1) A book called “The Structure of Flat Geometry” is in the process of being written to accommodate the below results with along with suitable introductory material.
It represents much of my work between 2017 and 2024. The following link gives a small preprint archive of these works: https://wordpress.com/post/conceptsofshape.space/1332 .
This includes detailed study of the role of Heron’s formula in the 3-body problem.
Alongside robustness tests of three kinds.
a) What other Heron-like formulae does the triangle support? Medians-Heron and Altitudes-Heron are well known, but I systematically studied triples of Cevians and obtained various infinite families of Heron-like formulae…
b) What happens to the large amount of significant 3 x 3 matrices arising from study of the triangle upon passing to the quadrilateral and the tetrahaedron of the 4-body problem?
c) What happens if it’s a triangle in some other simple space, such as on a sphere, the hyperbolic plane or in the Minkowski spacetime of Special Relativity? The first two have the same maximal amount of symmetry as flat space while having positive and negative constant-curvature respectively. while the last is maximal and flat to boot, its source of diversity being rather that its flat metric is indefinite rather than positive-definite…
Follow the above link for most details. Many research articles are linked to this link!
As regards b), 3 x 3, 4 x 4, 5 x 5, 6 x 6, 8 x 8 and 9 x 9 matrices ensue for the 4-body problem, as well as various oblong matrices. This gives a large number of further reasons why the 4-body problem is more generic than the 3-body problem! Even a few further details requiring the 5-, 6- and 8-body problems for first nontrivial occurrence were documented by me in 2017-2024. I have 2 further so far publicly undeclared little papers on this. Also my minimum-N review, while available on arXiv, is nowhere near up to date on this topic. For now it covers just what I knew in 2018, i.e. very near the start of this work! See below for a timescale of when I am updating, and wtiting sequels to, my Reviews…
“The Structure of Flat Geometry” itself considers the shape sphere formed by triangles modulo similarities. One of various proofs given, mentioned or set as Exercises in this book obtains this from Hopf’s little map from the 3-sphere to the 2-sphere. This generalizes to Hopf’s map from S^{2 n – 1} to CP^{n – 1} giving the shape space of N-a-gons modulo similarities. Where n = N – 1. At the Topological level, this polygon result is due to Smale, while at the Metric Geometry level, it is due to Kendall (for all that he proved it elsewise, also included in my book…) Let us refer to the triangle case, which is special by the CP^1 = S^2 coincidence conferring extra properties and rendering the geometry familiar, as Smale’s little theorem and Kendall’s little theorem. As proven by, among other methods, Hopf’s little map! My book then also shows that Smale’s little theorem quickly follows from Heron’s formula, and Kendall’s little theorem somewhat less quickly as well… To do this well, my Book also has a supporting Part on the Spherical Geometry.
My book manuscript now also contains a Part-sized Epilogue. Mostly on Affine and Projective Geometry. alongside how the Flat Geometry results covered by the book so far – Heron, Apollonius, Menelaus, Ceva, Stewart, Routh (with some new proofs and many generalizations of many of them) fare at the Affine and Projective level. Ceva and Menelaus are well-known to continue to serve here, but even they fall short at the Projective level. This motivates the Epilogue to briefly consider two further theorems – Pappus’ and Desargues’- that are not only Projectively significant but are indeed the only two major structural theorems in planar Projective Geometry… Corresponding work on their graphs, and the smallest Projective plane – the Fano plane – is also sketched. Finally, while the Affine and Projective counterparts of Kendall’s theory of similarity shape spaces is much harder, numerology extracted from it is used to successfully sniff out how the Affine and Projective theorems we cover generalize to higher spatial dimensions. This follows us already having found new generalizations of Apollonius’ and Stewart’s theorems in both this and other ways…
2) Further work on quadrilaterals requires Differential Geometry and CP^2 specifically, and so shall be in a subsequent Book that I shall start to write in 2027.
3) Various people have quipped that I am writing 4 books’ worth of material that could be called “The Structure of Flat Geometry” as well as 1 book’s worth of material on the kinematical structure of the N-body problem… So let’s explain what each of these are, and which part of this I have selected for the actual book called “The Structure of Flat Geometry”…
The selection is: use of Linear Algebra to give a new intertwined account of Flat Geometry and N-body problem kinematics. Which is followed by exploiting that spherical geometry is accessible at the Book’s level (we assume the reader has studied Maths or a STEM theory subject, including a sizeable course on Linear Algebra). So as to pass from Linear Algebra at the constellation, relative separation and eigenclustering (alias Jacobi coordinates) levels to Spherical Geometry at the preshape and shape space levels! Followed by coning our sphere to get to relational space: now quoteinting out Euclidean transformations rather than similarities… Beyond this point, we only use back of the envelope counts (numerous, including many new!)
