  Examples are given of the creation of closed timelike curves by choices of
coordinate identifications. Following G\"odel's prescription, it is seen that
flat spacetime can produce closed timelike curves with structure similar to
that of G\"odel. In this context, coordinate identifications rather than exotic
gravitational effects of general relativity are shown to be the source of
closed timelike curves. Removing the periodic time coordinate restriction, the
modified G\"odel family of curves is expressed in a form that retains the
timelike and spacelike character of the coordinates. With these coordinates,
the nature of the timelike curves is clarified. A helicoidal surface unifies
the families of timelike, spacelike and null curves. In all of these, it is
seen that as in ordinary flat spacetime, periodicity in the spatial position
does not naturally carry over into closure in time. Thus, the original source
of serious scientific speculation regarding time machines is seen to be
misconceived.
