Update on publishing an article

Discussion of shapes with curves and holes in various dimensions.

Re: Update on publishing an article

Postby PWrong » Wed Oct 29, 2014 12:33 pm

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.
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Re: Update on publishing an article

Postby PWrong » Mon Nov 10, 2014 6:28 am

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?
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Re: Update on publishing an article

Postby ICN5D » Fri Jan 14, 2022 1:31 am

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.
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Re: Update on publishing an article

Postby Marek14 » Fri Jan 14, 2022 6:28 am

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

For now, I'll leave this picture as a hint :)
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Re: Update on publishing an article

Postby ICN5D » Mon Jan 17, 2022 6:37 pm

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 ....
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Re: Update on publishing an article

Postby Marek14 » Mon Jan 17, 2022 8:23 pm

ICN5D wrote: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.
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