizes consist of large area versus log frequency plots have been sists of lithospheric plates that move over the and small populations, and that this differ- interpreted as evidence for discrete popu- relatively weak asthenosphere. Larger-scale ence in plate areas persists back in time at lations of plate sizes, but the sizes of litho- aspects of plate movement are the surficial least several tens of millions of years. spheric plates are also closely approxi- manifestations of mantle convection in the Mallard et al. (2016) employed spherical mated by a continuous density function in deeper Earth, but local processes of defor- models of mantle convection to examine which diameters of individual plates are mation may be only indirectly related to the geodynamical processes that drive the exponentially distributed; such size fre- regional stress fields. Because size frequen- tessellation of the Earth’s lithosphere and quencies are dependent only on the total cies from brittle fragmentation might be concurred with Bird (2003) and Morra et area and number of designated elements. manifest as self-similar (fractal) power laws al. (2013) that plate areas comprise several This implies that the spatial locations of (e.g., Davydova and Uvarov, 2013), it is distinct populations. Harrison (2016) pro- plate boundaries are controlled by a myr- important to know the relationships between posed an additional 107 plates as subdivi- iad of complicated and interrelated pro- lithospheric rheology and degree of frag- sions of the 52 plates proposed by Bird cesses such that the geographic occur- mentation, as well as the degree to which the (2003); he also interpreted several changes rence of any particular boundary is largely breakup of the lithosphere reflects the of slope in log-log plots of plate number indeterminate and thus spatially indepen- dynamics of mantle convection versus local versus area as reflecting the presence of dent of the proximity of other plate bound- interactions along plate boundaries. This is several size populations. aries. Observed breaks in slope on linear- particularly so as models of the evolution of While the designation of any particular ized size versus frequency plots are tectonic plates are extended over ever- region as a discrete “plate” is somewhat of merely coincidental and of themselves do increasing spans of geologic time (e.g., an evolving enterprise (e.g., Zhang et al., not support an interpretation of discrete Domeier and Torsvik, 2014; Matthews et al., 2017), the actuality of either single tectonic processes operating over distinct 2016; Merdith et al., 2017). (Sornette and Pisarenko, 2003) or multiple length scales. Although a purely random One approach to this question derives (Anderson, 2002; Bird, 2003; Mallard et distribution of plate boundaries also impli- from considerations of the numbers and al., 2016; Harrison, 2016) populations of cates a similar chance distribution of plate sizes of the tectonic plates. Anderson plate sizes is important to interpreting the sizes, some smaller plates are indeed clus- (2002), for example, noted that fracture manifestation of causative tectonic pro- tered along convergent boundaries in the patterns tend to self-organize such that cesses. If sizes of tectonic plates are read- southwestern Pacific. Such association of mud cracks, frozen ground, basalt col- ily characterized by a single frequency plates of similar (small) sizes suggests umns, and other natural features exhibit distribution, be it fractal or otherwise, an that locations of plate boundaries are best similar patterns. He argued that tectonic interpretation of multiple processes operat- described as reflecting nonhomogeneous plates therefore might consist of semi-rigid ing at distinct length scales is not sup- Poisson processes wherein probabilities larger polygons separated by (at times dif- ported. Conversely, if plate area-frequen- of reaching some plate boundary vary fuse) boundary zones of deformation sur- cies are best described by multimodal along any Earth-surface transect. Size faced by smaller elements. Bird (2003) distributions, then interpretations linking frequencies of continents, calderas, and presented a global data set interpreted as populations of large plates to processes many other geologic entities where embodying several fractal subpopulations occurring at convective length scales (e.g., dimensions are expressed as areal extent of plate sizes, each manifest as an approxi- Lenardic et al., 2006) and populations of exhibit similar size-frequency distribu- mately linear trend in log area versus log smaller plates to the generation of litho- tions, suggesting that lateral occurrences occurrence–frequency space. Sornette and spheric fragments at the edge of plate of their boundaries are also largely Pisarenko (2003) argued that plate sizes boundaries are entirely reasonable.

