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

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.

42 pages with 36 figures. Minor changes, a few new references

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Cited by in corpus (19)

• The Smallest Shape Spaces. I. Shape Theory Posed, with Example of 3 Points on the Line
• The Smallest Shape Spaces. III. Triangles in 2- and 3-d
• A Local Resolution of the Problem of Time. I. Introduction and Temporal Relationalism
• 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
• Spaces of Observables from Solving PDEs. I. Translation-Invariant Theory
• Geometry from Brackets Consistency
• Two new versions of Heron's Formula
• $N$-Body Problem: Minimal $N$'s for Qualitative Nontrivialities
• Maximal Angle Flow on the Shape Sphere of Triangles
• Monopoles of Twelve Types in 3-Body Problems
• 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