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For most of us, two degrees Celsius
is a tiny difference in temperature,
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not even enough to make
you crack a window.
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But scientists have warned that as
CO2 levels in the atmosphere rise,
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an increase in the Earth's temperature
by even this amount
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can lead to catastrophic effects
all over the world.
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How can such a small measurable
change in one factor
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lead to massive and unpredictable
changes in other factors?
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The answer lies in the concept of a
mathematical tipping point,
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which we can understand through the
familiar game of billiards.
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The basic rule of billiard motion is
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that a ball will go straight
until it hits a wall,
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then bounce off at an angle equal
to its incoming angle.
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For simplicity's sake, we'll assume that
there is no friction,
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so balls can keep moving indefinitely.
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And to simplify the situation further,
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let's look at what happens with only
one ball on a perfectly circular table.
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As the ball is struck and begins to move
according to the rules,
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it follows a neat star-shaped pattern.
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If we start the ball at
different locations,
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or strike it at different angles,
some details of the pattern change,
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but its overall form remains the same.
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With a few test runs, and some basic
mathematical modeling,
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we can even predict a ball's path
before it starts moving,
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simply based on its starting conditions.
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But what would happen
if we made a minor change
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in the table's shape
by pulling it apart a bit,
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and inserting two small straight edges
along the top and bottom?
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We can see that as the ball bounces
off the flat sides,
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it begins to move all over the table.
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The ball is still obeying the same rules
of billiard motion,
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but the resulting movement no longer
follows any recognizable pattern.
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With only a small change
to the constraints
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under which the system operates,
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we have shifted the billiard motion
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from behaving in a stable
and predictable fashion,
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to fluctuating wildly,
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thus creating what mathematicians
call chaotic motion.
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Inserting the straight edges into
the table acts as a tipping point,
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switching the systems behavior
from one type of behavior (regular),
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to another type of behavior (chaotic).
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So what implications does this simple
example have for the much more complicated
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reality of the Earth's climate?
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We can think of the shape of the table as
being analogous to the CO2 level
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and Earth's average temperature:
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Constraints that impact the
system's performance
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in the form of the ball's motion
or the climate's behavior.
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During the past 10,000 years,
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the fairly constant CO2 atmospheric
concentration of
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270 parts per million kept the climate
within a self-stabilizing pattern,
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fairly regular and hospitable
to human life.
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But with CO2 levels now at 400
parts per million,
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and predicted to rise to between
500 and 800 parts per million
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over the coming century,
we may reach a tipping point where
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even a small additional change
in the global average temperature
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would have the same effect as
changing the shape of the table,
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leading to a dangerous shift in the
climate's behavior,
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with more extreme and intense
weather events,
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less predictability, and most importantly,
less hospitably to human life.
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The hypothetical models that
mathematicians study in detail
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may not always look like
actual situations,
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but they can provide a framework
and a way of thinking
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that can be applied to help understand the
more complex problems of the real world.
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In this case, understanding
how slight changes
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in the constraints impacting a system
can have massive impacts
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gives us a greater appreciation for
predicting the dangers
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that we cannot immediately percieve
with our own senses.
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Because once the results do become visible,
it may already be too late.