Background Independence: $\mathbb{S}^1$ and $\mathbb{R}$ absolute spaces differ greatly in Shape-and-Scale Theory

Abstract: Kendall-type Shape(-and-Scale) Theory on $\mathbb{R}^d$ involves $N$ point configurations therein quotiented by some geometrically meaningful automorphism group. This occurs in Shape Statistics, the Classical and Quantum $N$-body Problem and as a model for many aspects of Generally Relativistic theories' Background Independence. Shape-and-Scale theory on the circle $\mathbb{S}^1$ is significant at the level of `rubber shapes' as 1 of only 3 classes of connected-without-boundary absolute spaces. It is also the first $\mathbb{T^d}$ and $\mathbb{RP}^d$ as well as the first sphere; spheres and tori are motivated by spatially-closed GR and $\mathbb{RP}^d$ by Image Analysis and Computer Vision. We now investigate the $\mathbb{S}^1$ case at the geometrical level. With $Isom(\mathbb{S}^1) = SO(2)$ itself a $\mathbb{S}^1$, the shape-and-scale $N$-body configuration spaces are systematically $\mathbb{T}^{N - 1}$. We show moreover that 3 points on the circle already suffices for major differences to occur relative to on the line $\mathbb{R}$. Scale is now obligatory. Totally antipodal configurations are as significant as the maximal collision. Topologically, partially antipodal configurations play an equivalent role to right angles: specifically a $d \geq 2$ notion on $\mathbb{R}^d$. Using up less and more arc than an antipodal configuration are the respective topological analogues of acute and obtuse triangles. The idea that quotienting out geometrical automorphisms banishes an incipient notion of absolute space is dead. Such indirect modelling is, rather, well capable of remembering the incipient absolute space's topology. Thus topological considerations of Background Independence have become indispensible even in mechanics models. In General Relativity, this corresponds to passing from Wheeler's Superspace to Fischer's Big Superspace.