GSA Today, v. 28, doi: 10.1130/GSATG358A.1. Copyright 2017, The Geological Society of America. CC-BY-NC. NUMBERS AND AREAS OF Plate Area (km2) TECTONIC PLATES 10,000 1,000,000 100,000,000 100 Detailed data on major tectonic bound- aries and areas of enclosed plates are sum- A n = 52 p = 0.045 % /km marized by Bird (2003), who presented the R2 = 0.87 characteristics and locations of 6,048 Bird (2003)

utuna Manus Gurnis et al. (2012) F points that serve to define 229 boundary Shetland

Galapagos segments delineating 52 lithospheric Fernandez plates. The relationship between plate area Juan 10 and frequency of occurrence is most con- B n = 20 South American Australian veniently represented as a cumulative fre- p = 0.025 % /km ceedence 2 Eurasian R = 0.93 Ex quency distribution in which plate area is North American plotted relative to size exceedance—the 10 Antarctic e South American number of plates equal to or larger than North American African any size in question; the Y-intercept East Antarctic Australian

ceedenc Paci c defines the total number (e.g., 52) of data Eurasian 1 Ex values (Fig. 1). E = n e(-Ap2)0.5 African Breaking and annealing P = 0.5 Nπ/T0.5 THE BROKEN SHEET FUNCTION Resamplings of Broken Sheets Paci c 1 Average breaking & annealing The general curvilinear form of log-log 1,000,000 100,000,000 Broken Sheet Function plate area frequency distributions is simi- 2 Inferred log-linear trends lar to size-frequency distributions for some Plate Area (km ) other area mosaics that include regions of Figure 1. Areas of tectonic plates relative to exceedance, the number of plate areas equal to or like sediment (lithotopes) across deposi- greater than the x-axis values. (A) Plate areas from Bird (2003). (B) Areas from Gurnis et al. (2012). Division into three apparent subpopulations in (A) (n = 52) and two subpopulations in (B) (n = 20) is tional surfaces (Wilkinson and based on seemingly linear trends (straight red lines). Light blue lines in (A) are 500 sets of plate Drummond, 2004), regions on global geo- areas (n = 52) resampled from a stable model time series wherein two random plates are annealed and another two randomly divided over thousands of iterations (see text). Heavy blue line in (A) is the logic maps (Wilkinson et al., 2009), sizes average area of many realizations of such annealed and broken plates. Light brown lines in (B) are of geopolitical subdivisions (McElroy et 500 sets of plate areas (n = 20) resampled from the broken sheet density function. Heavy brown line al., 2005), and taxonomic divisions of in (B) is the ideal broken sheet size frequency distributions predicated by the number of designated plates (20) and the total area of the sphere on which they exist (4 π steradians; ~510 × 106 km2). White organismal morphospace (Wilkinson, lines in (A) and (B) are the two series whose areas, by chance, are closest to observed sizes. Since 2011). Size-frequency distributions for there is no inherent division of sizes in these models, any subdivision based on the perception of differing slopes for straight-line segments is spurious. these tessellations reflect the partitioning of the total area into mosaics of sub-ele- ments wherein locations of boundaries, and