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Lorenz soon realised that while the computer was printing out the predictions to three decimal places, what is billiards it was actually crunching the numbers internally using six decimal places. Step 1: Using a pattern (download our Spooky Tic-Tac-Toe pumpkin and ghost patterns as a PDF), trace and cut out five pumpkins from the orange magnetic sheeting and five ghosts from the white magnetic sheeting. Cut out large petals and leaves from colored paper, and tape them around a large bowl (don't use glass). Mathematicians use the concept of a "phase space" to describe the possible behaviours of a system geometrically. The key to unlocking the hidden structure of a chaotic system is in determining its preferred set of behaviours - known to mathematicians as its attractor. The branch of fractal mathematics, pioneered by the French American mathematician Benoît Mandelbröt, allows us to come to grips with the preferred behaviour of this system, even as the incredibly intricate shape of the attractor prevents us from predicting exactly how the system will evolve once it reaches it. If released above the water it will fall, and if released underwater it will float.

Once there it clings to its attractor as it is buffeted to and fro in a literal sea of chaos, and quickly moves back to the surface if temporarily thrown above or dumped below the waves. The behaviour of the system can be observed by placing a point at the location representing the starting configuration and watching how that point moves through the phase space. Phase space may seem fairly abstract, but one important application lies in understanding your heartbeat. Keeping an eye on the asteroids is difficult but worthwhile, since such chaotic effects may one day fling an unwelcome surprise our way. Though we may not be able to predict exactly how a chaotic system will behave moment to moment, knowing the attractor allows us to narrow down the possibilities. This means that the ball will bounce infinitely many times on the sides of the billiard table and keep going forever. In mathematical billiards the ball bounces around according to the same rules as in ordinary billiards, but it has no mass, which means there is no friction. Suppose you want to play a game of billiards (or pool, or snooker, or whatever takes your fancy), but instead of playing on a rectangular table, you play it on an elliptical table.

In our next section, we'll show you how to play the Farm Fresh Dart game. An easy-to-make, fun-to-play variation on the game of pool can be played on the floor of any room. They not only provide excellent support but also add a touch of elegance to your billiard room. It is worth noting that the laws of physics that determine how the billiard balls move are precise and unambiguous: they allow no room for randomness. What at first glance appears to be random behaviour is completely deterministic - it only seems random because imperceptible changes are making all the difference. Then the game changes in a hurry. The objective is to pocket all your designated balls (either solids or stripes) and then sink the 8 ball to win the game. Now, before the game starts, the balls are racked all sorts of wrong. Since most gimbals are part of electronic systems, adding more complexity is not always the best choice. In 1887, the French mathematician Henri Poincaré showed that while Newton’s theory of gravity could perfectly predict how two planetary bodies would orbit under their mutual attraction, adding a third body to the mix rendered the equations unsolvable.

In this case, Lorenz’s equations were causing errors to steadily grow over time. This meant that tiny errors in the measurement of the current weather would not stay tiny, but relentlessly increased in size each time they were fed back into the computer until they had completely swamped the predictions. The starting weather conditions had been virtually identical. If the ball gets in the cup, the two hurry to switch places and equipment, starting the process again. The two predictions were anything but. Two weeks is believed to be the limit we could ever achieve however much better computers and software get. One fascinating aspect of mathematical billiards is that it gives us a geometrical method to determine the least common multiple and the greatest common divisor of two natural numbers. Mathematical billiards is an idealisation of what we experience on a regular pool table. A nice way to see this "butterfly effect" for yourself is with a game of pool or billiards. This fun and challenging game is played on a table with six pockets and fifteen balls.