In this approach, the usual problem with the radiative stability of the weak scale is trivially resolved: the ultraviolet cutoff of the theory is . How can the usual strength of gravitation arise in such a picture? A very simple idea is to suppose that there are n extra compact spatial dimensions of radius ~ R. The Planck scale of this (4 + n) dimensional theory is taken to be according to our philosophy. Two test masses of mass m1,m2 placed within a distance r << R will feel a gravitational potential dictated by Gauss's law in (4 + n) dimensions

On the other hand, if the masses are placed at distances r >> R, their gravitational flux lines can not continue to penetrate in the extra dimensions, and the usual 1/r potential is obtained,

so our e.ective 4 dimensional is

Putting and demanding that R be chosen to reproduce the observed yields

For cm implying deviations from Newtonian gravity over so- lar system distances, so this case is empirically excluded. For all , how- ever, the modification of gravity only becomes noticeable at distances smaller than those currently probed by experiment. The case n = 2 (R ~ 100µm-1mm) is particularly exciting, since new experiments will be performed in the very near future, looking for deviations from gravity in precisely this range of distances [11]. While gravity has not been probed at distances smaller than a millimeter, the SM gauge forces have certainly been accurately measured at weak scale distances. Therefore, the SM particles cannot freely propagate in the extra n dimension, but must be localized to a 4 dimensional submanifold. Since we assume that is the only short-distance scale in the theory, our 4-dimensional world should have a "thickness" in the extra n dimensions. The only fields propagating in the (4 + n) dimensional bulk are the (4 + n) dimensional graviton, with couplings suppressed by the (4 + n) dimensional Planck mass .