Update on publishing an article

[PWrong]

I talked to my PhD supervisor and my co-supervisor and showed them a brief summary of what we've done here.

The good news is that almost all of this is completely original; The concepts of rotatopes and toratopes are essentially new (although they are basically a subset of fibre bundles), many of the rotatopes and toratopes in 4D probably haven't been named and some might have never been defined. It's very likely that almost none of the higher dimensional rotatopes and toratopes have been defined before.

The bad news is it will be very hard to publish this in a serious journal, unless we can find a clear application to some other part of mathematics. Since all toratopes are homeomorphic to rotatopes, they're not very interesting from a topological point of view. Most work done in higher dimensions is topological. The homology groups I calculated years ago were difficult for me at the time, but apparently there is a theorem that would make them basically trivial to calculate. So we could prove a theorem about the "holes" of rotatopes and toratopes, but a topologist would say "that's just a trivial application of this more general theorem". Similarly with the volumes and surface areas. For rotatopes these are basically trivial, for toratopes they're difficult but not useful.

As I see it we have three options for going forward.

1. We submit it to a popular mathematics journal aimed at mathematics students and teachers (my supervisor suggested the "Mathematical Gazette"). In this case the article would include descriptions of the shapes and an essay on why they're interesting in general. We wouldn't need to come up with anything totally new in this case, although a general formula for volumes might be nice. We also wouldn't need to be so rigorous.

2. We try to find a concrete application to some area of mathematics or possibly physics. By an application I don't mean it has to be useful in the "real world", I just mean it should be able to help solve a problem that mathematicians are interested in, or provide a new way of looking at a known problem. As it stands, toratopes don't currently do that. It might be very hard for us to find an application, but what I can do is email the editors of some journals and find out if they can think of a potential application. One flaw with this is if we find an application for say, just the tiger, that wouldn't necessarily justify writing about anything other than the tiger and its application.

3. Give up on publishing this in journals in the near future and work on either completing the wiki or putting all the toratope information into the website.

[ICN5D]

Well, that's a good synopsis. It seems that finding an application would be a holy grail in a ways. Especially in physics. Hmmmm. I posted a little bit regarding Bose-Einstein condensates, and superposition. In all the rendering and exploring I've done, there's something fascinating about the cut arrays. That is, multiple objects in 3D that unify into one object, in higher dimensions. When a BEC exists in superposition, where this group of atom clouds is actually one particle, it seems like higher dimensions can unify this observation. Each cloud is an intercept of a single super-atom, which arcs up/down into higher directions above/below what w see in 3D. And, on this forum are the maths and concepts, behind objects that can do such a thing. May be coincidence, or a neat connection. Particle superposition is a very strange thing that has been observed. It's difficult to wrap one's mind around a group of objects behaving as one. Here is a mathematical way to describe it, by means of intercepts of a hyperdonut. I'm certainly no expert on the subject, just a toratope explorer. But, it is an interesting connection.

And, certainly the wiki can use some love. I haven't done any at all, just posts on a particular shape. No matter what, it still is good to work on it, be it in the formulas, definitions, visuals, etc. If I could just focus my mind away from 7D for a bit ......

[ICN5D]

Any word yet on an application to toratopes? Thinking about it, I feel it should be published anyways. Just an intro into the basic ideas and principles is enough to get people talking about it. Something I noticed on other forums, was that a published article was needed to start a thread or discussion on a subject. Then, we could get to work on the wiki and build it up with all the info posted here. It seems like just defining and documenting them is the application by default. More things can be discovered once the initial idea gets around.

[Marek14]

Maybe the trick is to show several problems which are all tied to toratopes one way or another. The combinatoric structure is one, then there are the surfaces and volume equations, then the transformation of square-root implicit equations into non-square-root ones... And toratopes gain in meaning by being in the intersection of all these problems.

[PWrong]

I think it will be very difficult to find a direct application. An application essentially means that someone else is trying to solve a problem, and they need rotatopes and toratopes to solve it, but they don't realise they need them. Having a connection between toratopes and some other area isn't enough if that connection is only one way.

