  We consider spin-polarized electrons in a single Landau level on a torus. The
quantum Hall problem is mapped onto a one-dimensional lattice model with
lattice constant $2\pi/L_1$, where $L_1$ is a circumference of the torus (in
units of the magnetic length). In the Tao-Thouless limit, $L_1\to 0$, the
interacting many-electron problem is exactly diagonalized at any rational
filling factor $\nu=p/q\le 1$. For odd $q$, the ground state has the same
qualitative properties as a bulk ($L_1 \to \infty$) quantum Hall hierarchy
state and the lowest energy quasiparticle exitations have the same fractional
charges as in the bulk. These states are the $L_1 \to 0$ limits of the
Laughlin/Jain wave functions for filling fractions where these exist. We argue
that the exact solutions generically, for odd $q$, are continuously connected
to the two-dimensional bulk quantum Hall hierarchy states, {\it ie} that there
is no phase transition as $L_1 \to \infty$ for filling factors where such
states can be observed. For even denominator fractions, a phase transition
occurs as $L_1$ increases. For $\nu=1/2$ this leads to the system being mapped
onto a Luttinger liquid of neutral particles at small but finite $L_1$, this
then develops continuously into the composite fermion wave function that is
believed to describe the bulk $\nu=1/2$ system. The analysis generalizes to
non-abelian quantum Hall states.
