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

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 $C{P}^{N-2}$ (with $C{P}^1=S^2$ 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).

arXiv comment17 pages, including 7 figures

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• Geometry from Brackets Consistency