This page is for people interested in my 2 papers. The links are to papers+supplements, and pre-publication until noted otherwise. Paper 1 is submitted May 2025, but referees should be aware of the general story of Paper 2, but the exposition in Paper 2 as at May 2025 still needs to be improved.
The supplementary appendices for both papers need much tidying. I have not included these in the May 2025 version of Paper 1.
Paper25_1: At given area and perimeter which tangential n-gons have the largest second moment.
Paper25_2: A bound on the torsional rigidity of tangential n-gons: regular optimises
tanIneq : For regular n-gons z_{*-}(n) increases with n
(Just a hideous inequality on tan: proof OK, but, I think, not worth publishing in a journal.)
Although I am happy with the results, it may well be that journals won't like how they are written. Owing to advancing age, I am keen to work with others who may be interested in the area.
I have a lot of related work. I hope to organise some of it into appendices for an arXiv preprint versions of the papers, i.e. each as the paper as submitted to the journal with appendices not in the journal version.
My hope is to get paper25_1 submitted a.s.a.p. , ?June 2025.
paper25_2 submission, perhaps late August 2025
Although I have a lot of good numerics (for the actual torsional rigidity) for tangential quadrilaterals, and comparison with bounds both my own and others, I would only proceed with work on this if it were with someone else, perhaps an MSc student. (zoom + overleaf + email).
I will not be taking much new work on, alone, while refereeing is in progress.
Tidying 3 years of accumulated Mathematica "bits" - deleting lots of the side issue items - will take time.
One item of related new work is at the Future Work? heading after the History heading
History
The lower bound is that in my paper
2021b "Torsional rigidity for tangential polygons" IMA Journal of Applied Mathematics.
(That lower bound was found by accident. Two separate approaches to asymptotics for small slip in slip flows suggested it, so, for me, my history on it starts with my 1993 paper with Alex McNabb in IMA Journal of Applied Maths.)
My 2020 numerics (torsional rigidity for isosceles triangles mostly) indicated that the lower bound Q- was very close to the actual torsional rigidity Q.
Further numerics (and exact solution at n=4), this time for regular n-gons, found that then the lower bound was spectacularly close. (Reported in Mathematics and Mechanics of Solids in 2022.)
At the Feb 22 ANZIAM national conference (remote presentations) I said I was going to (try to) show, at each fixed n, amongst tangential n-gons with inradius 1,
z- = (Q-)/Area^2
is largest at the regular n-gon. (Trivial at n=3.)
It is related to a very famous open question of Polya and Szego:
Is, at any fixed n, z=Q/Area^2 maximised at the regular n-gon?
Symmetrisation techniques settled it (around 1950 or a bit earlier) at n=3 and n=4. Saying anything that might help for n>=5 is obviously worthwhile.
I quickly decided that I probably wasn't going to get "all n", but n=5, or n<=6, would be good enough. tan(pi/n) entered formulae and this is reasonably nice for 3<= n<= 6.
Around Dec 22-Jan23 I had established, for regular n-gons, z-(reg-ngon) increasing in n. All n.(Not a very pretty proof.) Tends to the disk value as n tends to infinity.
As the 2021 lower bound was in terms of moments, getting inequalities on these and/or bounds and/or other information on the moments is important. Various results related to "EV theorems" felt relevant, but prior to 2023 were not in a form I could use.
Much of 22-23 was spent on two separate lines of enquiry - and attempts to write these up.
The first was just building on my proof at n=4 where I had been able to establish for the 2nd moment that "slender isosceles optimises", and doing lots of numerics. Once again the numerics indicated that the lower bound was close to the actual torsional rigidity. (Of course, not being able to get the 2nd moment result which was the block to being able to do n=5 and n=6 was frustrating.)
Quite early on I had been able to prove, for bicentric hexagons (and even earlier for bicentic quadrilaterals) that regular optimises z-. The starting point were papers by Radic.
This did lead to a conference paper (not on z- though) on bicentric hexagons.
Easiest place to get this is ResearchGate.
I spent much of 22-23 on problems where regular optimised for n<=4 but not above. Ghandihari and Graham, the latter on a tangential hexagon.
https://mathworld.wolfram.com/GrahamsBiggestLittleHexagon.html
and I found a bicentric hexagon, not regular, also "BiggestLittle"
I also put in a lot of numerical effort at z- at larger n, looking for regular not optimising my z-. I found no indication that regular doesn't optimise, but the numerics were not good enough esp. close to regular, to feel confident that there might not be a counter example. (I mostly looked at slender isosceles for counter examples.)
In January 2024 I came across Precupescu's EV theorem paper that could be used (just, and having tan(x) tending to infinity on the range I had to work on introduced technicalities in making it work).
So, using this EV result I finally had a "regular optimises" proof for n=5 and n=6.
In May 2024 I gave a (I think, good) seminar to ANZIAM-WA on the work, but just using Precupescu for the second moment result. (Many of the audience had heard bits in the occasional ANZIAM-WA meetings where we all could give 10 minute presentations.) (I did have some in the audience who wanted more on the background, but I had already done this at the 2022 conference, and I wanted to say what I had done, essentially my Paper 2.)
HOWEVER, the Precupescu paper was quite technical. I thought no applied maths journal is going to like it, and perhaps there are simplifications for my particular application. And, yes, there are. Hence Paper 1. (I have spent a lot of time, without success, looking for nice generalisations. I have spent even more time working on issues associated with biquadratics which have, as far as I can see now, absolutely no use for my Paper 1.)
Future work? bicentric polygons
I think I should keep a focus on the main question: Polya and Szego's "does regular optimise".
For z- I have proved this for n<=6.
But for larger n it just might be easier to prove it for bicentrics, say, i.e. a small subset of my tangential n-gons.
For n=3, 4, and 6 for bicentrics I have short proofs that regular optimises z-, so I will try to discover what is needed in connection with bicentric pentagons for a similar short proof that regular optimises z- . Yes, I know regular optimises at n=5 by paper25_2, but it is a shorter, neat proof I'm after. And, I think I will like the geometry associated with pentagons (Gauss, Kasner, others.)