Quadrilaterals in Shape Theory. II.
Alternative Derivations of Shape Space: Successes and Limitations
Edward Anderson -
We show that the recent derivation that triangleland’s topology and geometry is from Heron’s formula does not extend to quadrilaterals by considering Brahmagupta, Bretschneider and Coolidge’s area formulae. That -a-gonland is more generally (with 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 moreover also generalizes to -simplexlands being 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. -volume moreover provides a shape quantity for the -simplex, specified by the Cayley–Menger formula generalization of the Heron and della Francesca–Tartaglia formulae. While eigenvectors can be defined for the even- 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).
PACS: 04.20.Cv, 02.40.-k, Physics keywords: Background Independence, configuration spaces, 4-body problem.
Mathematics keywords: Euclidean, Projective and Differential Geometry, Shape Theory, spaces of simplexes, Distance Geometry, Shape Statistics.
Dr.E.Anderson.Maths.Physics *at* protonmail.com
The carrier space , alias absolute space in the physical context is an at least provisional model for the structure of space. Constellation space (reviewed in ) is then the product of copies of this carrier space,
modelling points on , or, if these points are materially realized, particles (classical, nonrelativistic). In Kendall’s Shape Theory [18, 21, 32] (see also [30, 34, 39, 40, 44, 45, 47, 51, 49, 50, 53, 54, 55, 57, 64]), we consider this for furthermore with similarities quotiented out. The corresponding shape spaces – reduced configuration spaces – are thus of the form
Kendall’s work remains rather more familiar in the Shape Statistics literature [18, 21, 30, 32, 34, 45, 51, 49], though related work in other fields has also appeared in e.g. Mechanics and Molecular Physics [27, 29, 35, 39, 44, 57], and in [39, 44, 43, 52] as regards modelling some aspects of General Relativity’s Background Independence [16, 15, 25, 26, 46, 52, 60, 63].
As we outline in Sec 2, diagonalizing the usual Heron’s formula was recently demonstrated  to suffice as first principles to derive both
2) the Hopf map , in the extended sense of Fig 1.a-b).
The current Article considers the extent to which these results generalize to Shape Theory with other spatial dimensions and point-or-particle numbers . There are two directions of generalization (Fig 1.e) away from triangleland in 2-
both topologically and metrically, with standard Fubini–Study metric. This generalization makes sense via the accidental topological-and-geometrical relation
2) -simplexes, for which Casson  showed that, at the topological level,
For the quadrilaterals, we show in Sec 3 that Coolidge’s area formula is sequentially more satisfactory than Brettschneider’s and Brahmagupta’s for this application, but still falls short of requirements to provide a derivation of the topology and geometry of the space of quadrilaterals. What works instead for -a-gons is Kendall’s extremization or the generalized Hopf map in the extended sense of Fig 1.c)-d) (both outlined in the Appendix), or a Mechanics reduction for which references are given in the Conclusion.
We moreover further explain our observation of non-extension in Sec 4, in terms of total mass-weighted area per unit moment of inertia ,
 which is a shape quantity, the sum being over the three two-Jacobi-vector subsystems supported by the quadrilateral. We further clarify the reason for this object’s featuring in the theory of quadrilaterals in Representation-Theoretic terms.
We next consider whether there is a Casson’s Theorem generalization of the Heron proof of ‘Kendall’s Little Theorem’ in Sec 5. For the 3-simplexes alias tetrahaedrons (for which Sec 8 provides the notation), volume both has a general formula in terms of separations – the della Francesca–Tartaglia formula [5, 6] – and provides a shape quantity
(for the mass-weighted volume). We provide the analogue of the Heron–Buchholz  matrix reformulation in terms of a ‘Tartaglia 3-array’. We then observe that eigenvectors are however not defined for this, by which the Heron derivation of the triangleland shape space does not generalize to tetrahaedronland either.
In Sec 6, we finally consider the general -simplexes, for which the -volume is given by the Cayley–Menger generalization [7, 10, 13, 19] of the della Francesca–Tartaglia formula. -volume moreover provides a shape quantity
and eigenvectors can be defined for the even- cases. The dimension count does not however work out for this to give on-sphere conditions corresponding to Casson’s result for the topology of the spaces of -simplexes. These observations made, we point to the string of dimensional coincidences behind the Heron derivation in Secs 4 and 6.
