This page will shortly be replaced by its own “brief living review” pdf article introducing the various Flat Geometry works unveiled so far.

The usual Heron’s formula is for the area of a triangle in terms of side-length data.

The Author’s publicly-declared work began here with the 2017 preprint “Two new versions of Heron’s Formula, which is updated here: https://wordpress.com/page/conceptsofshape.space/1244 . This covers how the sides-medians involution J gives a new Linear Algebra proof of the Heron’s formula with medians data, which, as far as we know is due to Hobson (1891). And even more interestingly how the eigentheory of the Heron’s formula encodes Hopf’s little map. With various details supplied in “The Fundamental Triangle Matrix Commutes with the Lagrange and Apollonius Matrices. With implications for deriving Hopf and Kendall’s Little Results.” https://wordpress.com/page/conceptsofshape.space/1308 .

The subsequent “The Fundamental Triangle Matrix” https://wordpress.com/page/conceptsofshape.space/1295 provides a brief new Linear Algebra proof of Heron’s formula itself. https://wordpress.com/page/conceptsofshape.space/1306 then shows that the name ‘Heron matrix’ – as occurs in the quadratic form version of Heron’s formula squared – is a major miss. For the very same matrix occurs in not only the cycles of the cosine rule but also of the triangle’s triangle inequalities! With seven subsequent interplays. So we call this matrix instead the Fundamental Triangle matrix, F .

There is substantial interplay in this work between Heron’s formula and Apollonius’ median-length Theorem. This can already be seen in the first article, since the sides-medians involution J is more primitively a description of Apollonius’ Theorem. This turns out to be a Linear-Algebraic counterpart of a couple of pages of Johnson’s 1929 book’s more traditional approach to Geometry. Also “A new Physical’ Proof of Apollonius’ Theorem” https://wordpress.com/page/conceptsofshape.space/1353 gives a new proof first and second moments methods.

We also need the Lagrange matrix L for the above interplays. This more basic and yet underlying matrix is an N-body problem matrix defined independently of spatial dimension. It occurs in the inertia quadric as viewed as a function of the N-body problem’s separations. Geometers might equivalently say n-simplex in arbitrary-dimensional flat space, where n := N – 1. We give a tour de force on this and various related matrices in “Lagrange Matrices: 3-Body Problem and General” https://wordpress.com/page/conceptsofshape.space/1436 . Diagonalizing this gives what have hitherto been known as relative Jacobi vectors but which we call eigenclustering vectors. It is using these variables on F ‘s own eigenvectors that yields the above recovery of Hopf’s little map.

This article is also a test-bed for the subject area/conceptual type colour-coded Testarossa package. Here covering both colour-coded text symbols and the rainbow vertical variant of Penrose birdtracks for details of all the arrays involved. Which is rather convenient given that indices pertaining to a large number of different spaces are involved. So the presentation involves multi-tensors with a large multiplicity, for which colours and candy-patterning does a far better job of clarifying than use of over a dozen different fonts of index would! The full Testarrosa elects to use an easily humanly distinguishable number of colours – 26 – with candy-patterning in its RVPBs extending the multi-tensoriality by 1 further order of magnitude. Testarossa will also be used in the next generation of presentations of the Author’s local classical resolution of the Problem of Time between Background-Independent Physics (such as GR) and Background-Dependent Physics (the rest of accepted Physics). Permitting both far greater clarity and conciseness in the face of the resolution requiring a multi-tensoriality of over 50.

Both can handle 26 different conceptual types of object/index at once. With the RVPB then using candy-patterning to handle subtypes, expanding capacity to 2 orders of magnitude. Two areas requiring candy-patterning are N-body problem configuration spaces, the simplest of which are in the above article. And types of constraint in Dirac-type constrained dynamical systems: a Principles of Dynamics application that I have been extending to local Lie Theory and local brackets algebras more generally. The above is however the first full use of Testarossa and Rainbow-VPB, the two 2021 Articles on Dirac-type constrained dynamics having just used prototypes (and were not even proofread for coloration errors). As such, this week’s Article has also provided the tech to clean up these two 2021 articles. It will also permit the Local Resolution of The Problem of Time to get nicer and more definitive updates, written faster than otherwise. I should even be able to grind out better versions of my four N-body problem reviews of 2017-2018. Gradually, for I will be prioritizing writing my Structure of Flat Geometry, Structure of Differential Geometry, Lie Theory, and Principles of Dynamics reworked using Local Lie Theory books. All four of these will use Rainbow-VPB, and the last two will use Testarossa as well.

