  We consider scenario in which our $(3+1)$-dim universe is actually a dense
stack of multiple parallel $(2+1)$-dim branes. For this purpose, we generalize
the DGP model to a multi-brane case. We solve for the propagation of the scalar
field and gravity in this setup. At short distances (high momenta) along the
branes interactions follow the $(2+1)$-dim laws, while at large distances (low
momenta) interactions follow the usual $(3+1)$-dim laws. This feature is
inherited from the original DGP model. In the direction perpendicular to the
brane, we show that interactions become $(3+1)$-dim at low momenta which are
unable to resolve the interbrane separation. Thus, this is one of the explicit
constructs of the "vanishing dimensions" scenario where high energy physics
appears to be lower dimensional rather than higher dimensional.