2 Ouline of Triangleland shape space from Heron’s formula
Definition 1 Consider an arbitrary triangle , denoted as in Fig 2. Using , to to denote , , will also be useful for us.
Definition 2 The semi-perimeter is
Definition 3 We use for short when unambiguous, to denote the area of .
Proposition 1 (Heron’s formula)
Remark 1 Squaring, introducing the tetra-area variable
and expanding out,
Remark 2 This can furthermore be recast in Linear Algebra terms as the quadratic form,
for squared separations (for triangles, separations are just side-lengths) 3-vector
and ‘Heron–Buchholz matrix’ 
Remark 3 Setting
the eigenvalues are with multiplicity 1, and with multiplicity 2.
The corresponding orthonormal eigenvectors are, respectively,
Remark 4 The diagonalizing variables are thus
In terms of these, Heron’s formula also takes the following form.
Diagonal Heron formula
Then upon introducing the rescaled ratio variables 
whose denominator is proportional to the moment of inertia, (21) becomes
the on 2-sphere condition.
Corollary 1 This gives ‘Kendall’s Little Theorem’: that the space of triangles is a 2-sphere.
In this way, Heron’s formula provides the Hopf map as well as ‘Kendall’s Little Theorem’, by which the above derivation gains its Heron–Hopf–Kendall monicker. That squares of expressions (24-26) add to 1 is standard:
Remark 9 Hopf quantities are the form taken by triangleland’s shape quantities [41, 44, 55]. is geometrically an anisoscelesness: a measure of departure from isoscelesness, the mass-weighted area per unit moment of inertia. is geometrically an ellipticity: a quantifier of whether the triangle is tall or flat. Anisoscelesness and ellipticity can moreover now be interpreted as the two eigenvectors of the Heron map other than the total moment of inertia. This comment is useful as regards the current Article’s assessment of whether quadrilateral area formulae similarly involve quadrilateralland’s shape quantities .
3 Quadrilateral area formulae
Definition 1 Consider an arbitrary quadrilateral denoted as in Fig 3.
We consider in particular formulae for the area A of quadrilaterals.
Remark 1 Setting , this returns Heron’s formula.
Remark 2 In expanded form,
Unlike for Heron’s formula, the second expanded form precludes dependence on alone. This feature simplified repackaging Heron’s formula as a quadratic form in squared variables .
Remark 3 The formula being solely for cyclic quadrilaterals is also a limitation.
Remark 4 For a general quadrilateral, SSSS – four sides – is not sufficient data . Thus no area formula exists without bringing in some further datum.
Remark 5 One alternative in bringing in more data is to involving angles between sides as data. This leads to the following.
Proposition 2 (Bretschneider’s formula)  (1842) For a quadrilatral,
Remark 6 As indicated in Fig 3, and are opposite angles. Thus in the cyclic case, their sum is by an elementary Theorem of Euclid’s.
then accounts for the last term of this vanishing in Brahmagupta’s formula.
Remark 7 Thus we interpret the new term conceptually as
Remark 8 There are however two reasons why Bretschneider’s area formula does not provide a useful extension of Heron’s formula for shape-theoretic use.
Reason 1 Involvement of the conceptually heterogeneous angular information, moreover in multiplicative fashion on one factor.
Reason 2 does not depend on the sides via their squares.
These matters are moreover remedied by making use instead of diagonal length data. This is homogeneous with side length data through both constituting separation data. Indeed, in separational alias Lagrangian and dual constellational approaches  – an accurate description of Shape Theory – no distinction is to be made between sides and diagonals, so modelling purely in terms of unqualified separations remains faithful to this.
Proposition 3 (Coolidge’s formula)  (1939) For a quadrilateral,
and the last term’s first factor offering not other ways of being zero. We have thus found another formulation for the cyclator,
Corollary 1 The expanded version of Coolidge’s formula is moreover
which is additionally manifestly of the form
Remark 10 Coolidge’s formula thus generalizes Heron’s formula in ways 1) and 2). Thus it admits the following re-expression.
Corollary 2 The tetra-area squared is given by the Coolidge quadratic form
Here, is the 6-vector
Remark 11 The Coolidge matrix’s eigenvalues and eigenvectors – solving
– are as follows. We can solve this separately for its and block, obtaining the eigenspectrum
The corresponding eigenvectors are
for -2, 2 and 4 respectively and e.g.
for the 3- 0 eigenspaces.