Item 2) above can be viewed as bringing in Differential Geometry and CP^2 geometry to continue the second half of the current Book with a sequal about the shape space of quadrilaterals. And to apply the generalized Hopf map more widely, much as “The Structure of Flat Geometry” does with Hopf’s little map under the aegis of this being accessible sphere mathematics… Since the quadrilaterals are in flat space, this is then a first use of Differential Geometry to study key arenas arising from flat Geometry. Items 4) and 5) below can then be viewed as a distinct use of Differential Geometry to work out the structure of Flat Geometry, now with some support from Lie Theory…
The fourth “Flat Geometry book” is taking the form, rather, of the Geometrical Graph Theory wing of a small online encyclopaedia, Online Encyclopaedia of Applied Graph and Order Theory | . This is in effect “the Combinatorial Structure of Flat Geometry”, which is by far most interesting for flat space’s Projective Geometry. Here we really go to town on the Fano, Pappus and Desargues graphs (So far 150 pages versus “The Structure of Flat Geometry” having a 5-page introductory Chapter on this in its Epilogue Part…) This small encyclopaedia wing is set to double in size by around 2030, by going after various other small and particularly beautiful Projective (and now occasionally also Affine) stuctures.
“The Structure of the N-body problem” can be viewed as very major updates to my existing 3-, 4- and minimum-N- body problem reviews. One and a half Parts of ‘The Structure of Flat Geometry” are about this, with consequences extending into providing new Flat Geometry results in several other Parts… This can be viewed as a roughly quarter-sized Introduction to this topic, suitable for 3rd year undergraduates. The updated Reviews, which are projected to come out between 2026 and 2028, are the corresponding quadruple sized account, now written for experts, with no holds barred as to the types of mathematics used. The combined manuscripts for this full-sized and much more advanced account currently run to over 600 pages (nor do they yet include many quadrilateral topics). By which they could indeed also be described as a further book-sized contribution, in the form of a long Series of Periodically Updating Reviews. The long-term idea here is that this Series would acquire further Reviews at some point on the other side of my “The Structure of Differential Geometry” ‘s account of the shape space of quadrilaterals, which is pitched to be accessible to fourth-year undergraduates…
4) From Lie Theory to a new Differential-Geometric Pillar of Flat Geometry
My local resolution of the classical Problem of Time between background-dependent and background independent Physics amounts to a rediscovery of a large chunk of local Lie Theory. It has however some extra kick, because a great result of Dirac’s for constrained dynamical systems was not spotted as a result generalizing to Lie Theory.
But it does generalize. As a second application of it, as a demonstration of working outside of constrained dynamics, I obtained the following.
C) That Conformal and Projective Geometry are the top geometries on flat space. And are a matter of choice: the special-conformal and special-projective generators do not like each other so oneis forced to choose the one or the other. In the Dirac-Lie approach, that these are the two top geometries and are incompatible choices drops out as two algebraic roots of a very simple expression. For an obstruction term, which needs to vanish in order to have a consistent algebra under Lie brackets.
Within the constrained dynamical systems context, two results of this type of note are as follows.
A) The Einstein Dilemma – Galilean or Lorentzian universal relativity locally becomes two of a set of three algebraic roots to get rid of an obstruction term in the Poisson brackets algebra. The third root is Carrollian relativity: the zero-speed-of-light option. This is the matter section result.
B) The gravitational sector result’s partner to the Lorentzian option fixes the value of the coefficient in the DeWitt supermetric to take the General Relativity (GR) value 1. This comes with two other algebraic root options (the Carrollian partner is Strong Gravity). But now also with a weak PDE solution factor. Whose solution is CMC: that the spatial slices are to be of constant mean curvature. These then decouple GR’s constaints as per Jimmy York Jr’s seminal work from the 1970s on GR’s initial-value problem (IVP).
By which Dirac’s Algorithm applied to the gravitational sector amounts to making the following choice. One has Strong Gravity (only models near singularities). Or Galilean Gravity (not what our world has either). Or the DeWitt supermetric, recovering GR in geometrodynamical form. Or CMC slicing, recovering GR a second time, this time in its conformogeometrodynamical form. Thus Dirac’s Algorithm gives 2 new derivations of GR within a single simple equation, about an obstruction term that needs to vanish, and can do so in 4 ways. 2 of which recover GR, each in a very different and technically valuable formulation.
