  We consider the lowest Landau level on a torus as a function of its
circumference $L_1$. When $L_1\to 0$, the ground state at general rational
filling fraction is a crystal with a gap--a Tao-Thouless state. For filling
fractions $\nu=p/(2pm+1)$, these states are the limits of Laughlin's or Jain's
wave functions describing the gapped quantum Hall states when $L_1\to \infty$.
For the half-filled Landau level, there is a transition to a Fermi sea of
non-interacting neutral dipoles, or rather to a Luttinger liquid modification
thereof, at $L_1\sim5$ magnetic lengths. This state is a version of the
Rezayi-Read state, and develops continuously into the state that is believed to
describe the observed metallic phase as $L_1\to \infty$. Furthermore, the
effective Landau level structure that emerges within the lowest Landau level
follows from the magnetic symmetries.
