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NUMBERS AND AREAS OF                                                         10,000                                        Plate Area (km2)   100,000,000
TECTONIC PLATES                                                                                                                                     n = 52
                                                                    A                                                              1,000,000
  Detailed data on major tectonic bound-                                                                                                      p = 0.045 % /km
aries and areas of enclosed plates are sum-                                                                                                       R2 = 0.87                  100
marized by Bird (2003), who presented the
characteristics and locations of 6,048        Bird (2003)                              Manus         Futuna  Shetland
points that serve to define 229 boundary      Gurnis et al. (2012)                        Galapagos             Fernandez
segments delineating 52 lithospheric
plates. The relationship between plate area                                                                  Juan                                                            10   Exceedence
and frequency of occurrence is most con-
veniently represented as a cumulative fre-    Exceedence        B                                                  n = 20                                   South American
quency distribution in which plate area is                                                                   p = 0.025 % /km                                   Australian
plotted relative to size exceedance—the                   10                                                                                                    Eurasian
number of plates equal to or larger than                                                                         R2 = 0.93
any size in question; the Y-intercept                                                                                                  North American
defines the total number (e.g., 52) of data                                                                      South American                                   Antarctic
values (Fig. 1).                                                                                                   North American
                                                                                                                   East Antarctic                  African
THE BROKEN SHEET FUNCTION
                                                                                                                           Australian         Paci c                         1
  The general curvilinear form of log-log                                                                                  Eurasian
plate area frequency distributions is simi-                          E = n e(-Ap2)0.5                                       African           Breaking and annealing
lar to size-frequency distributions for some                        P = 0.5 Nπ/T0.5                                                           Resamplings of Broken Sheets
other area mosaics that include regions of                                                                                     Paci c         Average breaking & annealing
like sediment (lithotopes) across deposi-                 1
tional surfaces (Wilkinson and
Drummond, 2004), regions on global geo-                             1,000,000                                100,000,000                      Broken Sheet Function
logic maps (Wilkinson et al., 2009), sizes
of geopolitical subdivisions (McElroy et                            Plate Area (km2)                                                          Inferred log-linear trends
al., 2005), and taxonomic divisions of
organismal morphospace (Wilkinson,            Figure 1. Areas of tectonic plates relative to exceedance, the number of plate areas equal to or
2011). Size-frequency distributions for       greater than the x-axis values. (A) Plate areas from Bird (2003). (B) Areas from Gurnis et al. (2012).
these tessellations reflect the partitioning  Division into three apparent subpopulations in (A) (n = 52) and two subpopulations in (B) (n = 20) is
of the total area into mosaics of sub-ele-    based on seemingly linear trends (straight red lines). Light blue lines in (A) are 500 sets of plate
ments wherein locations of boundaries,        areas (n = 52) resampled from a stable model time series wherein two random plates are annealed
and therefore the sizes of these elements,    and another two randomly divided over thousands of iterations (see text). Heavy blue line in (A) is the
are statistically independent.                average area of many realizations of such annealed and broken plates. Light brown lines in (B) are
                                              500 sets of plate areas (n = 20) resampled from the broken sheet density function. Heavy brown line
  Conceptually, if the sizes of sub-ele-      in (B) is the ideal broken sheet size frequency distributions predicated by the number of designated
ments are represented by linear distances     plates (20) and the total area of the sphere on which they exist (4 π steradians; ~510 × 106 km2). White
across them rather than areas, the distribu-  lines in (A) and (B) are the two series whose areas, by chance, are closest to observed sizes. Since
tion of distances between boundaries is the   there is no inherent division of sizes in these models, any subdivision based on the perception of
same as that arising from the classic one-    differing slopes for straight-line segments is spurious.
dimensional “broken stick” model of ran-
dom linear subdivision proposed by            exponentially distributed. The broken                                        per unit length of transect, itself expressed
MacArthur (1957), who suggested that          sheet distribution (e.g., Fig. 1) is the form                                as Equation (2),
ecological niches within a resource pool      that results when sizes of randomly parti-
could be broken up like a stick, with each    tioned sub-elements are plotted as areas,                                    	p = 0.5 np/A0.5,	(2)
piece of the stick representing a niche       rather than distances. The surface of the
occupied in the community. This one-          Earth can therefore be described as being                                    and A is the total surface or “sheet” area
dimensional style of division comprises an    subdivided into tectonic plates such that                                    being divided (for Earth, ~510 × 106 km2).
exponential distribution of separation mag-   plate sizes, as measured by the linear dis-
nitudes, the same as that arising from dif-   tances across them, are exponentially dis-                                     Comparisons to the 52 plate areas from
ferences between a series of ordered ran-     tributed and, hence, like the broken stick,                                  Bird (2003) and the 20 plate areas from
dom numbers. In the two-dimensional           are consistent with random subdivision.                                      Gurnis et al. (2012) yield R2 values of 0.87
manifestation of random division, a sce-                                                                                   and 0.93, respectively (Fig. 1); values of p,
nario that herein we refer to as the “broken    In the broken sheet size-frequency dis-                                    which correspond to the probability of
sheet” model, area-frequency distributions    tribution, size exceedance (E, the number                                    crossing a plate boundary, are ~0.045%
are those that would result when linear       of plates with areas greater than or equal                                   and 0.025% per kilometer, respectively.
distances between each boundary are           to some value) of any entity with some
                                              area (A) is defined by the relation in                                       VARIATION IN PLATE SIZE
                                              Equation (1),
                                                                                                                             If a broken sheet distribution describes
                                              	 E = n e(−Ap2)0.5,	(1)                                                      the sizes of lithospheric plates more effec-
                                                                                                                           tively than the multi-fractal (e.g., Bird,
                                              where n is the total number of entities, p is                                2003; Morra et al., 2013; Harrison, 2016)
                                              the incidence of boundary occurrence—                                        systems proposed earlier, then this system
                                              the probability of crossing some boundary                                    should also: (1) largely account for differ-
                                                                                                                           ences between these theoretical size

                                                          www.geosociety.org/gsatoday                                                                                                         5
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