Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Quadrilaterals in Shape Theory. II. Alternative Derivations of Shape
  Space: Successes and Limitations   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).

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The categories of this paper are: hep-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: q-fin