  In this work we study the space S(n) of positively oriented, simple n-gons
with labeled vertices up to oriented similarity and M(n) the moduli space of
simple n-gons as a quotient of S(n). We give a local description of S(n) and
M(n) around the regular n-gon and describe completely the two spaces when n = 4
using a topological Morse function. We also provide an asymptotic description
of the compactifications of S(4) and M(4) up to a complicated set at their
boundaries.