therefore the sizes of these elements, are statistically independent. exponentially distributed. The broken per unit length of transect, itself expressed Conceptually, if the sizes of sub-ele- sheet distribution (e.g., Fig. 1) is the form as Equation (2), ments are represented by linear distances that results when sizes of randomly parti- p = 0.5 np/A0.5, (2) across them rather than areas, the distribu- tioned sub-elements are plotted as areas, rather than distances. The surface of the tion of distances between boundaries is the and A is the total surface or “sheet” area Earth can therefore be described as being same as that arising from the classic one- being divided (for Earth, ~510 × 106 km2). subdivided into tectonic plates such that dimensional “broken stick” model of ran- Comparisons to the 52 plate areas from dom linear subdivision proposed by plate sizes, as measured by the linear dis- Bird (2003) and the 20 plate areas from MacArthur (1957), who suggested that tances across them, are exponentially dis- Gurnis et al. (2012) yield R2 values of 0.87 ecological niches within a resource pool tributed and, hence, like the broken stick, and 0.93, respectively (Fig. 1); values of p, could be broken up like a stick, with each are consistent with random subdivision. which correspond to the probability of piece of the stick representing a niche In the broken sheet size-frequency dis- crossing a plate boundary, are ~0.045% occupied in the community. This one- tribution, size exceedance (E, the number and 0.025% per kilometer, respectively. dimensional style of division comprises an of plates with areas greater than or equal exponential distribution of separation mag- to some value) of any entity with some VARIATION IN PLATE SIZE nitudes, the same as that arising from dif- area (A) is defined by the relation in If a broken sheet distribution describes ferences between a series of ordered ran- Equation (1), dom numbers. In the two-dimensional the sizes of lithospheric plates more effec- manifestation of random division, a sce- E = n e(−Ap2)0.5, (1) tively than the multi-fractal (e.g., Bird, nario that herein we refer to as the “broken 2003; Morra et al., 2013; Harrison, 2016) sheet” model, area-frequency distributions where n is the total number of entities, p is systems proposed earlier, then this system are those that would result when linear the incidence of boundary occurrence— should also: (1) largely account for differ- distances between each boundary are the probability of crossing some boundary ences between these theoretical size composite; another element is then e Intercept = 7.7 A B selected and randomly divided. Because 1.0 some minimum area should serve to sepa- rate lithospheric “plates” and smaller 0.6 Slope = -0.21; R2 = 0.97 structural elements, we assume a mini- Sampled Model mum area of 4,000 km2, about half the size 2 0.2 Slope = -2.31; R2 = 0.95 of the smallest (Manus, 8,117 km ) plate in log Exceedenc the Bird (2003) database. Moreover, owing 5.56.0 6.57.0 7.58.0 5.56.0 6.57.0 7.58.0 to constraints imposed by length scales of log Area (km2) mantle convection (e.g., Lenardic et al., 2006), we assume a maximum plate area C 6 2 1000 of 200 × 10 km , about twice the area of