We are then ready for the algebra formed by J , L and F ! Which we present in “Only 2 of the Fundamental Triangle, Lagrange and Apollonius Matrices are Independent. With ensuing Algebras, Irreducibles and Splits” https://wordpress.com/page/conceptsofshape.space/1310 .Since L‘s key property is that it is a projection, we also use the notation P for it. To be precise, the equal-masses L is a projection, while for arbitrary masses, L is superceded by a projection that S. S’anchez provided. By this stage we have found six routes to F , two of which are double-barrelled. In particular, F has meaning in all of Geometry, Algebra and Representation Theory. Partly for contrast, and partly for juxtaposition, there are also a large number of routes to Hopf’s little map. See for now in this regard the Figure at the top of Topology | , for the Author’s Invited Review “The Hopf Map 32 times in Geometry and Physics” has been delayed by the need to finish the book “The Structure of Flat Geometry”, which is largely based on the current Webpage’s list of new results.

https://wordpress.com/page/conceptsofshape.space/1436 further decomposes L , and, where distinct, P. Into elementary Linear-Algebraic operation matrices, such as positions-to-separations, picking a basis of separations, and then passing to the eigenbasis. As completed by mass-weighting, mass-unweighting, and, in one version, CoM (centre of mass) adjunction. This gives a Jacobi-type square root for the Euclidean metric on constellation space. And again for L and P , which can also be viewed as expressions for the induced metric on relative space. Such square-roots are Geometrically soldering forms, with in particular P = S^TS for the S’anchez soldering form (where T denotes transpose).

As a first robustness test, passing to arbitrary masses for a triangle takes us from Apollonius’s Median-Length Theorem to Stewart’s Cevian-Length Theorem. A truer name for Cevian is triangle-cotransversal. Here the cycle of Stewart’s Theorems is encoded by a Stewart matrix T. We then investigate the extent to which J‘s properties transcend to T and subcases in 3 articles: “Generalizing Heron’s Formula by use of Multi-Linear Algebra.I. Equi-Cevians and invertible Cevians” https://wordpress.com/page/conceptsofshape.space/1259 , “II. Mobius, Jacobi and Routh for concurrent Cevians” https://wordpress.com/page/conceptsofshape.space/1261 , and “III. Geometry can impart Sides-Cevians Nonlinearity” (2024), https://wordpress.com/page/conceptsofshape.space/1263 .

. This permits us to find various infinite-dimensional families of generalizations of Heron’s formula. Two are strong – involving F itself – the equi-Cevians and the altilarities (including the altitudes’ Heron formula as a special case). While others are weak: merely tensorially-related to F: concurrent Cevians, invertible Cevians, unimodular Cevians. An ensuing 4-Areas Theorem for Affine Geometry is here: https://wordpress.com/page/conceptsofshape.space/1658 .

Passing instead to equal-mass quadrilaterals and tetrahaedrons (3-simplices), there is a Jacobi-H versus -K ambiguity in eigenclustering network. In the H case, our moments method is found to return https://wordpress.com/page/conceptsofshape.space/1297 . Euler’s 4-Body Theorem (more widely, and yet less correctly, known as Euler’s Quadrilateral Theorem). in fact the opposite side-lengths 3-cycle here manages to return F as well. In the K-case, however, a new Theorem arises: “Eigenclustering-Length Exchange Theorems. II. Straight-P_3 alias K-counterpart of Euler-H, https://wordpress.com/page/conceptsofshape.space/1299. In its incipient form, it relates a linear combination of the squares of the spike and handle lengths (viewing the K as an axe). One can however solve for these separately, returning Apollonius for the spike and a striking analogue of Apollonius for the handle. This pattern is extended to the K(N)-eigenclustering for arbitrary N in “III. The Bottom Series: Straight-P_n”, https://wordpress.com/page/conceptsofshape.space/1301and “Infinite Series of Generalizations of Apollonius’ Theorem”,https://wordpress.com/page/conceptsofshape.space/2723 .

