Topological Shape Theory

Abstract

Kendall's Similarity Shape Theory for constellations of points in the carrier space ${\mathbb{R}}^n$ was developed for use in Probability and Statistics. It was subsequently shown to reside within (Classical and Quantum) Mechanics' Shape-and-Scale Theory, in which the points are interpreted as particles and the carrier space plays the role of absolute space. In other more recent work, Kendall's Similarity Shape Theory has been generalized to affine, projective, conformal and supersymmetric versions, as well as to ${\mathbb{T}}^n$, ${\mathbb{S}}^n$, ${\mathbb{RP}}^n$ and Minkowski spacetime carrier spaces. This has created a sizeable field of study: generalized Kendall-type Geometrical Shape(-and-Scale) Theory. Aside from offering a wider range of shape-statistical applications, this field of study is an exposition of models of Background Independence of relevance to the Absolute versus Relational Motion Debate, and the Foundations and Dynamics of General Relativity and Quantum Gravity. In the current article, we consider a simpler type of Shape Theory, comprising relatively few types of behaviour: the Topological Shape Theory of rubber shapes. This underlies the above much greater diversity of more structured Shape Theories; in contrast with the latter's (stratified) manifolds shape spaces, the former's are just graphs. We give examples of these graphs for the smallest nontrivial point-or-particle numbers, and outline how these feature within a wider range of Geometric Shape Theories' shape spaces, and are furthermore straightforward to do Statistics, Dynamics and Quantization with.

34 pages including 25 figures

References in corpus (20)

• Some remarks on the notions of general covariance and background independence
• Conditional probabilities with Dirac observables and the problem of time in quantum gravity
• Nonparametric statistics on manifolds with applications to shape spaces
• Foundations of Relational Particle Dynamics
• Problem of Time and Background Independence: the Individual Facets
• Explicit partial and functional differential equations for beables or observables
• On Types of Observables in Constrained Theories
• The Smallest Shape Spaces. I. Shape Theory Posed, with Example of 3 Points on the Line
• Consciousness and the Quantum
• 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
• Relational mechanics of shape and scale
• Configuration Spaces in Fundamental Physics
• TRiPoD (Temporal Relationalism incorporating Principles of Dynamics)
• Manifolds of Projective Shapes
• Alice in Triangleland: Lewis Carroll's Pillow Problem and Variants Solved on Shape Space of Triangles
• Two new versions of Heron's Formula
• 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

Cited by in corpus (4)

• $N$-Body Problem: Minimal $N$'s for Qualitative Nontrivialities
• Background Independence: ${\mathbb{S}}^1$ and $\mathbb{R}$ absolute spaces differ greatly in Shape-and-Scale Theory
• Rubber Relationalism: Smallest Graph-Theoretically Nontrivial Leibniz Spaces
• Quadrilaterals in Shape Theory. II. Alternative Derivations of Shape Space: Successes and Limitations