y Data “smaller plate” slope the largest (Pacific, 104 × 106 km2) plate. 800 Data “larger plate” slope Given these two constraints, repeated 600 Model all regression slopes annealing and division of members of the 400 population rapidly results in model size Frequenc 200 frequencies that are both stable with respect to numbers of iterations and indis- 0.60 0.66 0.72 0.78 0.84 0.90 0.96 tinguishable from the observed frequency Log-log Slope R2 distribution of modern plate areas (Fig. D 1A). The range of permissible area fre- 600

y Data di erence in slopes quencies afforded by this simple model of 500 Model di erences in slopes repeated random annealing and fragmenta- 400 tion completely overlaps the observed sizes 300 of Bird’s (2003) 52 plates. 200 Frequenc 100 VERACITY OF PLATE SUBPOPULATIONS 00.4 0.81.2 1.622.42.8 3.23.6 44.4 A single “broken sheet” hypothesis for Differences in log-log Slopes the generation of a continuum of plate Figure 2. Correlations and slopes of apparent linear trends in measured and model plate area fre- quencies. (A) Plate areas from Gurnis et al. (2012) exhibiting an apparent inflection in slope at log sizes must also account for the widely held area ~7.7 (~50 × 106 km2); slope difference is 2.11. (B) Two model area frequency distributions, each perception that plate areas somehow com- comprising 20 randomly delimited plate areas with a total of 510 × 106 km2; red and blue lines repre- prise two or more subpopulations, each sent the two best-fit log-linear regressions that account for the largest amount of plate size vari- ance. (C) Frequency distribution of R2 values of 1,000 models of 20 randomly delimited plate areas scaled to some distinct tectonic processes. (light yellow bars) compared to R2 values of smaller (red bar, red line in [A]) and larger (blue bar, blue We suggest that what appear to be “popu- line in [A]) “populations” in the Gurnis et al. (2012) data. (D) Frequency distribution of apparent dif- ferences in slopes of “smaller” (e.g., red lines in [B]) and “larger” (e.g., blue lines in [B]) plate areas lation-specific” segments in log-size ver- among 1,000 models of 20 randomly delimited plate areas (tan bars) compared to that defined by sus log-exceedance plots (Fig. 1) are no smaller (red line in [A]) and larger (blue line in [A]) plate “populations” in the Gurnis et al. (2012) data (brown bar). Note that area-exceedance correlations of “small” and “large” plate areas in the more than coincidental trends in a sparsely observed data as well as differences in these slopes all fall well within the range of values expected sampled continuum of plate areas. Two for the sparse sampling of a continuous broken sheet distribution of plate areas. issues are relevant to the veracity of divid- ing and interpreting curvilinear log-log frequency distributions and those observed boundaries. Modern plate size frequencies data arrays on the basis of apparent among measured plate areas; and (2) yield are a snapshot of the time-integrated geo- straight line segmentation. First, any results that are in agreement with the logic histories of the growth and decline model that includes a greater number of apparent grouping of plate areas into the in the numbers and sizes of all constitu- subdivisions and a greater number of several subpopulations based on apparent ents of the global plate population (e.g., parameters (each line segment being linear trends in log-log plot of area versus Morra et al., 2013). described by some slope and intercept) will exceedance (e.g., Fig. 1). With respect to A straightforward model of such pro- certainly result in better agreement with differences between theoretical and cesses might simply presume that the data than one with fewer parameters (only observed plate areas, it seems apparent observed lithospheric plate area frequency the number of plates and total area com- that increases in the size of any particular distribution is a natural consequence of prise the broken sheet representation). plate might occur fairly continuously both the random division and random However, benefits from increases in good- through marginal accretion during sea- annealing of members of some initial pop- ness-of-fit are balanced by costs in model floor spreading or more abruptly during ulation of plate areas. We effect such a complexity (e.g., Akaike, 1974), and the development of tectonic sutures at simulation with a population of n = 52 greater numbers of model parameters run convergent margins, and that decreases plates (e.g., Bird, 2003), each with an ini- counter to the heuristic perception that might occur continually during subduc- tial area of 9.8 × 106 km2 (A = 510 × 106 simpler is better. Furthermore, any array tion, or relatively episodically during the km2). From this group, one pair is selected representing some sparsely sampled curvi- development of rifted or transform at random and annealed into a single linear distribution will unavoidably exhibit runs of apparent linearity (e.g., Fig. 2). Are the several subpopulations of plate areas Randomly-distributed suggested by Bird (2003), Morra et al. Plate Centers 12 95% (2013), Harrison (2016), and Mallard et al. 50% 5% (2016) statistically distinct from apparently 10 America n linear runs manifest in the sparse sampling Average = 8 ± 2.5 of a broken sheet? 