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

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 $G$.

86 pages with 53 figures. Improved presentation of section introducing Jacobi coordinates, improved Figures and a few further references

References in corpus (18)

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Cited by in corpus (20)

• The Smallest Shape Spaces. II. 4 Points in 1-d Suffices to have a Complex Background-Independent Theory of Inhomogeneity
• The Smallest Shape Spaces. III. Triangles in 2- and 3-d
• Specific PDEs for Preserved Quantities in Geometry. I. Similarities and Subgroups
• Alice in Triangleland: Lewis Carroll's Pillow Problem and Variants Solved on Shape Space of Triangles
• A Local Resolution of the Problem of Time. I. Introduction and Temporal Relationalism
• Spaces of Observables from Solving PDEs. I. Translation-Invariant Theory
• Two new versions of Heron's Formula
• Geometry from Brackets Consistency
• $N$-Body Problem: Minimal $N$'s for Qualitative Nontrivialities
• Monopoles of Twelve Types in 3-Body Problems
• Maximal Angle Flow on the Shape Sphere of Triangles
• Shape (In)dependent Inequalities for Triangleland's Jacobi and Democratic-Linear Ellipticity Quantitities
• Specific PDEs for Preserved Quantities in Geometry. II. Affine Transformations and Subgroups
• Specific PDEs for Preserved Quantities in Geometry. III. 1-d Projective Transformations and Subgroups
• Background Independence: ${\mathbb{S}}^1$ and $\mathbb{R}$ absolute spaces differ greatly in Shape-and-Scale Theory
• Topological Shape Theory
• Rubber Relationalism: Smallest Graph-Theoretically Nontrivial Leibniz Spaces
• Quadrilaterals in Shape Theory. II. Alternative Derivations of Shape Space: Successes and Limitations
• Isotropy Groups and Kinematic Orbits for 1 and 2-$d$ $N$-Body Problems
• A Local Resolution of the Problem of Time. VIII. Assignment of Observables