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surface. The Earth’s evolving tectonic
state, particularly with respect to conti-
nental fragmentation, serves to influence
ocean currents, atmospheric composition,
and circulation, as well as balances of
incoming and outgoing radiation; the
location of elevated terrain suitable for
the development of glacial ice forces cli-
mate change, which in turn serves to
modulate rates of geochemical cycling
through atmospheric and oceanic reser-
voirs (e.g., DeConto, 2009). Wilson
cycle–scale changes in degree of conti-
nental amalgamation and dispersal have
been invoked as causal drivers for a wide
variety of large-scale processes ranging
from changes in continental freeboard
(e.g., Whitehead and Clift, 2009) to cli-
matic and biogeochemical cycling (e.g.,
Nance and Murphy, 2013) to global
marine animal diversity (e.g., Zaffos et al., just academic interest. The study of many
2017). As such, the broken sheet model
serves as a first-order metric for the quan-
tification of changes in extents of conti-
nental aggregation over geologic time.
From a practical point of view, the bro-
ken sheet function or a derivative thereof
can serve as a useful metric in describing
size frequencies in many systems where
entity size is measured as some area, and
where log size versus log exceedance
(cumulative count) comprise curvilinear
arrays in log-log space, as is the case with
respect to some compilations of calderas
(e.g., Geyer and Martí, 2012), impact cra-
ters (e.g., Hergarten and Kenkmann,
2015), and earthquake magnitudes (e.g.,
Kagan, 2002). With respect to litho-
spheric plates, Mallard et al. (2016) have
recently noted that specifics concerning
how plate sizes relate both to properties of
the lithosphere and processes of underly-
ing mantle convection are poorly under-
stood. In order to address these questions,
they employ three-dimensional spherical
models of mantle convection that combine
pseudo-plasticity and variations in viscos-
ity in order to generate plate-like behavior
(Fig. 4A). Pseudo-plasticity is realized
through a yield stress that characterizes
the plastic limit at which concentrated
strain produces plate boundaries. Their
models produce plate size–frequency dis-
tributions that serve to more intimately
relate styles of lithosphere fragmentation
to processes of mantle convection.
Perhaps not surprisingly, lower values of
yield stress correspond to greater degrees
of fragmentation. But this relation is more
readily quantified when transition prob-
abilities are determined for degrees of
fragmentation at differing yield stresses.
Specifically, transition probability (Tp)
decreases (lithosphere fragmentation
increases) with increasing yield stress (Ys)
as: Tp = 0.108 Ys−1.14; R2 = 0.95 (Fig. 4).
From a utilitarian perspective, the broken
sheet function therefore appears to effec-
tively capture degrees of lithospheric tes-
sellation under differing rheological con-
ditions and therefore affords a potentially
useful metric for describing the evolution
of crustal deformation over the entire
span of Earth’s geologic history.
From a more philosophical point of
view, understanding the nature of size
frequencies of tectonic plates, continents,
and other entities is perhaps of more than
geologic features commonly generates
quite dissimilar interpretations, and these
disagreements often arise from inherently
different perceptions of our world. On the
one hand, a deterministic view links the
origins of observed phenomena to unique
and discernable causes in an explainable
way, while on the other, a more stochastic
perspective argues that the concatenation
of multiple intricate geologic processes
gives rise to a large degree of randomness
and generally unresolvable levels of com-
plexity in the natural world. Where one
falls on this spectrum bears directly on
how one interprets the numbers and sizes
of tectonic plates and the reality of pro-
posed linkages between a rather deter-
ministic understanding of regional
motions of the asthenosphere and those
more complex and decidedly less predict-
able processes of local deformation.
ACKNOWLEDGMENTS
Details of this analysis profited from discus-
sions with many individuals in the Department of
Earth Sciences at Syracuse University; input from
Joe Kula, Jim Metcalf, and Scott Miller was par-
ticularly valuable. We thank Jerry Dickens for
encouragement to write the paper and Claire
Mallard for sharing data on numbers and areas of
plates from her three-dimensional spherical mod-
els of mantle convection. Peter Bird, Christopher
Harrison, Linda Ivany, and Greg Hoke read drafts
of the manuscript and offered many helpful com-
ments and suggestions.
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