Remark 12 These eigenvectors admit the following geometrical interpretation.
Note the dimensional mismatch: at best this gives a 5-sphere, without recourse for a second restriction down to .
4 Shape quantities for the quadrilateral
Preshape space itself is then
both topologically and metrically a sphere. In 1-, moreover, since there are no continuous rotations, preshape space is equivalent to shape space:
In 1-, the relative Jacobi scalars are Euclidean invariants. Let us normalize these using
which furthermore plays the role of  radius of preshape space radius This leaves us with quantities111This can also be found in the theory of internal rotations, alias democracy transformations in the Molecular Physics literature.
subject to the on- condition
being subject to the on- condition
For , are Euclidean invariants. There are
distinct such, and we can form a matrix out of them: the Euclidean matrix
For , this matrix has 3 independent elements and two invariants,
The second equality here is Lagrange’s identity, and is the mass-weighted area of the triangle. For , has six distinct elements and three invariants:
The first equality in the last equation is a generalization of Lagrange’s identity; this third invariant is moreover zero in 2-.
Normalizing by gives the similarity matrix
The trace is just a number . For , the similarity matrix has just one other invariant,
For , the similarity matrix has two other invariants,
Remark 1 In general, such matrix elements are not geometrically independent. One can see them as supplying an associated linear space of conserved quantities. They are not moreover the only possibilities.
has equiareal  significance. Being antisymmetric, this has
components. For , it has only one component, whereas for , it has 3. In all cases,
by ’s antisymmetry. For , the only other invariant is
whereas for , there are two:
gives moreover an of quantities. These can be repackaged as one scale invariant, , and an
of shape quantities. In 2-, the latter pick out an adjoint representation of the shape space isometry group ,
This , due to its corresponding to similarity geometry, is made by normalizing the by , so these are . These are 3 anisoscelesnesses, 3 areas, 1 ellipticity, and a linear combination of 2 ellipicities (corresponding to the ‘hypercharge’  in the Particle Physics counterpart).
In 2-, for (summing over all subsystem areas) is available to form a sextet of shape quantities , this last quantity being one which commutes with all the other shape quantities.
’s can also be viewed as the Casimir  corresponding to the repackaging of the . This concludes our Representation Theoretic insight into volume’s replacement, noting moreover that it is not area of the whole figure but rather the (square root of) the sum of squares of subsystem areas. Thus the ’remarkable extra commuting quantity’ description of this in the Molecular Physics literature  has been clarified by a combination of basic Geometry and elementary Representation Theory.
Problem 1 Observe that total area for is not among the shape quantities.
For use below and in the Conclusion, the -a-gon extension: has an
of shape quantities.
of these form an affinely significant restricted representation which carries
further Casimirs for even, or
General- -matrices, now with
symmetric nondiagonal anisoscelesnesses partnering antisymetric areas, represent these. Let us furthermore check how many pure and how many linear combination ellipticity objects these have in general, i.e. what the generalization of the and the hypercharge are. For even, we have
linear combinations of ellipticities. On the other hand, for odd, we have
ellipticities and an equal number of linear combinations of ellipticities.
Problem 2) The triangle’s 3 aligned 3- spaces are 6, 5 and 8- for quadrilateralland and so do not match up. 222The Veronese embedding provides a further relation between shape quantities [17, 42]; this projective-geometric approach moreover generalizes to the Veronese–Whitney embedding for higher -a-gons, a technique already well-known in the Shape Statistics literature [45, 51, 49]. Thus on this occasion, projective methods are more powerful than area formulae; for triangles, an area formula suffices but for quadrilaterals a projective method is needed.
More generally, for the -a-gon, these three numbers are
These can only coincide pairwise if one of the following holds.
So even the only possible pairwise coincidences have to take the triangle value , with occasional occurrence of the relationally trivial cases . Finally, it is clear that the only full solution admitted in . In this way, the triangle case’s Heron working is already unique at the basic combinatorial level.
5 Tetrahaedron set-up and Della Francesca–Tartaglia volume formula
Definition 1 Consider an arbitrary tetrahaedron denoted as in Fig 3. We set this up such that , , meet at one vertex and , , concur pairwise at the other three vertices , , .
We consider in particular formulae for the area A of quadrilaterals.
