I have simulated gravity in both three and two dimensions. You can play with these simulations here:

• Newtonian Gravity
• 2D Gravity

Explanations of these simulations can be found below.

The N-Body Problem

Newton’s law of universal gravitation states that objects attract each other via the inverse-square law:

\[F = \frac{Gm_1m_2}{R^2}\]

Here \(F\) is the force experience by one of the objects, \(m_1\) and \(m_2\) are the masses of the objects, \(R\) is the distance between the objects, and \(G\) is a constant. We can use this rule to predict the motion of stars and planets. For the case of two planets, this motion can be found explicitly, from which we can derive Kepler’s Laws describing orbital motion. When we have three or more planets, the motion becomes chaotic and we cannot solve the problem exactly. This problem is known as the N-body problem, and has troubled mathematicians for centuries.

When trying to simulate more than two bodies, we have to use approximate, numerical methods to solve the problem. These are used to study all sorts of physical systems — the motion of planets around a star, the motion of stars in a galaxy, or the motion of galaxies through the universe. I have made my own N-Body simulation.

2D Gravity

If the universe had two, rather than three, spatial directions, then gravity would act very differently. Newton’s law of universal gravitation would be modified to:

\[F = \frac{Gm_1m_2}{R^1}\]

This may not look so different from our previous equation, but the inverse, rather than the inverse-square, makes all the difference. In a 2D universe, it takes an infinite amount of energy too escape from the gravitational field of a planet. You can observe this fact by playing around with my simulation here. Because the planets cannot escape, you can watch the simulation run forever!