The Smallest Shape Spaces. I. Shape Theory Posed, with Example of 3 Points on the Line

TL;DR

This paper develops a shape theory framework focusing on configuration spaces of points, exploring their topological and geometric properties, with detailed analysis of 3 points on a line as a foundational example.

Contribution

It introduces a shape-theoretic approach to configuration spaces, emphasizing topological, combinatorial, and geometric structures, and highlights the importance of the Aufbau Principle in shape analysis.

Findings

Shape spaces are represented as graphs, enabling combinatorial analysis.

Topological features of shape spaces are crucial for understanding automorphisms.

Mirror image and indistinguishability quotients affect shape symmetries and dynamics.

Abstract

This treatise concerns shapes in the sense of constellations of points with various automorphisms quotiented out: continuous translations, rotations and dilations, and also discrete mirror image identification and labelling indistinguishability of the points. We consider in particular the corresponding configuration spaces, which include shape spaces and shape-and-scale spaces. This is a substantial model arena for developing concepts of Background Independence, with many analogies to General Relativity and Quantum Gravity, also with many applications to Dynamics, Quantization, Probability and Statistics. We also explain the necessity of working within the shape-theoretic Aufbau Principle: only considering larger particle number $N$, spatial dimension $d$ and continuous group of automorphisms $G$ when all the relatively smaller cases have been considered. We show that topological shape spaces are graphs, opening up hitherto untapped combinatorial foundations both for these and for topological features of the more usually-considered spaces of metric shapes. We give a conceptual analysis of inhomogeneous Background Independence's clustering and uniformness aspects. We also consider the fate of shape spaces' (similarity) Killing vectors upon performing the mirror image and particle indistinguishability quotientings; this is crucial for dynamical and quantization considerations. For now in Part I we illustrate all these topological, combinatorial, differential-geometric and inhomogeneity innovations with the example of 3 points in 1-$d$. Papers II to IV then extend the repertoire of examples to 4 points in 1-$d$, triangles (3 points in 2- and 3-$d$) and quadrilaterals (4 points in 2-$d$) respectively. The quadrilateral is a minimal requirement prior to most implementations of the third part of the shape-theoretic Aufbau Principle: adding further generators to the automorphism group $G$.