The Smallest Shape Spaces. II. 4 Points in 1-d Suffices to have a Complex Background-Independent Theory of Inhomogeneity

TL;DR

This paper analyzes shape spaces for 4 points in 1-d, revealing complex topological and metric structures that model inhomogeneity and uniformity, with implications for cosmology, background independence, and quantum gravity.

Contribution

It provides a detailed topological and metric analysis of shape spaces for 4 points in 1-d, including tessellations, quantifiers, and automorphism groups, advancing understanding of background independence.

Findings

Shape space graphs are highly complex topologically.

Metric shape spaces are pieces of spheres with significant tessellations.

Various configurations realize inhomogeneity and uniformity distinctly.

Abstract

The program of understanding Shape Theory layer by layer topologically and geometrically -- proposed in Part I -- is now addressed for 4 points in 1-$d$. Topological shape space graphs are far more complex here, whereas metric shape spaces are (pieces of) spheres which admit an intricate shape-theoretically significant tessellation. Metric shapes covers a far wider range of notions of inhomogeneity: collisions, symmetric states, mergers and uniform states are all distinctly realized in this model. We furthermore provide quantifiers for the extent to which various ways which configurations maximally and minimally realize these. Some of the uniform states additionally form cusps and higher catastrophes in the indistinguishable-particle and Leibniz shape spaces. We also provide shape-theoretically significant notions of centre for the indistinguishable-particle and Leibniz shape spaces. 4 points in 1-$d$ constitutes a useful and already highly nontrivial model of inhomogeneity and of uniformity -- both topics of cosmological interest -- and also of background independence: of interest in the foundations of physics and in quantum gravity. We finally give the automorphism groups of the topological shape space graphs and the metric shape space (pieces of) manifolds, which is a crucial preliminary toward quantizing the indistinguishable-particle and Leibniz versions of the model.