We investigate how to make the surface of a convex polyhedron (a
polytope) by folding up a polygon and gluing its perimeter shut, and the
reverse process of cutting open a polytope and unfolding it to a polygon. We
explore basic enumeration questions in both directions: Given a polyon, how
many foldings are there? Given a polytope, how many unfoldings are there to
simple polygons? Throughout we give special attention to convex polygons, and
to regular polygons. We show that every convex polygon folds to an infinite
number of distinct polytopes, but that their number of combinatorially distinct
gluings is polynomial. There are, however, simple polygons with an exponential
number of distinct gluings.
In the reverse direction, we show that there are polytopes with an exponential
number of distinct cuttings that lead to simple unfoldings. We establish
necessary conditions for a polytope to have convex unfoldings, implying, for
example, that among the Platonic solids, only the tetrahedron has a convex
unfolding. We provide an inventory of the polytopes that may unfold to regular
polygons, show that, for n > 6, there is essentially only
one class of such polytopes.