It is shown using first physical principles and without relying on stochastic fading channel models, that the number of spatial degrees of freedom available in planar wireless networks embedded in a three-dimensional propagation environment is limited by the spatial size of the cut that divides the environment into two parts. Specifically, in the case of propagation inside a cylinder of height h and base area n, containing n communicating source-destination node pairs, the number of available channels is at most proportional to h√n and hence, as the number of nodes increases, the per-user information capacity must follow an inverse square-root of n law.