  We report on the results of a study of the motion of a four particle
non-relativistic one-dimensional self-gravitating system. We show that the
system can be visualized in terms of a single particle moving within a
potential whose equipotential surfaces are shaped like a box of pyramid-shaped
sides. As such this is the largest $N$-body system that can be visualized in
this way. We describe how to classify possible states of motion in terms of
Braid Group operators, generalizing this to $N$ bodies. We find that the
structure of the phase\textcolor{black}{{} space of each of these systems
yields a large variety of interesting dynamics, containing regions of
quasiperiodicity and chaos. Lyapunov exponents are calculated for many
trajectories to measure stochasticity and previously unseen phenomena in the
Lyapunov graphs are observed.