The motivation that I was thinking of is this: If there was a society that lived in 4 or more dimensions, what shapes would four dimensional children learn about in school? Rotatopes and toratopes answer that question. I think when we explain it that way, we can justify discovering these things without a direct application.

So I think we should go for the option of publishing in a recreational maths journal. I've written up a brief document that I gave to my supervisors. Who else here can write in LaTeX?

[Marek14]

I think it will be very difficult to find a direct application. An application essentially means that someone else is trying to solve a problem, and they need rotatopes and toratopes to solve it, but they don't realise they need them. Having a connection between toratopes and some other area isn't enough if that connection is only one way.

The motivation that I was thinking of is this: If there was a society that lived in 4 or more dimensions, what shapes would four dimensional children learn about in school? Rotatopes and toratopes answer that question. I think when we explain it that way, we can justify discovering these things without a direct application.

So I think we should go for the option of publishing in a recreational maths journal. I've written up a brief document that I gave to my supervisors. Who else here can write in LaTeX?

The 4D world idea is interesting. For example: what shape should the wheels of a 4D cart have? Wheels in 3D are basically short, fat cylinders. What properties would cubinder spherinder or duocylinder wheels have? And of course, the tires on those wheels would be various toratopes

[ICN5D]

Maybe the trick is to show several problems which are all tied to toratopes one way or another. The combinatoric structure is one, then there are the surfaces and volume equations, then the transformation of square-root implicit equations into non-square-root ones... And toratopes gain in meaning by being in the intersection of all these problems.

That's exactly what I've been thinking. These shapes open the door to a lot of interesting math applications alone, especially in the combinatorics of polynomials. Toratopes haven't been defined before, which makes those equations never written or solved, in the way we're trying to do. I imagine that showing the process of solving a degree-16 polynomial into perfect roots of four torus quartics may be of interest, and value.

Then there's the topology side of it. Sure, these are specific examples of a more generalized theory, but they haven't been defined! There could be something new to discover there. Also worth noting is some kind of a generalized Pappus's Centroid Theorem, and how it applies to 2-spheres and over.

The motivation that I was thinking of is this: If there was a society that lived in 4 or more dimensions, what shapes would four dimensional children learn about in school? Rotatopes and toratopes answer that question. I think when we explain it that way, we can justify discovering these things without a direct application.

Yep, I agree. It's like Basic 4D Geometry 101. That way, it'll immortalize the names: spherinder, cubinder, duocylinder, etc. Lots of study has gone into polytopes, but not so much anything curved with circular edges. Especially not any kind of toratope. We could show how a tiger and ditorus can be S¹xS¹xS¹ , but are distinctly different. Defining them in fiber bundles makes the notation quite a bit easier to read and translate. It also helps to read T³ and think 3-torus, and S¹xC2 , and think 'inflated Clifford Torus' , for quicker access to a visual in the mind. The n-torus has been studied, but has it been defined implicitly, or graphed in 3D intercepts? Those are the questions I feel we're answering here.

[PWrong]

We need to distinguish carefully between a problem in toratopes that can be solved using another area of mathematics, and a problem in another area of mathematics that can be solve using toratopes. It's easy to find the former, but much harder to find the latter.

Then there's the topology side of it. Sure, these are specific examples of a more generalized theory, but they haven't been defined! There could be something new to discover there.

The topology of toratopes isn't that interesting, because toratopes are homeomorphic to rotatopes in a higher dimension. And the embedding dimension doesn't matter in topology. The topology of rotatopes isn't especially interesting because they're just products of spheres. Spheres are well understood and products are well understood. The article could have a short section about the topology and homology groups, but there's actually not that much to say.

Also worth noting is some kind of a generalized Pappus's Centroid Theorem, and how it applies to 2-spheres and over.