The main item here is however that eigenclustering networks are in 1 : 1 correspondence with the unlabelled rooted binary trees; see E. Anderson and A. Ford’s Graph and Order Theory of N-Body Problem's Eigenclustering Networks", https://wordpress.com/page/conceptsofshape.space/2090 . And for N > 2, there is 1 ELET (Eigenclustering Length-Exchange Theorem) per eigenclustering network. These are more conveniently indexed by the corresponding defoliated trees, which are now at-most binary. Apollonius is then the P_2 path, Euler is P_3-bent, and its K-analogue is P_3 straight . K(N) is then the P_n-straight case. `IV. 5-Body Problem” , https://wordpress.com/page/conceptsofshape.space/1512 provides the 5-Body cases of the ELET. While E. Anderson and A. Ford inter-relates multiple representations of eigenclusterings up to N = 10.

E. Anderson, K. Everard and A. Ford extend this approach to generalize Stewart’s Theorem to 1 Theorem per at-most binary tree on > 1 vertex. See E. Anderson, K. Everard and A. Ford extend this approach to generalize Stewart’s Theorem to 1 Theorem per at-most binary tree on > 1 vertex. See “Stewart’s Theorem generalized to 1 Theorem
per Eigenclustering Network per N-Simplex, https://wordpress.com/page/conceptsofshape.space/2721 and “II. K(N) alias straight-P_n, https://conceptsofshape.wordpress.com/wp-admin/post.php?post=2759

For quadrilaterals, the ELET matrices are not multiplicatively compatible with the Lagrange matrix. Thus they do not supply a ounterpart of the sides-medians involution in the triangle matrix algebra. However, Ptolemy’s Theorem steps up as an alternative source of involutors. See E. Anderson and K. Everard, “Ptolemy’s Theorem and Inequality: from a Linear Algebra point of view”, https://wordpress.com/page/conceptsofshape.space/1938 (2024) , E. Anderson, “Linear Algebra and Inequalities from combining Ptolemy’s Theorem and Inequality
with Euler’s Quadrilateral Theorem”, https://wordpress.com/page/conceptsofshape.space/1967 . “4-Body Problem: Ptolemy–Lagrange Algebra”, (2024) https://wordpress.com/page/conceptsofshape.space/2121 . And E. Anderson and K. Everard,“Linear Algebra of Cyclic Quadrilaterals: Ptolemy, Diagonal, Area, and Circumradius Formulae”,
https://wordpress.com/page/conceptsofshape.space/1971 (2024) .Which is also Preprint 1 on https://institute-theory-stem.org/geometry/ .

The last of these also entertains the extent to which Brahmagupta’s area formula for cyclic quadrilaterals generalizes Heron’s formula for triangle area. A Bretschneider counterpart – generalizing to an area formula for convex quadrilaterals – is around 2 days from being completed. Some preliminaries for both can be found at https://wordpress.com/page/conceptsofshape.space/1237 .Quadrilateral area formulae do not in any case generalize some aspects of Heron’s formula. Some go rather to the della Francesca and Tartaglia tetrahaedron volume formula and its Cayley-Menger determinant generalizations. While other fall to the quadrilateral Casimir, which is, rather, the square root of the sum of the areas of the three constituent triangle subsystems… We are also a different 2 days away from being able to present an Article on this Casimir…

A. Ford showed that a further reason for the triangle matrix algebra’s nice behaviour is that all the matrices therein are combinatorial matrices. See “Triangle, but not most Quadrilateral, Matrices are Combinatorial” Preprint 2 on https://institute-theory-stem.org/geometry/ . I gave a Linear Algebra counterpart of this in “Eigentheory of Combinatorial Matrices: a More General Meaning for the CoM-Relative Split” (2024), Book need: high
https://wordpress.com/page/conceptsofshape.space/2280 .

Some of these works also recover Smale’s little Theorem: that the space of triangles modulo similarities is topologically a sphere. And Kendall’s little Theorem: that it is metric-geometrically a sphere as well. In each case deducing these from just Heron’s formula.

