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 surfa
ces 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 phasetextcolor{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.
