Stable Orbits are Impossible in More Than Three Dimensions

If one assumes that a few hundred million years in stable orbit are necessary for planetary life to evolve, the number of space dimensions is also fixed by our existence. That is because, according to the laws of gravity, it is only in three dimensions that stable elliptical orbits are possible. Circular orbits are possible in other dimensions, but those, as Newton feared, are unstable. In any but three dimensions even a small disturbance, such as that produced by the pull of the other planets, would send a planet off its circular orbit and cause it to spiral either into or away from the sun, so we would either burn up or freeze. Also, in more than three dimensions the gravitational force between two bodies would decrease more rapidly than it does in three dimensions. In three dimensions the gravitational force drops to 1/4 of its value if one doubles the distance. In four dimensions it would drop to 1/8, in five dimensions it would drop to 1/16, and so on. As a result, in more than three dimensions the sun would not be able to exist in a stable state with its internal pressure balancing the pull of gravity. It would either fall apart or collapse to form a black hole, either of which could ruin your day. On the atomic scale, the electrical forces would behave in the same way as gravitational forces. That means the electrons in atoms would either escape or spiral into the nucleus. In neither case would atoms as we know them be possible.

The force of gravity gets weaker the more dimensions you add to the Universe, making stable planetary orbits impossible.