  Starting with a flat sheet of paper, points can be constructed as the
intersection of two folds. The set of constructible points clearly depends on
which folds are admissible. In this paper, we study the situation where a fold
is admissible if its slope is admissible and it contains an already constructed
or a generator point. We give an explicit characterization of the set of
constructible points. We also state several criteria for this set to be a ring.
This answers questions originally raised by Erik Demaine and discussed by
Butler et al. and Buhler et al.
