  We say that a collection Gamma of geodesics in the hyperbolic plane H^2 is a
modular pattern if Gamma is invariant under the modular group PSL_2(Z), if
there are only finitely many PSL_2(Z)-equivalence classes of geodesics in
Gamma, and if each geodesic in Gamma is stabilized by an infinite order
subgroup of PSL_2(Z). For instance, any finite union of closed geodesics on the
modular orbifold H^2/PSL_2(Z) lifts to a modular pattern. Let S^1 be the ideal
boundary of H^2. Given two points p,q in S^1 we write pq if p and q are the
endpoints of a geodesic in Gamma. (In particular pp.) We show that is an
equivalence relation. We let Q_Gamma=S^1/ be the quotient space. We call
Q_Gamma a modular circle quotient. In this paper we will give a sense of what
modular circle quotients `look like' by realizing them as limit sets of
piecewise-linear group actions
