For most of us, two degrees Celsius is a tiny difference in temperature, not even enough to make you crack a window. But scientists have warned that as CO2 levels in the atmosphere rise, an increase in the Earth's temperature by even this amount can lead to catastrophic effects all over the world. How can such a small measurable change in one factor lead to massive and unpredictable changes in other factors? The answer lies in the concept of a mathematical tipping point, which we can understand through the familiar game of billiards. The basic rule of billiard motion is that a ball will go straight until it hits a wall, then bounce off at an angle equal to its incoming angle. For simplicity's sake, we'll assume that there is no friction, so balls can keep moving indefinitely. And to simplify the situation further, let's look at what happens with only one ball on a perfectly circular table.

As the ball is struck and begins to move according to the rules, it follows a neat star-shaped pattern. If we start the ball at different locations, or strike it at different angles, some details of the pattern change, but its overall form remains the same. With a few test runs, and some basic mathematical modeling, we can even predict a ball's path before it starts moving, simply based on its starting conditions. But what would happen if we made a minor change in the table's shape by pulling it apart a bit, and inserting two small straight edges along the top and bottom? We can see that as the ball bounces off the flat sides, it begins to move all over the table. The ball is still obeying the same rules of billiard motion, but the resulting movement no longer follows any recognizable pattern.

With only a small change to the constraints under which the system operates, we have shifted the billiard motion from behaving in a stable and predictable fashion, to fluctuating wildly, thus creating what mathematicians call chaotic motion. Inserting the straight edges into the table acts as a tipping point, switching the systems behavior from one type of behavior (regular), to another type of behavior (chaotic). So what implications does this simple example have for the much more complicated reality of the Earth's climate? We can think of the shape of the table as being analogous to the CO2 level and Earth's average temperature: Constraints that impact the system's performance in the form of the ball's motion or the climate's behavior.

During the past 10,000 years, the fairly constant CO2 atmospheric concentration of 270 parts per million kept the climate within a self-stabilizing pattern, fairly regular and hospitable to human life. But with CO2 levels now at 400 parts per million, and predicted to rise to between 500 and 800 parts per million over the coming century, we may reach a tipping point where even a small additional change in the global average temperature would have the same effect as changing the shape of the table, leading to a dangerous shift in the climate's behavior, with more extreme and intense weather events, less predictability, and most importantly, less hospitably to human life. The hypothetical models that mathematicians study in detail may not always look like actual situations, but they can provide a framework and a way of thinking that can be applied to help understand the more complex problems of the real world.

In this case, understanding how slight changes in the constraints impacting a system can have massive impacts gives us a greater appreciation for predicting the dangers that we cannot immediately percieve with our own senses. Because once the results do become visible, it may already be too late..