Tetra-area is not just the Heron quantity but also the first nontrivial Cayley–Menger quantity. Its generalization is
For even, the numerical factor , whereas for odd . For the tetrahaedron case, let us define
for V the volume.333The Introduction uses the convention that mass-weighted versions of objects are denoted in Greek letters, so there is the mass-weighted analogue of V here. Della Francesca–Tartaglia’s formula is then as per Fig 6.a). Expanding out, this can moreover be rewritten as
the separations 6-vector and the totally-symmetric Tartaglia 3-tensor, with components as per Fig 6.b).
Straightforwardly, there are no isotropic tensors of rank 3 in dimension 6, so questions of diagonalization are moot.
6 -simplices and their Cayley–Menger formulae
We finally extend consideration to a simplex: we show that Heron to Cayley–Menger generalization does not extend the Little Kendall’s Theorem from Heron’s formula proof. Secs 2 and 3 cover the first two nontrivial such: and 3.
The -dimensional Cayley–Menger formula is as per Fig 7.
Remark 1 This can be rewritten as
the separations -vector, for
the number of separations for point-or-particles, and
the Cayley–Menger -array, of which (17) and Fig 5.b) provide the explicit form for the first two.
By appearing contracted with equal factors, we can take the Cayley–Menger -array to be totally symmetric:
In forming an infinite series of totally symmetric tensors, these bear some analogy to multipole expansion tensors . Those are however spatial tensors whereas Cayley–Menger tensors are configuration space tensors. It follows that the monopole expansion resides within a fixed dimension (usually ), whereas Cayley–Menger tensors increase in dimension according to
We are to next contemplate diagonalizing these, i.e. solving
for the -dimensional matchingly totally symmetric isotropic tensor. This gives separate even and odd cases.
On the one hand, for odd , there are no rank- isotropic tensors in dimension , so (104) is moot.
On the other hand, for even- (> 0), we do have an eigenvalue problem, producing
eigenvalues. This is to be compared with the Casson sphere corresponding dimensionally to
quantities summing to 1. [ remains available as a scale quantity along the Casson diagonal.] So, for even, we would require
which can only be solved for by
Thus, even just combinatorially, the extent of the Cayley–Menger–Casson coincidence is just the Heron–Kendall–Casson coincidence yielding the 2-sphere of triangles.
It was recently shown  that for triangle constellations in 2-, the form taken by the corresponding shape space – a sphere – can be derived from Heron’s formula. This is a fourth derivation of this result: ‘Kendall’s Little Theorem’. The other three are Kendall’s extremization  and using the Hopf map (both outlined in the Appendix), and setting up an indirectly-formulated similarity mechanics action for the problem and conducting Lagrangian-level reduction  (or similar [44, 52]). The Hopf map itself is moreover derived from Heron’s formula in the process. Its three coordinate functions – mass-weighted tetra-area, anisoscelesness and ellipticity in the shape theoretic context – arise as the subject of Heron’s formula and the two non-unit eigenvectors of the Heron–Buchholz matrix.
In the current article, however, we show that this fourth derivation is a one-off, firstly in the sense that not even quadrilateral constellations admit an analogous derivation from an area formula. While the passage from Brahmagupta’s formula to Bretschneider’s and then to Coolidge’s is shown to be progressively more suitable for Shape Theory, even Coolidge’s formula fails to yield a comparable result. One underlying reason for this is that the shape space of quadrilaterals is (and that of -a-gons is more generally: Kendall’s Theorem). This is somewhat more structurally complex than , so some methods of arriving at this 2- space are ‘spherical’ rather than ‘projective’ and so fail to generalize to projective cases. All the other three derivations mentioned above moreover do generalize to -a-gons, with the Hopf case proceeding via one of the usual generalizations of the Hopf map, as per Fig 1.c-d). The -a-gon versions of these derivations are covered likewise in the Appendix and [32, 39, 44, 52]. Another underlying reason is because total area of a figure (after mass-weighting and dividing by the matching power of the moment of inertia) does not constitute a shape quantity in 2- for . It is, rather, the sum of squares of the triangle subsystems supported by a given choice of Jacobi coordinates which, upon being dressed in this manner, constitutes a shape quantity . In the process of analyzing this, we render clear that this ’remarkable extra commuting quantity’ – as described in the Molecular Physics literature  – has been given a clear explanation rooted in both basic Geometry and elementary Representation Theory.
