Rubber Relationalism: Smallest Graph-Theoretically Nontrivial Leibniz Spaces

Abstract

Kendall's Similarity Shape Theory for constellations of N points in the carrier space ${\mathbb{R}}^d$ as quotiented by the similarity group was developed for use in Probability and Statistics. It was subsequently shown to reside within Mechanics' Shape-and-Scale Theory, in which points are interpreted as particles, carrier space plays the role of absolute space, and the Euclidean group is quotiented out. Let us jointly refer to Shape(-and-Scale) Theory as Relational Theory, and to its reduced configuration spaces as relational spaces. We now consider a less structured version: the Topological Relational Theory of `rubber configurations'. This already encodes some features of the much more diverse Geometrical Relational Theories. In contrast with the latter's (stratified) manifold relational spaces, the former's are graphs: much simpler to treat; their edges encode topological adjacency. We concentrate on Leibniz spaces, corresponding to indistinguishable points and mirror-image identification. These are moreover the building blocks of the distinguishable and (where possible) mirror-image distinct cases' relational spaces. For connected manifold without boundary carrier spaces, there are just 3 'rubber relationalisms: $\mathbb{R}$, ${\mathbb{S}}^1$, and a joint one for all carrier spaces with $d\ge 2$. For $d\ge 2$, rubber configurations are in 1:1 correspondence with partitions, with ${\mathbb{S}}^1$ and $\mathbb{R}$ giving successive refinements. We find that generic and maximal configurations are universally present as cone points, as are binaries in the first 2 cases. Deconing leaves us with residue graphs containing the N-specific information. We provide graph-theoretical nontriviality criteria for which N = 6, 6 and 5 are minimal across these models, and stronger such for which N = 8, 8 and 6 are minimal, and outline GR topology-change analogue-model and N-body problem applications.

35 pages including 31 figures

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• $N$-Body Problem: Minimal $N$'s for Qualitative Nontrivialities
• Quadrilaterals in Shape Theory. II. Alternative Derivations of Shape Space: Successes and Limitations
• Comparative Theory of Background Independence