8 n = 52 In order to address that question, we 6 consider the tabulation of 20 plate areas Number Antarctic, North from Gurnis et al. (2012) used by Morra et American Eurasian al. (2013) to define two subpopulations of 4 Pacifi c South plate sizes (Figs. 1B and 2A). As noted, Australian 2 African these plate sizes are closely approximated North Bismarck by areas in which diameters are exponen- tially distributed—the broken sheet func- 024681012141618202224 tion (Fig. 1B). If we randomly draw a sam- ple of 20 areas from such a theoretical Neighbor Centroids Within 5,000 km population, by chance the resultant array Figure 3. Number of neighbor plate centers within 5,000 km of plate centroids (bars) and Monte Carlo simulation of numbers of neighbors apparent among 500 model sets of randomly distributed will exhibit some number of apparent lin- plate centers (yellow envelope). Names of several larger plates and those with fewest (African) and ear runs in log plate area versus log most (north Bismarck) neighbors are above appropriate bars. Note that 16 plates have 17 or more neighbors (darker red bars), a density that reflects a spatial association of smaller plates; all of exceedance space (Fig. 2B). In order to these occur in the southwestern Pacific. quantify the degree of spurious linearity apparent in such randomly sampled popu- lations, we repeatedly calculate the slopes and correlation coefficients for the two homogeneous as well; that is, the numbers hypothesis that the distribution of plate linear trends that most closely match the of plates within some distance of the cen- sizes is truly random. Those observed sample areas in such a model array, deter- troid of any other plate might exhibit a uni- plates with 17 or more neighbors within mine the sample (plate) number and area modal distribution. Conversely, if plates 5,000 km (Fig. 3) are located exclusively in where the intersection between these two were geographically associated by size, the southwestern Pacific; no other part of linears occurs, and calculate the differ- then numbers of neighbors within some the Earth exhibits a statistically significant ences in their slopes. Based on this exer- distance of any plate center might exhibit concentration of tectonic plates. cise, it becomes apparent that the R2 values some sort of multimodal frequency distri- of all spurious linear arrays (Fig. 2C), as bution, with smaller distances separating DISCUSSION well as the differences between their smaller plate centroids, and larger dis- Numbers and Sizes of Tectonic Plates slopes (Fig. 2D), comprise populations that tances separating larger. Among the Bird completely overlap similar parameters (2003) data, the closest pair of centroids is In general, the numbers and areas of derived from the Gurnis et al. (2012) data. that of the North Bismarck and Manus modern tectonic plates are closely repli- Because we cannot reject the null hypoth- plates off Papua New Guinea (235 km); no cated by the distributions expected when esis that all of these areas were drawn from plates have any neighbors within a smaller locations of boundaries are largely inde- the same size-frequency distribution, any distance. Conversely, the most widely sep- pendent; to a first approximation, the proposition that they somehow exemplify arated centers are those of the South Earth’s lithospheric surface is randomly several distinct subpopulations of plate American and Philippine Sea plates subdivided. That several smaller plates areas becomes untenable. Proposed linear (19,412 km); at that (or any greater) dis- exhibit geographic association also sug- runs of plate areas are entirely consistent tance, all centroids have 51 neighbors. gests that actual fragmentation is more with the sparse sampling of a broken sheet. Taking 5,000 km as a working distance, accurately characterized as a spatially het- the frequency of so-defined neighbors for erogeneous Poisson process; the probabil- GEOGRAPHIC CLUMPING OF any one of the Bird (2003) plates ranges ity of crossing some plate boundary varies TECTONIC PLATES from 1 to 23 (Fig. 3). Moreover, Monte with geographic position, being higher Given that a broken sheet model of plate Carlo simulations show that the number of across the southwestern Pacific. fragmentation, wherein geographic loca- neighbors within 5,000 km of randomly Understanding the reasons for differing tions of plate boundaries are randomly distributed centroids in fact does comprise numbers and sizes of tectonic plates is distributed across the Earth’s surface, is in a Gaussian distribution with a mode of ~8 important from a number of perspectives. good agreement with observed areas of neighbors, the modal number expected for As noted, difference in plate areas might plates (Fig. 1), we might then ask if the the centroids of 52 plates haphazardly dis- implicate different processes in their evo- Earth’s plates also exhibit random geo- persed across ~510 × 106 km2 of the Earth’s lution, with larger plates being carried and graphic dispersal. If the distribution of surface (Fig. 3). Because several plates transported by mantle convection and plate boundaries was laterally homoge- exhibit numbers of neighbors that fall well smaller ones undergoing greater amounts neous, then it follows that the areas of above 95% confidence limits for randomly of brittle deformation along regions of con- those plates should be spatially placed centroids, we can reject the null vective convergence. The greatest 2.0 AB100 MPa 150 MPa 1.5

