Update on publishing an article

[PWrong]

I've read a lot more about separating PDEs. Things are actually simpler than they seem in a way, and much more complicated in another way.

There are two equations I'm interested in: The Helmholtz equation and Laplace's equation. Laplace's equation is simpler, which means there are more nice solutions available. Some coordinate systems are special in that these equations can be "separated" in them.

In 3D, the Helmholtz equation is separable in 11 different coordinate systems. But it turns out that they're all degenerate forms of ellipsoidal coordinates. These are related to the quadrics, but they don't map one to one with them. In fact it actually seems a bit subjective what you count as a "different" coordinate system.

The Laplace equation is separable in all of these, but there's also a different form of solution that permits two new coordinate systems: toroidal and bispherical. However these are also special cases of "cyclidal coordinates". In fact, I just realised that bispherical coordinates look just like toroidal coordinates, except that the coordinate surfaces are those dodgy torii that intersect themselves. I can hardly find anything about cyclides anywhere, but they're basically a generalization of the torus. The equation reminds me of the expanded equation of the torus. So they're a special type of quartic surface.

The method they use (at least in the old textbook I'm reading) for Laplace's equation is bizarre. It involves something called "pentaspherical coordinates" and a 5D version of Laplace's equation.

Now here's my conjectures for 4D:

1. The Helmholtz equation will be separable in hyperellipsoidal coordinates and all its degenerate forms, but nothing else.

2. Laplace's equation will be separable in all of these, plus a large number of 4D cyclidal coordinate systems, which will relate to toratopes in some way.

3. Possibly it will also be separable in some coordinate systems corresponding to octic (eighth order) surfaces, and maybe others. I think the toratopes ((II)(II)) and (((II)I)I) are both octic surfaces.

[PWrong]

I talked to my supervisor again. He recommended some people I can talk to who know more about PDEs than he does. I'll write up a paper, post it on arxiv, and send it to these people and ask what they think.

Here's some questions I'll mention in my paper, along with an explanation of toratopes.

1. Can it be proven that Helmholtz is separable only in quadric coordinates, and Laplace only in cyclidal coordinates?

2. Classification of cyclides and cyclidal coordinates in 3D

3. Classification of quadrics and quadric coordinates in nD, together with solutions to the Helmholtz equation.

4. All the toratopic coordinate systems in nD. Solutions to the Laplace equation for just the quartic (cyclide) ones.

5. If the answer to 1 is yes, then can anything interesting be done with other coordinate systems?

[ICN5D]

Hey guys, I just wanted to revive this ancient topic we once talked about. So, about 7 years ago, I made a post on my little subreddit /r/hypershape of a table of toratopes up to 7D, with symbols, equations and my modified sphere bundle notation. I wasn't trying to do anything with it really, since it was new and cool at the time (and still is..). But, apparently it did help someone with their research in some way. I got contacted by a guy at MIT (a mathematical physics researcher) who wants to cite me on this reddit post.

I haven't heard back from him yet, and still don't know what was important about the post. But the fact still remains: toratopes are novel and interesting, and may have applications in some way. The guy even asked me why this stuff is still outside of mainstream maths. He has a point, you know! One of the struggles on this thread was finding an interesting enough application to justify the time, if I'm not mistaken. I completely understand this part. So many things to work on and never enough time, right?

But maybe that doesn't matter. Maybe just the information of toratopes is good enough, with no worries about any such application. Mathematicians know all about topological sphere bundles and combinatorial tree graphs. But they may not know about the application of the 2 fields: The combinatorial variations of sphere bundles, which is what toratope notation defines.

I might even try to write a paper myself and submit it to a recreational math journal, who knows man. It would be some kind of breakdown on how to use this novel notation, how it relates to algebraics, topology and combinatorics, and maybe even some stuff on multi-complex numbers (in some solutions), at the very least. Peer reviewed is peer reviewed and published is published, regardless of what the journal is. At least this way we can write the article on wikipedia.

[Marek14]

Actually, I'm trying to gather info necessary for an article as well.

For now, I'll leave this picture as a hint

[ICN5D]

Well right on man, that's good to hear! Nice to see that you're taking up the challenge. I'm guessing it has to do with insane tiling patterns ....

[Marek14]

Well right on man, that's good to hear! Nice to see that you're taking up the challenge. I'm guessing it has to do with insane tiling patterns ....

It's not just any pattern. This is a 14-Archimedean tiling - periodic tiling made from regular polygons with 14 different vertex configurations, which is the maximum possible number.

[StephenWiggins]

I believe I have applications for toratopes in the area of reaction dynamics in theoretical chemistry. Many reaction dynamics problems can be treated in terms of an autonomous, Hamiltonian vector field. Consider a three degree of freedom Hamiltonian system. The phase space is 6 dimensional, but the dynamics occurs in the 5 dimensional level set of the Hamiltonian (energy surface). It is desired to seek a codimension one (i.e. 4 dimensional ) “dividing surface” (DS). A DS has the property that all trajectories evolving from reactants to products must pass through it (there are more details, but they are not so important at this level). The DS that I have been analyzing has the structure of a four dimensional torus, which appears to be a ditorus. Such dividing surfaces appear to arise naturally in a variety of systems. I have found it frustrating that there is not a literature that can be cited on toratopes.

Spherinders also arise naturally in reaction dynamics problems.

[StephenWiggins]

Here is an Arxiv link to the paper on reactive islands that uses the spherinder structure

https://arxiv.org/abs/2104.05798

The paper was published in Physica D

https://www.sciencedirect.com/science/a ... 8921001330

There is probably a pay wall, but the Arxiv version is freely available.

I published this paper before I knew about this website, which I stumbled across when trying to find a mathematical literature to describe the structure of four dimensional tori in recent work. That paper is under review in the International Journal of Bifurcation and Chaos, where I did cite this website.

[mr_e_man]

The kind of torus usually discussed on this site (including the "ditorus") is a 3D manifold embedded in 4D Euclidean space (or a 4D solid with such a 3D boundary).
I think you want something one dimension higher. See the right half of http://hi.gher.space/wiki/List_of_toratopes

[StephenWiggins]

Appendix B of this paper

https://www.worldscientific.com/doi/epd ... 7423500888

refers to the material on this website. This should now satisfy Wikipedia’s criteria for adding material on toratopes to their website. I believe this could be useful to many people.

[ICN5D]

This should now satisfy Wikipedia’s criteria for adding material on toratopes to their website.

Hi Stephen,

It is very cool to see a ditorus equation being used in theoretical chemistry. Very unexpected place to see hyperdonuts, I'd say!

As for getting toratopes on wikepedia, well I wish it were that simple. I think the editors are going to be (highly) resistant. Only the ditorus is mentioned in the article, which is sort of already "known" to academia. The name ditorus is the only new reference here.

What someone will likely have to do, is go through the effort of detailing the entire class of toratopes. Toratopes are a specific application of combinatorics and topology. Someone has to show how they relate to tree graph combinations from the A000669 sequence, and how that relates to variations of a sphere bundle. This will definitely satisfy the wikipedia requirements.

[ICN5D]

All right, my question is for Stephen, if he ever comes back to see this:

In the paper, is your application exclusive to the ditorus equation only, or could it work properly with the tiger equation as well?

I understand that your approach was very computationally expensive, since the ditorus equation expands into one sick-nasty giant polynomial. I was just (naively) curious if the 4D tiger could apply, or was even considered at all. I won't claim to understand anything in the paper , or the well-detailed underlying principles that went into the "chance discovery" of the ditorus equation, but I do recognize it!