The thing about PCT is that it's almost kind of a mnemonic. It saves you doing difficult integrals in the case of simple shapes that come up again and again. It's very useful for engineers. But the most general version of PCT possible is the theory of multiple integrals and coordinate transformations. This can find you the volume of any shape in any dimension. That's what I was trying to do before. But I wasn't inventing any new maths, I was just applying well known techniques to a new set of shapes. If I wanted to, I could do much more complicated things, like solve a partial differential equation over a tiger, or analyse a billiard flow on its surface. The tools to do this already exist, it's just difficult to apply them because the shapes are complicated and hard to visualise.

There's a joke that seems relevant here.

An engineer, a physicist and a mathematician are staying in a hotel.
The engineer wakes up and smells smoke. He goes out into the hallway and sees a fire, so he fills a trash can from his room with water and douses the fire. He goes back to bed.
Later, the physicist wakes up and smells smoke. He opens his door and sees a fire in the hallway. He walks down the hall to a fire hose and after calculating the flame velocity, distance, water pressure, trajectory, etc. extinguishes the fire with the minimum amount of water and energy needed.
Later, the mathematician wakes up and smells smoke. He goes to the hall, sees the fire and then the fire hose. He thinks for a moment and then exclaims, "Ah, a solution exists!" and then goes back to bed.

This is how mathematicians will see the volumes of toratopes problem. The shape is there, the tools for calculating its volume are there, so what's the problem?

[PWrong]

What might be interesting is the coordinate transforms themselves. Rotatopic coordinates would be very useful to anyone working with any shapes with some kind of symmetry in higher dimensions. It might be good to look at hyperbolic rotatopic coordinates and other kinds of systems. As far as I know nothing like that has been done. But we still have the problem that if any mathematician needed it, they could easily derive it.

[ICN5D]

Ah, yes. I completely see where you're coming from, now. No one really cares if there's a known way to solve a problem. They care about finding an unknown way, to an unsolved problem.

• Is there any application to just pure math in polynomial structure?
- If so, toratopes represent a very high degree of symmetry in both implicit and explicit. What also seems interesting are the maths behind deriving the intercepts. Are there known methods to finding them, in degree-8 or 16? How about finding the 16 circles intercept of a degree-32? Or, is that just another example of known stuff?

I like the idea of toratopic coordinates. Strange, I was thinking about that today, actually. If it's possible to locate yourself on the surface of a torus using an x,y grid, then one could do the same on the 3-surface of a 3-torus or tiger. I wonder how that would relate to the Villarceau sections, and the tangent points? Marek noticed in the 12-tangent cut of double tiger (((II)(II))(II)) , that you can inscribe an icosahedron using those twelve points as vertices. I have no idea what that means, but sounds interesting. Higher dimensional Villarceau sections are also completely unexplored.

I honestly can't think of any unsolved problems that would relate to toratopes. My exposure to academics is very small, which is my disadvantage. But, I'll keep an eye out for anything that sounds relevant.

[PWrong]

Apparently I posted about solving partial differential equations in higher dimensions over nine years ago. Maybe I was on to something. There are 13 coordinate systems in which Laplace's equation is separable. I've found some articles discussing Laplace's equation in 4D Cartesian and hyperspherical coordinates, but nothing else. Duocylindrical coordinates might have been done somewhere. Spherindrical and cubindrical might have been done and will be too trivial to be interesting. But knowing what we know here, maybe I could enumerate all of the coordinate systems in nD in which this equation (and other PDE's) have a solution. I imagine that would be pretty interesting. I don't know if it's a question any mathematicians would really like to have answered though.

The main problem now is, I already have an interesting postdoc project which I'm about to start working on any day now. But I'll see what my supervisors think of this idea.

[PWrong]

What also seems interesting are the maths behind deriving the intercepts. Are there known methods to finding them, in degree-8 or 16? How about finding the 16 circles intercept of a degree-32? Or, is that just another example of known stuff?

It seems like this falls in the category of questions about toratopes, and not interesting to anyone who isn't already interested in toratopes.

