  We study the intersection ring of the space $\M(\alpha_1,...,\alpha_m)$ of
polygons in $\R^3$. We find homology cycles dual to generators of this ring and
prove a recursion relation in $m$ (the number of steps) for their intersection
numbers. This result is analog of the recursion relation appearing in the work
of Witten and Kontsevich on moduli spaces of punctured curves and on the work
of Weitsman on moduli spaces of flat connections on two-manifolds of genus $g$
with $m$ marked points. Based on this recursion formula we obtain an explicit
expression for the computation of the intersection numbers of polygon spaces
and use it in several examples. Among others, we study the special case of
equilateral polygon spaces (where all the $\alpha_i$ are the same) and compare
our results with the expressions for these particular spaces that have been
determined by Kamiyama and Tezuka. Finally, we relate our explicit formula for
the intersection numbers with the generating function for intersection pairings
of the moduli space of flat connections of Yoshida, as well as with equivalent
expressions for polygon spaces obtained by Takakura and Konno through different
techniques.
