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3. What are the symmetries of the arithmetic billiard path (as a geometrical figure)? 2. For which numbers does the arithmetic billiard path end in the corner opposite to the starting point? If you would like to play the animation again, double click the refresh button in the top right corner. Jumping up and down to try to move the Earth is like mounting a fan on a sailboat, pointing the fan at the sail, and expecting the boat to move forwards. It is possible to use a solar sail to steer the Earth into the Sun. Maybe the Sun is going Red Giant and you miss the days when lead didn't melt in direct sunlight. Since the Sun carries the vast majority of the entire mass of the solar system, any force which moves it is likely to drag all of the planets along with it. Ceres, the solar system's largest asteroid, has less than 1/40,000th the mass of Earth; the Moon, a mere 1/80th. These objects are the heaviest you're likely to find - there are heavier moons and entire planets you could consider using, but to be honest from this point of view it looks more like using a succession of hundreds, thousands or tens of thousands of smaller asteroid impacts would be a better bet.

You could build an engine at either pole and this wouldn't have any effect, but anywhere else and the constantly changing angle of thrust will cause the Earth to behave somewhat like a loose Catherine Wheel-type firework. Maybe. Or better yet: the Earth already has a standing magnetic field; perhaps we could construct a cylinder of cable around it, and pass current to move it using Lorentz forces. This meant that tiny errors in the measurement of the current weather would not stay tiny, what is billiards but relentlessly increased in size each time they were fed back into the computer until they had completely swamped the predictions. The best we can do for three bodies is to predict their movements moment by moment, and feed those predictions back into our equations … In this case, Lorenz’s equations were causing errors to steadily grow over time. Billiards method. Clonk the Earth with something big and heavy, causing it to alter course.

Lorenz famously illustrated this effect with the analogy of a butterfly flapping its wings and thereby causing the formation of a hurricane half a world away. Which means the distance the Earth moves when everybody jumps will be one trillionth of the distance that all the people jumped: that is to say, 10-11 metres, or about half the radius of a hydrogen atom. A 40-tile rug, using the same drawgroups as the typical 9-tile - that means simply choose your favorite rugs, export their sprites from tmog, and import them into this one. What this all means is that nobody will use it to make rugs because there're too many sprites(haha). So just click on the pics above to go to the Rugs page, where you may download unattractive base-object rugs with multiple tiles. If the system is jolted somehow, it may find itself on an altogether different attractor called fibrillation, in which the cells constantly contract and relax in the wrong sequence.

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. A chaotic system will also move predictably towards its attractor in phase space - but instead of points or simple loops, we see "strange attractors" appear - complex and beautiful shapes (known as fractals) that twist and turn, intricately detailed at all possible scales. 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. You could reuse the same asteroid again and again, looping it around a few gas giants and back to gain lots more kinetic energy from those gas giants in the same way that Earth just gained velocity from the rock. It involves asteroids, like the above method, only instead of direct impacts, this time we just steer them past the Earth, allowing rock and planet to exchange a little momentum, with the result of an Earth moving on a slightly different track and an asteroid moving on a significantly different one.