1.5 1.0 1.0 0.5 0.5

200 MPa 250 MPa

log Exceedence CD

1.0 log Exceedence E 1.0

0.6 0.5

/km) 0.5 -3 0 0.5

678 678 0.4 log Area (km2) log Area (km2)

100 MPa 0.3 150 MPa 200 MPa ransition Probability (1

T 250 MPa 0.2 Tp = 0.108 Ys-1.14 R2 = 0.95

100 150 200250 Yield Stress (MPa) Figure 4. (A–D) Convection in 3D spherical models of mantle convection (inset Earth models) and the logarithm of plate size versus log exceedance (cumulative count) (colored lines) for yield stresses of 100, 150, 200, and 250 MPa from Mallard et al. (2016). (E) Transition probabilities for broken sheets calculated for the 18 arrays of plate area and number (colored lines in [A–D]). Relations between model yield stresses and transitions probabilities defined an approximate power law relation between stress at model plate boundaries and degree of plate fragmentation. clustering of plates is around the equatorial mechanisms of plate formation and is geographic association of larger and North Bismarck plate off Papua New destruction might exhibit a degree of vari- smaller plates, merely enforces the suppo- Guinea (23 neighbors within 5,000 km). ation along a theoretical length scale. At a sition that those factors responsible for the Association of smaller plates in the south- longer scale, it is generally accepted, for generation of differing plate numbers, western Pacific could reflect a greater example, that larger plates bearing conti- sizes, and locations, both in time and in importance of such processes of plate frac- nental crust tend to aggregate over cold space, must reflect a concatenation of ture, particularly when overriding plates downwellings, leading to overheating of many complex processes, but that these are oceanic lithosphere, in ways that are the mantle, which in turn gives rise to the processes also operate with differing geo- not associated with convergence when the tensional fragmentation of continents (e.g., graphic and/or temporal intensity across overriding plate is continental lithosphere, Gurnis, 1988). Conversely, at a shorter the Earth’s surface. as is found along the eastern Pacific. scale, subduction zones may tend to pro- However, fragmentation alone is a uni­ duce smaller plates (e.g., Mallard et al., IMPLICATIONS directional process that serves to abruptly 2016), particularly when subduction- As a first approximation, the Earth’s decrease plate sizes and somewhat obvi- related back-arc volcanism develops into lithosphere generally comprises a ran- ates considerations of size changes that oceanic spreading centers (e.g., Bird, domly broken sheet wherein the occur- might arise through subduction, spreading, 2003); microplates may also be produced rences of plate boundaries are spatially or suturing. along seafloor spreading centers when independent but somewhat geographically Although identification of “populations” propagating rifts pass by each other (e.g., clustered across the southwestern Pacific. of large and small plates defined by appar- Hey et al., 1985). Other processes of plate This suggests that the processes of ent linear runs in log size versus log generation and destruction are less sensi- spreading, suturing, fragmentation, and exceedance space may indeed be spurious, tive to length scale; plates of any size can subduction, which ultimately result in this does not preclude an interpretation be amalgamated during continental colli- differing plate areas as well as contiguous that plate-center convection and plate-mar- sion and/or destroyed by subduction. areas of granitic crust (i.e., continents), gin tectonism have differentially influ- That plate areas are in general agree- are irreconcilably complex while also enced plate size histories. There is much ment with the premise that the Earth’s lith- exhibiting some degree of spatial and support for the premise that the osphere is randomly fragmented, yet there temporal heterogeneity across the Earth’s surface. The Earth’s evolving tectonic readily quantified when transition prob- Anderson, D.L., 2002, How many plates?: Geology, state, particularly with respect to conti- abilities are determined for degrees of v. 30, p. 411–414, https://doi.org/10.1130/0091- nental fragmentation, serves to influence fragmentation at differing yield stresses. 7613(2002)030<0411:HMP>2.0.CO;2. Bird, P., 2003, An updated digital model of plate ocean currents, atmospheric composition, Specifically, transition probability (Tp) boundaries: Geochemistry Geophysics and circulation, as well as balances of decreases (lithosphere fragmentation Geosystems, v. 4, p. 1–52, https://doi.org/ incoming and outgoing radiation; the increases) with increasing yield stress (Ys) 10.1029/2001GC000252. location of elevated terrain suitable for as: Tp = 0.108 Ys−1.14; R2 = 0.95 (Fig. 4). Davydova, M., and Uvarov, S., 2013, Fractal statis- tics of brittle fragmentation: Frattura ed Integrità the development of glacial ice forces cli- From a utilitarian perspective, the broken Strutturale, v. 24, p. 60–68, https://doi.org/10.3221/ mate change, which in turn serves to sheet function therefore appears to effec- IGF-ESIS.24.05. modulate rates of geochemical cycling tively capture degrees of lithospheric tes- DeConto, R.M., 2009, Plate tectonics and climate through atmospheric and oceanic reser- sellation under differing rheological con- change, in Gornitz, V., ed., Encyclopedia of voirs (e.g., DeConto, 2009). Wilson ditions and