In going from height h down to height h/2, we proceed by dividing the (x,y)-plane into pieces, one for each relevant hemisphere, together with a residual piece. We consider each piece separately: The tubes that begin over a piece remain over that same piece throughout this stage of their descent. This is precisely what we tried to do before, only now the piece we associate to each hemisphere extends beyond the base of the hemisphere, and the hemispheres we have to consider and the pieces we associate to them change at each step. As before, we will make the tubes vertical over the residual piece. What we have to figure out is how to associate a piece to each relevant hemisphere, and how to cut up the space above it. Of course these two problems are really the same: What piece we associate to a hemisphere depends on how we plan to cut up the space above it.