I like the idea of toratopic coordinates. Strange, I was thinking about that today, actually. If it's possible to locate yourself on the surface of a torus using an x,y grid, then one could do the same on the 3-surface of a 3-torus or tiger.

Toroidal coordinates are more complicated than you might expect.
https://en.wikipedia.org/wiki/Toroidal_coordinates
We could probably generalise these to toratopes, but we'd have to understand the original first.

[PWrong]

Here's a quick list of coordinate systems that probably exist in 4D. I don't know what any of these would be like or whether they'd be separable, but it's clear there are a lot of them.

Rotatopic coordinates: Cartesian, hyperspherical, spherindrical, cubindrical and duocylindrical.

For any 3D coordinate system we can simply add a w axis:
confocal ellipsoidal, confocal paraboloidal, conical, elliptic cylindrical, oblate spheroidal, paraboloidal, parabolic cylindrical, prolate spheroidal

We can combine any pair of 2D coordinate systems: Polar, elliptic, parabolic, bipolar.
For example, "elliptic elliptic coordinates" or "polar bipolar coordinates".

Then there are probably many other combinations of these quadric based coordinate systems.

Coordinates based on the "bi" prefix:
In 3D, bispherical comes from rotating bipolar around the x axis (which contains the focii), while toroidal comes from rotating bipolar around the y axis (which separates the focii).

In 4D, we could have
bihyperspherical (the obvious extension of bipolar, bispherical, ...)
bispherindrical (bispherical x I)
bicubindrical (bipolar x I x I)
bipolar bipolar

toroidal cylindrical (toroidal x I)
bispherical rotated around the plane containing the focii,
bispherical rotated around the plane separating the focii
Bitoroidal? That's probably a thing.
I assume one of these has a spheritorus isosurface while the other has a torisphere isosurface.
There are probably also systems with ((21)1) and (22) as isosurfaces.

[PWrong]

I can't find out if there are provably only 13 systems, or if these are just the only ones with known solutions. I might have a go at one that isn't separable and see what happens.

[ICN5D]

That looks interesting, and yes, is more complex than I was thinking. I wonder what a tiger coordinate system would offer, in some other application? Perhaps some kind of electrical engineering, or physics? Seeing as how a tiger contains properties of Hopf fibration, who knows what else it can do.


There's some other things I think about, in relation to electron orbitals. Quickfur pointed out to me how an orbital is some kind of 6D standing wave, with a real and complex component. The real component intercepts our 3D plane, with another duplicate of it in the "complex plane". If the real is identical, the orbitals still behave differently when the complex is different.

From what I've seen by exploring these toratopes, is something very similar, if not exactly what this is describing. Starting in 5D, we have enough room for a toratope to completely wrap around a 3D plane, and contain it like a thin needle. While in this empty cut, one can move out of the center of the hole, to an intercept of the ring.

This will make half (or less) of the array of intercepts appear, with the other half displaced in a higher dimension, and thus in the complex plane. Every one of those gifs I made for ((((II)I)I)((II)I)) is scanning while turning into an empty cut, the last sequence is the empty. In them, you see individual clusters of tori appear, evolve, and vanish, in multiple sequences. Those are the ring intercepts that are stacked in a higher dimension, cut at another angle.

So, perhaps the two orbital components are like the ring intercepts of a toratope. What we see is the intercepting orbital. What we don't see is the second copy, that exists in a higher dimension of space, that we know nothing about. Kind of interesting to think about.

[PWrong]

One of the main applications for any uncommon coordinate system is to solve PDE's where the initial conditions can be simply expressed in that coordinates. So, suppose you know what the temperature is everywhere on the surface of a tiger, then you could use tiger coordinates to solve the heat equation and find the temperature on the inside of the tiger. But this kind of application isn't great, because again we're just using toratopes to solve a problem about toratopes.

However in general it might be a good idea. PDEs have a huge range of applications, and some of them require four or more spatial dimensions. I doubt PDE's have been studied in higher dimensions except in cartesian and hyperspherical coordinates, so there's a lot to discover here. The main problem is that PDEs are very difficult. They're not my area of expertise, although I kind of enjoy them.

