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Kendall-type Shape(-and-Scale) Theory on $\mathbb{R}^d$ involves $N$ pointconfigurations therein quotiented by some geometrically meaningful automorphismgroup. This occurs in Shape Statistics, the Classical and Quantum $N$-bodyProblem and as a model for many aspects of Generally Relativistic theories'Background Independence. Shape-and-Scale theory on the circle $\mathbb{S}^1$ issignificant at the level of `rubber shapes' as 1 of only 3 classes ofconnected-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 motivatedby spatially-closed GR and $\mathbb{RP}^d$ by Image Analysis and ComputerVision. 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 showmoreover that 3 points on the circle already suffices for major differences tooccur relative to on the line $\mathbb{R}$. Scale is now obligatory. Totallyantipodal configurations are as significant as the maximal collision.Topologically, partially antipodal configurations play an equivalent role toright angles: specifically a $d \geq 2$ notion on $\mathbb{R}^d$. Using up lessand more arc than an antipodal configuration are the respective topologicalanalogues of acute and obtuse triangles. The idea that quotienting outgeometrical automorphisms banishes an incipient notion of absolute space isdead. Such indirect modelling is, rather, well capable of remembering theincipient absolute space's topology. Thus topological considerations ofBackground Independence have become indispensible even in mechanics models. InGeneral Relativity, this corresponds to passing from Wheeler's Superspace toFischer's Big Superspace.

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