 
Summary: ATrainTrack?
Lee Mosher
354 NOTICES OF THE AMS VOLUME 50, NUMBER 3
On a surface S, train tracks approximate simple
closed curves just as partial quotients of contin
ued fraction expansions approximate rational num
bers. The simple closed curve on the 4punctured
sphere (see photograph on cover and p. 356) that,
in about 1972, was painted on the wall of the UC
Berkeley math department by William P. Thurston
and Dennis Sullivan is approximated by the train
track shown in Figure 1(a). To visualize the ap
proximation, blur your eyes so that parallel strands
of the curve merge into branches of the train track
and so that diverging strands split apart at switches
of the train track. Train tracks were introduced by
Thurston in the late 1970s as a means of studying
simple closed curves and related structures on
surfaces.
In general, the surface S should be of finite