The above material, and more like it, is to be published in my book “The Structure of Flat Geometry”, which should be out within 2025.

My new freely-available book is out here: Applied Combinatorics – Concepts of Shape

Abstract Sides and medians are both Jacobi coordinate magnitudes. These furthermore enter equably into the spherical coordinates on Kendall’s shape sphere and the Hopf coordinates. This motivates treating medians on the same footing as sides in the geometry of triangles and the ensuing Shape Theory. In this article, we consequently reformulate inequalities for the medians in terms of shape quantities, and proceed to find inequalities bounding the mass-weighted Jacobi coordinates. This work moreover identifies the 4/3 – powers of which occur frequently in the theory of medians – as the ratio of Jacobi masses.

One of the Hopf coordinates is tetra-area. Another is anisoscelesness, which parametrizes whether triangles are left-or-right leaning as bounded by isoscelesness itself. The third is ellipticity, which parametrizes tallness-or-flatness of triangles as bounded by regular triangles. Whereas tetra-area is clearly cluster-choice invariant, Jacobi coordinates, anisoscelesness and ellipticity are cluster-choice dependent. And yet they can be ‘democratized’ by averaging over all clusters. Democratized ellipticity moreover trivializes, due to ellipticity being the difference of base-side and median second moments, whose averages are equal to each others. Thus we introduce a distinct linear ellipticity’ quantifier of tallness-or-flatness of triangles whose democratization is nontrivial, and find inequalities bounding this.

Some of this article’s inequalities are shape-independent bounds. Others’ bounds depend on the isoperimetric ratio and arithmetic-to-geometric mean ratio of the sides: shape variables.

Abstract We systematically consider simple relational variables — relative variables, ratio variables and dilatational variables — for Graph Theory. We apply these to simplifying graph inequalities and elucidating a large number Graph-Theoretically significant probability-valued variables. This material has further use in developing network stucture quantifiers. It represents interaction between Similarity Geometry, and basic Shape Theory in the sense of Kendall, with Graph Theory.

Abstract Dirac based his theory of constrained systems on Linear Algebra foundations. It is a brackets-algebraic consistency procedure with multiple outcomes, including new constraints dropping out and redeclaring brackets becoming necessary (Dirac brackets). This procedure has not yet been edited, however, to caution about and remove scaffolding structures that turn out to not in general be brackets-algebraically consistent. We perform this task here. Our main innovation is moreover demonstration that substantial progress can be made from placing Dirac’s Algorithm on Linear Algebra and Order Theory foundations. For chains, lattices, posets, and digraphs abound therein For instance, in the simpler version of the algorithm, its iterations form a chain, of which its Dirac brackets updating steps are a subchain. Its consistent algebraic structures, meaningful notions of weak equality, of appended Hamiltonians and of observables form bounded lattices. Many key notions – such as Dirac’s extended Hamiltonian or Dirac observables – are identified as extrema of these lattices, cementing their permanence. Others are however revealed to be but simplest examples of middles. By this, e.g. Kuchar observables are in general to be replaced by a poset of algebraicially-consistent middling A-observables.

In the harder – path dependent – (previously called bifurcating or branching) version of the algorithm, moreover,
iterations and Dirac brackets types become digraph-valued. What previously was a lattice of consistent constraint subalgebraic structures now becomes a competing lattice, described overall by a semi-lattice, with weak equalities, appended Hamiltonians and observables following suit. Order Theory conceptualization thus remains both lucid and under control within this harder case. Such Order-Theoretic considerations furthermore transcend to extended variants and to Temporal Relationalism implementing variants. And to the Generalized Lie Algorithm: a vast generalization of scope in which to apply Dirac’s insights from constrained dynamical systems to wherever Lie Theory is applicable.

Article dedicated to the memory of Niall o’Murchadha (see the Acknowledgments section if visiting here in this regard).

Based on material presented at the 2021 Summer school on Combinatorial and Topological Applications to Fundamental Physics.

13/10/2025. Proposition 2 is true, but the proof given above is nonsense.

