[Paper Review] Rubber Relationalism: Smallest Graph-Theoretically Nontrivial Leibniz Spaces

Edward Anderson|arXiv (Cornell University)|May 9, 2018

Topological and Geometric Data Analysis30 references3 citations

TL;DR

This paper introduces rubber relationalism—a topological abstraction of Leibniz spaces in N-body problems—by modeling indistinguishable, mirror-image-identified point configurations as graphs encoding topological adjacency. It identifies the smallest nontrivial cases as N=5 for ℝ, N=6 for ℝᵈ (d≥2), and N=6 for 𝕊¹, with residue graphs revealing universal cone-point structures and N-specific information.

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 $d \geq 2$. For $d \geq 2$, 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.

Motivation & Objective

• To develop a topological analogue of Shape Theory by replacing metric-based relational spaces with graph-based 'rubber configurations' that encode only topological adjacency.
• To identify the smallest nontrivial relational spaces under Leibniz quotienting (indistinguishability and mirror-image identification) using graph-theoretic criteria.
• To determine minimal N for which rubber Leibniz spaces become nontrivial across three universal carrier space models: ℝ, ℝᵈ (d≥2), and 𝕊¹.
• To analyze the role of generic and maximal configurations as cone points and to extract N-specific information via deconing operations.
• To establish connections between rubber relationalism and applications in background-independent physics, including GR topology change and the N-body problem.

Proposed method

• Modeling N-point configurations on carrier spaces (ℝ, ℝᵈ, 𝕊¹) as graphs where edges represent topological adjacency, forming 'rubber relational spaces'.
• Applying Leibniz quotienting—identifying indistinguishable points and mirror images—to reduce configuration spaces to minimal relational graphs.
• Using partition theory to show that rubber configurations in d≥2 dimensions correspond bijectively to set partitions, with ℝ and 𝕊¹ cases providing successive refinements.
• Defining 'residue graphs' by removing cone points (generic and maximal configurations), thereby isolating N-specific topological information.
• Applying graph-theoretical nontriviality criteria to determine minimal N for nontriviality: N=6,6,5 for the three models, with stronger criteria yielding N=8,8,6.
• Leveraging lattice theory and Hasse diagrams to interpret quotiented configuration spaces as bounded lattices, with antitone duality linking group actions to quotient structures.

Experimental results

Research questions

• RQ1 What is the smallest number of points N for which rubber Leibniz spaces become graph-theoretically nontrivial in ℝ, ℝᵈ (d≥2), and 𝕊¹?
• RQ2 How do generic and maximal configurations manifest as cone points in rubber relational graphs, and what is their role in the topology of the configuration space?
• RQ3 To what extent do residue graphs after deconing encode N-specific information about the configuration?
• RQ4 How do the ℝ, ℝᵈ (d≥2), and 𝕊¹ models differ in their refinement of partition structures, and what universality classes emerge?
• RQ5 Can rubber relationalism serve as a simplified model for topology change and the N-body problem in background-independent theories?

Key findings

• The smallest nontrivial rubber Leibniz space occurs at N=5 for the ℝ carrier space, with the 'helm' graph as the residue graph.
• For d≥2, the minimal nontrivial case is N=6, with the 'submarine' graph as the residue graph, representing the joint universal class for ℝᵈ.
• For the 𝕊¹ carrier space, the minimal nontrivial case is also N=6, with the 'aircraft carrier' graph as the residue graph.
• Generic and maximal configurations universally appear as cone points in all three models, with deconing yielding residue graphs that encode N-specific relational structure.
• Stronger nontriviality criteria identify N=8,8,6 as minimal for the three models, indicating a higher threshold for nontriviality under stricter conditions.
• The rubber relational spaces for d≥2 are in one-to-one correspondence with set partitions, and the full relational structure is captured by lattice-theoretic duality between group actions and quotiented spaces.