Quadrilaterals in Shape Theory. II. Alternative Derivations of Shape Space: Successes and Limitations

TL;DR

This paper explores the geometric and topological properties of shape spaces for polygons and simplices, highlighting limitations of Heron's formula for quadrilaterals and extending the theory to higher dimensions with Cayley-Menger formulas.

Contribution

It clarifies the limitations of Heron's formula for quadrilaterals, generalizes shape spaces to higher dimensions using Cayley-Menger formulas, and discusses the topological structure of these spaces.

Findings

Triangle shape space is topologically a 2-sphere (S^2).

Heron's formula does not extend to quadrilaterals for shape space derivation.

Cayley-Menger formulas provide shape quantities for higher-dimensional simplices.

Abstract

We show that the recent derivation that triangleland's topology and geometry is $S^2$ from Heron's formula does not extend to quadrilaterals by considering Brahmagupta, Bretschneider and Coolidge's area formulae. That $N$-a-gonland is more generally $CP^{N - 2}$ (with $CP^1 = S^2$ recovering the triangleland sphere) follows from Kendall's extremization that is habitually used in Shape Theory, or the generalized Hopf map. We further explain our observation of non-extension in terms of total area not providing a shape quantity for quadrilaterals. It is rather the square root of of sums of squares of subsystem areas that provides a shape quantity; we clarify this further in representation-theoretic terms. The triangleland $S^2$ moreover also generalizes to $d$-simplexlands being $S^{d(d + 1)/2 - 1}$ topologically by Casson's observation. For the 3-simplex - alias tetrahaedron - while volume provides a shape quantity and is specified by the della Francesca-Tartaglia formula, the analogue of finding Heron eigenvectors is undefined. $d$-volume moreover provides a shape quantity for the $d$-simplex, specified by the Cayley-Menger formula generalization of the Heron and della Francesca-Tartaglia formulae. While eigenvectors can be defined for the even-$d$ Cayley-Menger formulae, the dimension count does not however work out for these to provide on-sphere conditions. We finally point out the multiple dimensional coincidences behind the derivation of the space of triangles from Heron's formula. This article is a useful check on how far the least technically involved derivation of the smallest nontrivial shape space can be taken. This is significant since Shape Theory is a futuristic branch of mathematics, with substantial applications in both Statistics (Shape Statistics) and Theoretical Physics (Background Independence: of major relevance to Classical and Quantum Gravitational Theory).