1. The word “commutative” went missing from the working. This was pointed out in October 2025 by K. Everard. As in the result being alluded to is that the product of 2 commutative projectors is itself a projector.
2. But I then remembered working out some years ago that the Dirac projectors that can be abstracted from each Dirac bracket are not in general commutative. So the domain of validity of the proof by projectors does not include the Dirac bracket.
3. Now as to an alternative proof making no mention of projectors. By Sniatycki 1974, Dirac brackets are Poisson. One can then just as well form a Dirac bracket from ‘a Poisson bracket that so happens to be the Dirac bracket obtained from some previous Poisson bracket’!

As an advanced notification, I will have some time for projectors (Dirac and more basic) in Lent 2026. I will tidy up this preprint then. Either add material to my Dirac projector note or write a second note. Write an extra chapter for my Linear Mathematics book. And co-write a separate note with K. Everard on some finer details of commuting projectors.

Abstract

We apply Shape Theory and Order Theory to spaces of graphs. We concentrate on
small examples which are minimal for exhibiting various nontrivialities.

Abstract

We consider the sense in which Dirac brackets are projections of Poisson brackets. In the process, we define a Dirac tensor, a Dirac projector and a Dirac–Jacobi tensor.

Dirac algebroid formed by General Relativity’s constraints

Seminar 1) Interplay between foliations and algebroids
Seminar 2) Combinatorial improvements in understanding of Dirac-type Algorithms for constrained systems. 
Seminar 3) Using differential geometric flow methods to better understand the subsequent Problem of Observables
Seminar 4) how  deriving spacetime from space rests on cohomology

This Summer School is on each Saturday for the next four Saturdays (starting August 21st 2021).

Contact dr.e.anderson.maths.physics *at* protonmail.com to join , subject to approval and to abiding by the Summer School’s rules.

ABSTRACT: We consider global issues with Relationalism: a first piece of Background Independence that resolves various Problem of Time facets and is locally implemented by Lie derivatives. Spacetime Relationalism and Configurational Relationalism are quite similar, though the second of these requires supplementing with a heterogeneous Temporal Relationalism. Avoiding both zeroes and Killing vectors is involved, as are some fibre bundle effects, including bibundles, monopoles on configuration space and the Gribov effect. Most of Relationalism’s globality follows from quotient configuration spaces being stratified manifolds. Nine distinct strategies for dealing with stratified manifolds are compared. Compactness and metric-space guarantees for nice (in particular Hausdorff) stratified manifolds are provided. Fibre bundles do not suffice for stratified manifolds, so general bundles, differential spaces, presheaves and sheaves are brought in instead. Hausdorff paracompact (HP) spaces continue to afford simplifications, with some support when local compactness and second-coutability apply. Relational Particle Mechanics, Gauge Theory and GR examples of these various global issues are provided.

[79 pages, including 25 figures. Second of a series of six Fundamental Physics and Applied Topology Articles.]

ABSTRACT: The Problem of Time is due to conceptual gaps between General Relativity and the other observationally-confirmed theories of Physics; it is a major foundational issue in Quantum Gravity. The Problem of Time’s multiple facets were mostly pointed out over 50 years ago by Wheeler, DeWitt and Dirac. These facets were subsequently classified by Kuchar and Isham. They argued that the lion’s share of the problem consists of interferences between facets. They also posed the question of in which order the facets should be approached. By further considering the nature of each facet at the local classical level, the Author showed the facets to be two copies of Lie Theory — spacetime and canonical — with a Wheelerian 2-way route therebetween. This solves the facet ordering question. The resulting mathematical framework turns out moreover to be consistent enough to smooth out all local classical facet interferences as well.

It would furthermore be preferable if all of the Background Independence aspects, resultant Problem of Time facets, and strategies to resolve these, were treated in a globally well-defined manner. The current article begins to address this by classifying what is meant by `global’. Be this at the level of mathematical structure (Topology, Differential Geometry, Lie Theory, PDEs, Functional Analysis). At the level of which spaces the modelling actually requires (space, spacetime, configuration space, phase space, space of spacetimes…). Or at the level of each aspect of time, space or Background Independence more generally. We also include globalization strategies and justification of A Local Resolution of the Problem of Time being possible in the first place. This is based on Hausdorff paracompact spaces, which for now support a Shrinking Lemma.

[50 pages including 14 figures. First Article in a series of six on fundamental Physics and Applied Topology.]