Gravity in n spatial dimensions

What you are seeing: a test particle orbiting a heavy mass under the generalized force $\vec F = -k\,\vec r/r^{d-1}$ that you would get if space had $d$ rather than $3$ spatial dimensions. Drag $d$ from 2 to 6 and watch what happens to the orbit. Only $d = 3$ gives the closed Kepler ellipse; $d = 2$ precesses, $d = 4$ is marginal, $d \ge 5$ plunges