Heron’s formula and generalizations are the current term’s discussion topic at the Applied Combinatorics and Topology discussion group. Further articles on Heron’s formula are currently being finalized, and shall also be linked here once completed for public view.

So far…

1) https://wordpress.com/page/conceptsofshape.space/1244 updates my 2017 preprint “Two new versions of Heron’s Formula.

2) https://wordpress.com/page/conceptsofshape.space/1295 provides a brief new LinearAlgebra proof.

3) https://wordpress.com/page/conceptsofshape.space/1306 shows that the name ‘Heron matrix’ is a major miss, since 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 it Fundamental Triangle matrix, F .

There is substantial interplay in this work between Heron’s formula and Apollonius’ Theorem. For which a column of new articles has also started to appear, as follows.

A.1) https://wordpress.com/page/conceptsofshape.space/1353: a new physical proof. Which proceeds via a Linear Algebra formulation in terms of the Apollonius matrix O .

A.2) https://wordpress.com/page/conceptsofshape.space/1364: metric and flow results for sides-ratio data on the triangleland shape sphere.

We also need the Lagrange matrix L for our interplay. This is an N-body problem matrix independent of spatial dimension. We give a tour de force on this and various related matrices here:

L) https://wordpress.com/page/conceptsofshape.space/1436

Which is also a test-bed for the subject area/conceptual type colour-coded Testarossa package.

And for the “Rainbow Vertical Penrose Birdtracks” figure-factory for use in Multi-Tensor Analysis.

Both can handle 26 different conceptual types of object/index at once. With the Rainbow-VPB 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.

4) We are then ready for the algebra formed by O , L and F ! This is here:

https://wordpress.com/page/conceptsofshape.space/1310 .

Because O‘s key property is that it is an involution, and L‘s is that it is a projection, we also use the notation J , P, F . 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.

We even pick up a further occurrence of F in Linear-Algebraicizing Euler’s Quadrilateral Theorem! This is here:

https://wordpress.com/page/conceptsofshape.space/1297 .

The main item here is however that our new proofs of Apollonius and EQT are the 3-path and 3-star tree versions of proofs applicable to all trees. Corresponding to all eigencluster shapes (alias relative Jacobi coordinate shapes) for all N-body problems. This is furthermore a dimension-independent result, thus holding for the usual 3-d context of the N-body problem as well as for the 2-d one of N-a-gons. We follow this up in the following three articles:

https://wordpress.com/page/conceptsofshape.space/1299

https://wordpress.com/page/conceptsofshape.space/1301

https://wordpress.com/page/conceptsofshape.space/1512

5) Here is a first robustness test of the Heron’s formula proof of Kendall’s Little Theorem and the smallest Hopf map https://wordpress.com/page/conceptsofshape.space/1237 . By considering quadrilateral area formulae; these fail in many ways, and are argued to not be the right generalization to try from a technical point of view. The quadrilateral Casimir is, rather, the square root of the sum of the areas of the three constituent triangle subsystems… We will at some point then re-try this robustness analysis using this instead…

6) Here is a new 1-parameter family (the equi-Cevian triples) of Heron’s formulae in a very strong sense (for equi-Cevians) As well as a new 3-parameter family in a looser sense (for invertible Cevians). https://wordpress.com/page/conceptsofshape.space/1259 .

The 3-parameter family’s Theorem is given a new form here, alongside a further form for a 2-parameter family: the concurrent triples of Cevians https://wordpress.com/page/conceptsofshape.space/1261 .

An ensuing 4-Areas Theorem for Affine Geometry is here: https://wordpress.com/page/conceptsofshape.space/1658 .