Result C) is then the first major result following from my “Lie-Dirac Algorithm”. Pointing to the possibility that other areas of the Theory of STEM to which Lie Theory is applicable shall also possess foundational results of this kind.
For now, my work on this consists of writing a book “the Principles of Dynamics reformulated using local Lie Theory”, subtitled “which solves the local classical Problem of Time”. And a book “Local Lie Theory”, in which the general Lie-Dirac case is given, alongside its flat geometry example C) as a further “Pillar of Geometry” loosely in Stillwell’s sense. And the following two further Pillars of Geometry that come from Lie Theory, which, while largely not new unlike C), have been rather overlooked.
5) Two further Differential-Geometric Pillars of Flat Geometry: Geometrical Invariant theories whose Physical counterparts are theories of Classical Observables
Both Lie and Cartan developed different theories of geometrical invariants. Cartan’s is well known, but Lie’s is not. They are so different that Cartan’s uses differential Calculus while Lie’s uses integral Calculus! What has been largely overlooked is that either of these approaches amount to giving Geometry the direct analogue of a working theory of classical observables. In contrast, Physicists often mention classical observables but rather seldom touch upon what they are. And when they do, they usually refer to unrelated conceptualizations, without realizing that authors like Lie and Cartan had already done more specific, and very convincing work on this matter.
Physicists are more likely to be aware of Dirac’s work on observables, which does cover some needs. Physicists who are experts on classical observables usually compare Dirac’s observables with those of another physicist, Bergmann. A few even know to compare Dirac and Kuchar observables.
But I showed in 2013 that Kuchar observables in GR amount to passing from a 2-path of unconstrained and fully constrained = Dirac observables to a 3-path with Kuchar observables as the middle point. Between 2013 and 2021 I developed that notions of obsevables for a system form a bounded lattice, with Kuchar observables being a first instance of a nontrivial middle. But the correct generalization of this to arbitrary theories involves there being a nontrivial poset middle. Unconstrained and Dirac play the roles of top and bottom element here. My 2017 Book “the Problem of Time” includes a first few examples from studying physical theories. While my 2021 article https://conceptsofshape.space/2021/10/14/diracs-algorithm-for-constrained-systems-from-the-perspective-of-order-theory/ in honour of my friend the late Professor Niall o’ Murchadha turns this on its head. By considering that finding systems with Order-Theoretically interesting constraint-and-observables lattices (these are a dual pair) as one source of determining which models should be used in developing Theoretical Physics.
The first step suggested is to take Henneaux and Teitelboim’s treatise (a valuable 1992 expansion on, and examples collection for, Dirac’s constrained dynamical systems work, which he last reviewed in 1964). And reappraise it in Order-Theoretic terms. The second is to fill in the gaps: which interesting Order Theory features do not yet have any physical model system that represents them. At the level of the Principles of Dynamics of Mechanical systems — the staple examples and counterexamples in Henneaux and Teitelboim — no further physical justification is needed for bringing in Order-theoretically interesting models. But on the longer run, we want models and theories that are both Physically well motivated and Order-Theoretically distinct from each other.
In my 2017 book ‘The Problem of Time”, the prototype of Order-Theoretically interesting distinctions between theories that was given is GR versus Supergravity. In fact, GR-as-Geometrodynamics is Order-Theoretically trivial in the sense that its Order Theory structure is the “Kuchar” 3-chain, whose middle is trivial as a poset. While Supergravity is Order-Theoretically nontrivial in this way. This was accompanied by the observation that, while many other theories are no different from GR, or a subset of GR, as far as Problem of Time resolution goes, Supergravity is markedly different. The Order Theory distinction is tied to the super-generator being the square root of the Hamiltonian constraint (a 1970s result of Teitelboim’s). The idea of using having a resolved classical Problem of Time as a selection principle among physical theories (“Comparative Theory of Background Independence”) covers a far larger arena than just GR versus Supergravity however. By which no finer details of Supergravity are planned in either of my “Lie books”.
The reader may be wondering what a Differential-Geometric Pillar of Flat Geometry is. These a priori being two different types of geometry! If you are trained aa a continuum mathematican or Theoretical Physicist, you will however know of one such. Killing vectors give a systematic derivation of isometry groups such as flat Euclidean space’s and flat Minkowski spacetime’s My point is then that generalized Killing vectors (similarity, affine, conformal… as well as metric and thus isometry-related) are but one of four such Pillars. The other three being Lie’s integral theory of observables, Cartan’s differential theory of observables and my Pillar using my “Lie-Dirac” generalization of the Dirac Algorithm. My two “Lie Books” will put these four differential-geometric pillars of flat geometry on an equal footing, as well as showing how their Principles of Dynamics counterparts resolve the local classical Problem of Time. I am hoping that one of these two books will be a submitted manuscript by the end of 2030.
