The number
of facets forming an intersection of n cylinders is

F(n) = n*(n-1)*2

F(1)=1

F(2)=4

Remember

the Sum i=1 to n of i is n*(n+1)/2

hence

F(n) = n*(n-1)*2

=
4* (Sum i=1 to n-1 of i)

= 4* (Sum i=1 to n-2 of i) +
4*(n-1)

= F(n-1) + 4*(n-1)

The n th
cylinder forms 4 new facets with each of the already present n-1 cylinders. He
hits the other cylinder at two opponent places and splits there for two
different directions one facet in two.

__P.S. __

Often the
situation is degenerateted and very small facets are even vanishing so that
F(n) gives an upper limit for the number of facets.