  Hard sphere systems are often used to model simple fluids. The configuration
spaces of hard spheres in a three-dimensional torus modulo various symmetry
groups are comparatively simple, and could provide valuable information about
the nature of phase transitions. Specifically, the topological changes in the
configuration space as a function of packing fraction have been conjectured to
be related to the onset of first-order phase transitions. The critical
configurations for one to twelve spheres are sampled using a Morse-theoretic
approach, and are available in an online, interactive database. Explicit
triangulations are constructed for the configuration spaces of the two sphere
system, and their topological and geometric properties are studied. The
critical configurations are found to be associated with geometric changes to
the configuration space that connect previously distant regions and reduce the
configuration space diameter as measured by the commute time and diffusion
distances. The number of such critical configurations around the packing
fraction of the solid-liquid phase transition increases exponentially with the
number of spheres, suggesting that the onset of the first-order phase
transition in the thermodynamic limit is associated with a discontinuity in the
configuration space diameter.