Some dimensional coincidences supporting special features of the triangle case, and its Heron derivation, were also provided in Sec 4. These rely on scaled triangles having 3 independent relational coordinates, 3 relative separations and 3 shape quantities. Moreover, the relational space of scaled triangles is (conformal to) : the space in which spheres are most commonly modelled as extrinsically residing within. But these threes become
for -agons, for which we showed that even just double coincidences imply that , or the relationally trivial 0 or 1, and is the sole triple solution.
We also explained that the triangeland is also the first nontrivial simplexland alias basisland: the
diagonal which Casson showed to be topologically . Thus ‘Kendall’s Little Theorem’ may additionally be viewed as the unique case in which both Kendall’s Theorem and Casson’s Theorem apply. This means that whether derivations of ‘Kendall’s Little Theorem’ admit Casson’s Theorem generalizations should also be checked.
The next case along on the simplexland diagonal is tetrahaedronland. This admits the della Francesca–Tartaglia volume formula, as is most easily seen from recasting Heron’s formula as a determinant and generalizing dimensionally. This can be reformulated in terms of a (new, as far as the Author is aware) ‘totally symmetric Tartaglia 3-tensor’ in the 6- separation space supported by tetrahaedral constellations. This however does not admit eigenvalues and eigenvectors since there are no isotropic tensors of rank 3 in 6-.
Our next observations are that the entire simplexland diagonal is populated by the hypervolume formulae of Cayley–Menger [7, 10, 13, 19]. That these can be reformulated in terms of for the (also new, as far as the Author is aware) infinite series of Cayley–Menger totally-symmetric -tensors. That mass-weighted -hypervolume divided by a matching power of the moment of inertia is a suitable shape quantity for each . That the even- cases among these do possess the right dimensions-and-rank isotropic tensors to have eigenvalue problems. These eigenvalue problems, however, do not have the right dimensionality to produce on- conditions, other than in the case in which the Heron derivation of ‘Kendall’s Little Theorem’ is recovered.
 and the current article demonstrate that combining basic undergraduate Geometry and Linear Algebra is still capable of producing new results. Taking the stance that Shape Theory is a futuristic branch of both Statistics and Background-Independent Physics, we note moreover that maximally straightforward derivations of its simplest results is of considerable pedagogical significance. Heron’s formula is widely known in schools across the globe, and at least in the U.K., eigenvalues and eignevectors are within what is taught to mathematically-inclined high school students in their final year. So a derivation of ‘Kendall’s Little Theorem’ that the space of triangles is a sphere based on just these elements renders Shape Theory presentable to mathematically-inclined high-school students. Kendall’s extremization in turn lies within the grasp of undergraduates majoring in mathematics, whereas the Hopf map and Mechanics reduction workings are more commensurate with what is covered in graduate school (be that Geometry, Geometrical Methods, or a partly geometrical course in Dynamics or Statistics). It is therefore interesting to see how far one can take each route to Shape Theory, with the current Article revealing that the technically-simplest of all known derivations of a nontrivially shape-theoretic result not extending beyond the first case of triangles. The current article thus represents a caveat in the most introductory exposition of Shape Theory, that while knowing Heron’s formula and eigenvalue–eigenvector workings suffices to obtain the triangleland sphere, other kinds of derivations are required for the larger shape spaces.
Let us finally note that the Hopf map generalizing along the -a-gonlands plays a deep underpinning role in these being more geometrically understood than the simplexland diagonal. This is with particular reference to Kendall’s Theorem being a metric as well as topological result, whereas Casson’s Theorem is just topological. This, and accumulated knowledge about the metric geometry and associated linear methods for , makes -a-gons far more amenable to physical study than simplexlands. There is one sense in which this is unfortunate: that tetrahadreons – the 4-body problem in 3- – is more directly relevant to nature than -a-gons in 2-. One sense in which this does not matter: -a-gons will do as a model arena of many aspects of classical and quantum GR’s dynamics, problem of time and background independence. And one sense in which this is fortunate: that -a-gons provide a ‘shape representation’ for quantum-information-theoretic qu-its [37, 48].
Acknowledgments I thank Chris Isham and Don Page for previous discussions. Reza Tavakol, Malcolm MacCallum, Enrique Alvarez and Jeremy Butterfield for support with my career.