The Smallest Shape Spaces. III. Triangles in 2- and 3-d

Abstract

This is an innovative treatise on triangles, resting upon 1) 3-body problem techniques including mass-weighted relative Jacobi coordinates. 2) Part I's detailed layer-by-layer topological and geometrical study of Kendall-type shape spaces - configuration spaces of all possible shapes - which, for triangles, are (pieces of) spheres. 3) Hopf mathematics. Triangles are moreover prototypical through being the smallest models which carry relative-angle as well as length-ratio information. Both 1) and 3) produce insightful new versions of Heron's formula, 3)'s simultaneously providing new foundations for 2). Medians, and regular triangles bounding between tall and flat triangles, also play prominent roles. Right triangles form three kissing cap-circles on the shape sphere, from which a shape-theoretic answer to the well-known conundrum of what is the probability that a triangle is obtuse very readily follows: 3/4. The differential-geometric aspects of this answer moreover generalize to numerous variant problems. Hopf mathematics additionally gives a general bundle section interpretation to Kendall's iconic spherical blackboard of vertex-unlablelled mirror-image-identified triangles, and of its two variants where one of these two conditions are dropped. We attribute a monopole to each of these spaces and to the full shape sphere, one due to Dirac, one to Iwai and the other two are new to this paper. We finally make insightful comparison of triangles in 2-$d$ with a) Part II's 4 points on the line. b) Triangles in 3-$d$, which are particularly significant as the smallest model exhibiting stratification. Stratified manifold-sheaf pairs - sheaves adding useful local and global structure to general bundles - lie at the heart of Shape Theory's future development.

84 pages and 59 figures. This v2 has several substantial new sections and 9 new figures

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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. II. 4 Points in 1-d Suffices to have a Complex Background-Independent Theory of Inhomogeneity
• 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