$N$-Body Problem: Minimal $N$'s for Qualitative Nontrivialities

We review the $N$-Body Problem in arbitrary dimension $d$ at the kinematical level, with modelling Background Independence in mind. In particular, we give a structural analysis of its reduced configuration spaces, decomposing this subject matter into basic Topology, Geometry, Group Theory, Linear Algebra, Graph Theory and Order Theory. At the metric level, these configuration spaces of shapes form basic geometric series for 1- and 2-$d$: $\mathbb{S}^{N - 2}$ and $\mathbb{CP}^{N - 2}$, though there are no more such series for $d \geq 3$. $d \geq 3$ also sees an onset of stratification. Casson's diagonal, for which $N = d + 1$, plays a critical role which we explain in simple Linear Algebra terms; these are moreover topologically spheres. $N = d$ and $N = d + 2$ have further significance as well, the latter as regards a counting notion of genericity of the isotropy groups and kinematical orbits realized. These observations twin the $N$ = (3, 4, 5) progression in qualitative complexity that almost all $N$-Body Problem work for concrete $N$ concentrates on with the much better-known $N$ = (2, 3, 4) progression in 2-$d$: from intervals to triangles to quadrilaterals: a large source of intuitions and mathematical analogies. We furthermore provide an Order-Theoretic genericity criterion, which is almost always bounded by the double-slope $N = 2 d + 1$ line, though an accidental relation pushes $N$ up by 1 to 8 in 3-$d$. We finally consider rubber shapes, for which the configuration spaces are graphs: much simpler than stratified manifolds, and yet containing quite a few of metric-level shapes' qualitative features. This singles out $N = 5$ and 6 in 1-$d$ and $N = 6$ and 8 in $\geq$ 2-$d$ for the onsets of various graph-theoretical nontrivialities.