6) Global matters
On the longer run, obstruction terms mean cohomology, and “Comparative Theory of Background Independence” means comparing cohomology theory on a large number of categories. That GR gives a working such means that the metric-differential-geometric category is in. But might we be able to dismiss using less, or more, structure than this for our Physics?
This is moreover not the only selection principle. GR also works because Teitelboim showed that the Hamiltonian constraint of GR encodes refoliation invariance. Thus resolving one of the 8 local Isham-Kuchar Problem of Time facets. So to get a local Problem of Time resolution for different stuctures than metric differential structure, we need to know what the category’s counterpart of foliations are. Refoliation invariance involves independence from intermediary objects in evolving from A to B. So I called the general-category version of refoliation-invariance RIO invariance (IO for intermediary object). There is then the question, for whichever categories and theories that support cohomologically nice enough brackets algebras, as to whether they additionally manage to be RIO-invariant. This is probably harder to deal with than cohomology on many categories. Which is a sharply-posed Mathematical question, while we do not even know whether RIO-invariance is sharply posed, much less how to compute answers to questions about RIO. This suggests not looking too far afield for now. E.g. what is a foliation in other types of differential geometry. What is it in discrete geometry models (a large variety of which have been used in alternative theories of quantum gravity). What is it for just topological manifolds in place of differential ones.
The above two paragraphs are an account of what I will probably be spending quite a lot of my research time on in the early 2030s. That quantum operator algebras are brackets algebras which have their own brackets-Dirac Algorithm gives some further long-term direction. Though Isham often warned me that one cannot really do QM until one has a grasp of what the classical model does globally. When the classical model is flat space, or Minkowski spacetime, one is exempt from this: the Killing vectors do all the work. But in more general cases it matters. The problem with this is that I have already documented over 100 facets of the Global Classical Problem of Time, and there’s nothing nice and simple like local Lie Theory to hold these together. Many globalize in different ways! No framework that I am aware of can incorporate all of this globality at once. In other words, at the global classical level, the Problem of Time returns with a vengeance, hydra-like with many more heads than the Isham-Kuchar classification’s 8 local heads. I observed these 100+ facets between 2018 and the present date (I last counted them in 2024).
They caused of me to feel I had made a good choice in 2013: to spend half of my subsequent time on Topology of all kinds. The good news being that Topology is general enough that no matter what subject one studies far enough, the same kinds of Topology drop out of it. So becoming an expert in Topology is useful in all parts of the Theory of STEM rather than in just whichever part one gets far enough with first to encounter Topology. As such, there are many other areas of the Theory of STEM that I can now contribute to as an Applied Topologist.
The usual opening, for a person with a Differential Geometry background, is that this is one of two very different commonly-encountered easier cases. The other is Graph Theory. My second book “Applied Combinatorics” contains much new research in its Part VIII. Firstly along the lines of applying Graph Theory to the arenas formed by the objects from all branches of basic Combinatorics. Secondly, using that Order Theory carries further conceptual and theoretical strength, giving many new results when applied to these arenas.
I next turned Graph and Order Theory on Linear Algebra (including Calculus’ Linear-Algebraic form) in my nearly-finished book “Linear Mathematics. Vector Spaces, Linear Methods and Tensors Throughout STEM” that I spent much of 2023 writing. Several research articles indicating some of the new results thus obtained should be appearing in 2024. Graph Theory and Linear Algebra are now the main two engines in getting new results for Flat Geometry. The above Heron and quadrilateral work is mostly Linear-Algebra driven, but still has some Graph Theory input, and other parts of “The Structure of Flat Geometry” are more graph-driven. This being an easier book, it probably will not get as far as setting Order Theory loose on Flat Geometry. But my first “Lie book” shall compensate in this regard, given what is said above about brackets algebras and invariants/observables. So on the medium run, I am using Graph Theory and Order theory on everything else. And on the long run, I will be using actual (as opposed to Differential-Geometric, Graph or Order subcases of) Topology to study many different areas of the Theory of STEM.
While I cannot hope to solve the 100+-facet global classical problem of time for many years (if at all, even partially), I have in the meanwhile started to make public my list of its facets. This is somewhat analogous to Isham and Kuchar listing 8 local problem of time facets in 1992 and 1993, while just using one further box for “all extra things global”.
Concepts of Shape 