therefore affords a potentially Paleoclimatology and Ancient Environments: Amsterdam, Springer-Verlag, p. 784–798, cycle–scale changes in degree of conti- useful metric for describing the evolution https://doi.org/10.1007/978-1-4020-4411-3_188. nental amalgamation and dispersal have of crustal deformation over the entire Domeier, M., and Torsvik, T.H., 2014, Plate tecton- been invoked as causal drivers for a wide span of Earth’s geologic history. ics in the late Paleozoic: Geoscience Frontiers, variety of large-scale processes ranging From a more philosophical point of v. 5, p. 303–350, https://doi.org/10.1016/j.gsf from changes in continental freeboard view, understanding the nature of size .2014.01.002. Geyer, A., and Martí, J., 2012, Applying Benford’s (e.g., Whitehead and Clift, 2009) to cli- frequencies of tectonic plates, continents, law to volcanology: Geology, v. 40, p. 327–330, matic and biogeochemical cycling (e.g., and other entities is perhaps of more than https://doi.org/10.1130/G32787.1. Nance and Murphy, 2013) to global just academic interest. The study of many Gurnis, M., 1988, Large-scale mantle convection marine animal diversity (e.g., Zaffos et al., geologic features commonly generates and the aggregation and dispersal of superconti- 2017). As such, the broken sheet model quite dissimilar interpretations, and these nents: Nature, v. 332, p. 695–699, https://doi.org/ 10.1038/332695a0. serves as a first-order metric for the quan- disagreements often arise from inherently Gurnis, M., Turner, M., Zahirovic, S., DiCaprio, L., tification of changes in extents of conti- different perceptions of our world. On the Spasojevic, S., Müller, R.D., Boyden, J., Seton, nental aggregation over geologic time. one hand, a deterministic view links the M., Manea, V.C., and Bower, D.J., 2012, Plate From a practical point of view, the bro- origins of observed phenomena to unique tectonic reconstructions with continuously clos- ken sheet function or a derivative thereof and discernable causes in an explainable ing plates: Computers and Geosciences, v. 38, p. 35–42, https://doi.org/10.1016/j.cageo.2011 can serve as a useful metric in describing way, while on the other, a more stochastic .04.014. size frequencies in many systems where perspective argues that the concatenation Harrison, C.A.G., 2016, The present-day number of entity size is measured as some area, and of multiple intricate geologic processes tectonic plates: Earth, Planets, and Space, v. 68, where log size versus log exceedance gives rise to a large degree of randomness no. 37, p. 3–14, https://doi.org/10.1186/s40623 (cumulative count) comprise curvilinear and generally unresolvable levels of com- -016-0400-x. Hergarten, S., and Kenkmann, T., 2015, The num- arrays in log-log space, as is the case with plexity in the natural world. Where one ber of impact craters on Earth—Any room for respect to some compilations of calderas falls on this spectrum bears directly on further discoveries?: Earth and Planetary Science (e.g., Geyer and Martí, 2012), impact cra- how one interprets the numbers and sizes Letters, v. 425, p. 187–192, https://doi.org/10 ters (e.g., Hergarten and Kenkmann, of tectonic plates and the reality of pro- .1016/j.epsl.2015.06.009. 2015), and earthquake magnitudes (e.g., posed linkages between a rather deter- Hey, R.N., Naar, D.F., Kleinrock, M.C., Phipps Morgan, J.W., Morales, E., and Schilling, J.G., Kagan, 2002). With respect to litho- ministic understanding of regional 1985, Microplate tectonics along a superfast spheric plates, Mallard et al. (2016) have motions of the asthenosphere and those seafloor spreading system near Easter Island: recently noted that specifics concerning more complex and decidedly less predict- Nature, v. 317, p. 320–325, https://doi.org/ how plate sizes relate both to properties of able processes of local deformation. 10.1038/317320a0. Kagan, Y.Y., 2002, Seismic moment distribution the lithosphere and processes of underly- ACKNOWLEDGMENTS revisited: I. Statistical results: Geophysical ing mantle convection are poorly under- Journal International, v. 148, p. 520–541, https:// stood. In order to address these questions, Details of this analysis profited from discus- doi.org/10.1046/j.1365-246x.2002.01594.x. they employ three-dimensional spherical sions with many individuals in the Department of Lenardic, A., Richards, M.A., and Busse, F.H., models of mantle convection that combine Earth Sciences at Syracuse University; input from 2006, Depth-dependent rheology and the hori- zontal length scale of mantle convection: Journal pseudo-plasticity and variations in viscos- Joe Kula, Jim Metcalf, and Scott Miller was par- ticularly valuable. We thank Jerry Dickens for of Geophysical Research, v. 111, B07404, ity in order to generate plate-like behavior encouragement to write the paper and Claire https://doi.org/10.1029/2005JB003639. (Fig. 4A). Pseudo-plasticity is realized Mallard for sharing data on numbers and areas of MacArthur, R.H., 1957, On the relative abundances through a yield stress that characterizes plates from her three-dimensional spherical mod- of birds: Proceedings of the National Academy the plastic limit at which concentrated els of mantle convection. Peter Bird, Christopher of Sciences of the United States of America, v. 43, p. 293–295, https://doi.org/10.1073/pnas strain produces plate boundaries. 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