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NUMBERS AND AREAS OF
TECTONIC PLATES
Detailed data on major tectonic bound-
aries and areas of enclosed plates are sum-
marized by Bird (2003), who presented the
characteristics and locations of 6,048
points that serve to define 229 boundary
segments delineating 52 lithospheric
plates. The relationship between plate area
and frequency of occurrence is most con-
veniently represented as a cumulative fre-
quency distribution in which plate area is
plotted relative to size exceedance—the
number of plates equal to or larger than
any size in question; the Y-intercept
defines the total number (e.g., 52) of data
values (Fig. 1).
THE BROKEN SHEET FUNCTION
The general curvilinear form of log-log
plate area frequency distributions is simi-
lar to size-frequency distributions for some
other area mosaics that include regions of
like sediment (lithotopes) across deposi-
tional surfaces (Wilkinson and
Drummond, 2004), regions on global geo-
logic maps (Wilkinson et al., 2009), sizes
of geopolitical subdivisions (McElroy et
al., 2005), and taxonomic divisions of
organismal morphospace (Wilkinson,
2011). Size-frequency distributions for
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-
ments wherein locations of boundaries,
and therefore the sizes of these elements,
are statistically independent.
Conceptually, if the sizes of sub-ele-
ments are represented by linear distances
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
same as that arising from the classic one-
dimensional “broken stick” model of ran-
dom linear subdivision proposed by
MacArthur (1957), who suggested that
ecological niches within a resource pool
could be broken up like a stick, with each
piece of the stick representing a niche
occupied in the community. This one-
dimensional style of division comprises an
exponential distribution of separation mag-
nitudes, the same as that arising from dif-
ferences between a series of ordered ran-
dom numbers. In the two-dimensional
manifestation of random division, a sce-
nario that herein we refer to as the “broken
sheet” model, area-frequency distributions
are those that would result when linear
distances between each boundary are
10,000 Plate Area (km2) 100,000,000
n = 52
A 1,000,000
p = 0.045 % /km
R2 = 0.87 100
Bird (2003) Manus Futuna Shetland
Gurnis et al. (2012) Galapagos Fernandez
Juan 10 Exceedence
Exceedence B n = 20 South American
p = 0.025 % /km Australian
10 Eurasian
R2 = 0.93
North American
South American Antarctic
North American
East Antarctic African
Australian Paci c 1
Eurasian
E = n e(-Ap2)0.5 African Breaking and annealing
P = 0.5 Nπ/T0.5 Resamplings of Broken Sheets
Paci c Average breaking & annealing
1
1,000,000 100,000,000 Broken Sheet Function
Plate Area (km2) Inferred log-linear trends
Figure 1. Areas of tectonic plates relative to exceedance, the number of plate areas equal to or
greater than the x-axis values. (A) Plate areas from Bird (2003). (B) Areas from Gurnis et al. (2012).
based on seemingly linear trends (straight red lines). Light blue lines in (A) are 500 sets of plate
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
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
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
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.
exponentially distributed. The broken per unit length of transect, itself expressed
sheet distribution (e.g., Fig. 1) is the form as Equation (2),
that results when sizes of randomly parti-
tioned sub-elements are plotted as areas, p = 0.5 np/A0.5, (2)
rather than distances. The surface of the
Earth can therefore be described as being and A is the total surface or “sheet” area
subdivided into tectonic plates such that being divided (for Earth, ~510 × 106 km2).
plate sizes, as measured by the linear dis-
tances across them, are exponentially dis- Comparisons to the 52 plate areas from
tributed and, hence, like the broken stick, Bird (2003) and the 20 plate areas from
are consistent with random subdivision. Gurnis et al. (2012) yield R2 values of 0.87
and 0.93, respectively (Fig. 1); values of p,
In the broken sheet size-frequency dis- which correspond to the probability of
tribution, size exceedance (E, the number crossing a plate boundary, are ~0.045%
of plates with areas greater than or equal and 0.025% per kilometer, respectively.
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
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