The hard sphere system is discussed with the focus on geometry. Three types of exact relations are derived: (i) purely definitional; (ii) definitional and geometric; and (iii) definitional, geometric, and thermodynamic. This careful separation allows a critical study of the statistical thermodynamics of hard spheres. In particular, a geometric Gibbs equation is derived containing only the fundamental geometric parameters of the system, and a set of exact lsquo;lsquo;rulesrsquo;rsquo; can be derived from this equation restricting the behaviors of the fundamental parameters over the entire density range. The geometric Gibbs equation (and the rules) are useful in the derivation of the thermodynamic equation of state, but such application is reserved for a following paper.
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