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

Figure 1. Orbit traces under the generalized central force; the trail accumulates many revolutions. Method: velocity-Verlet on the inverse-power central force.

dimension d3.00

initial r₀1.00

eccentricity f1.05

trail length1500

WHAT TO TRY

Set the dimension d to 3: the inverse-square force gives closed, stable elliptical orbits. This is the only dimension where bound orbits both exist and stay closed.
Push d to 4 or higher: the force steepens to r^-(d-1) and orbits spiral inward to a plunge or fly apart, the reason stable planetary systems need exactly three space dimensions.
Tune the eccentricity and initial radius: in 3D the orbit precesses only if you break the inverse-square law, the deep link between the force law and orbital closure.