Quickfur pointed out to me how an orbital is some kind of 6D standing wave, with a real and complex component.

Well, kind of, but it's very different from a 6D toratope. A toratope has 6 real values that depend on 5 real variables, whereas a wavefunction has 2 real values that depend on 3 real variables. So there isn't that much of an analogy to be made. It's cool to think about, but not rigorous enough for our purposes.

[PWrong]

I'm struggling to figure out how to even describe tiger coordinates. If toroidal is bipolar rotated around the separating axis, what are tiger coordinates?

Perhaps an easier question: is the tiger just a torus rotated in a particular way?

[ICN5D]

What is a PDE ?


This also reminds me of what I read in Brian Greene's book, The Elegant Universe. He talked about how they rewrote some thermo equations in an 8D context, making it closely resemble general relativity. Not sure how that applies. Could they have solved a heat equation for some toratope-like shape, that can also be made from a relativity equation? One thing toratope intercepts do well, is unite in higher dimensions, into one shape and one equation.

Yes, tiger is a torus rotated along plane zw, around plane xz, from the orientation ((xy)z), into ((xy)(zw)) . The torus is revolved around a circle that lays flat in the zw plane. The other plane is xw, which makes ditorus ((xw)y)z). I guess in some sense, it's bi-toroidal around the separating axis, like ditorus, in a different orientation.

[PWrong]

Partial differential equation.

He talked about how they rewrote some thermo equations in an 8D context, making it closely resemble general relativity. Not sure how that applies.

Yes, this sort of thing happens all the time. That particular case might not apply to us, but if we develop the tools for solving PDEs in higher dimensions, that could have a wide range of applications and we wouldn't have to find one specifically.

Here's a guess for what tiger coordinates might be, based on the toroidal coordinates.


If this is correct, then surfaces of constant tau should be non-intersecting tigers, with the three radii depending on a, b and tau.

[PWrong]

One application of toroidal coordinates is in electromagnetism. We haven't looked at electromagnetism in 4D in a long time, but maybe tigers would arise naturally there.

[Marek14]

I'm struggling to figure out how to even describe tiger coordinates. If toroidal is bipolar rotated around the separating axis, what are tiger coordinates?

Perhaps an easier question: is the tiger just a torus rotated in a particular way?

Yes, of course. All 4D toratopes occur as rotations of 3D ones (and this is true in higher dimensions as well).

Generally, an n-dimensional toratope can be rotated around a (n-1)-dimensional hyperplane into (n+1)-dimensional toratope. So you pick the one dimension in toratopic notation that is not included in the hyperplane and replace its symbol I with either II (if the rotation hyperplane bisects the toratope) or (II) (if it's outside of the toratope). So:

Glome (IIII) is a bisecting rotation of a sphere around a plane.
Torisphere ((III)I) is a bisecting rotation of a torus around a vertical plane.
Ditorus (((II)I)I) is a nonbisecting rotation of a torus around a vertical plane.
Tiger ((II)(II)) is a nonbisecting rotation of a torus around a horizontal plane.
Spheritorus ((II)II) is a bisecting rotation of a torus around a horizontal plane, OR a nonbisecting rotation of a sphere around a plane.

Same relations hold higher:

Pentasphere (IIIII) is a bisecting rotation of a glome around a hyperplane.
41-torus ((IIII)I) is a bisecting rotation of a torisphere around a vertical hyperplane.
311-ditorus (((III)I)I) is a bisecting rotation of a ditorus around a major/medium/minor hyperplane.
Tritorus ((((II)I)I)I) is a nonbisecting rotation of a ditorus around a major/medium/minor hyperplane.
Tiger torus (((II)(II))I) is a nonbisecting rotation of a ditorus around a major/major/minor hyperplane.
221-ditorus (((II)II)I) is a nonbisecting rotation of a torisphere around a vertical hyperplane or a bisecting rotation of a ditorus around a major/major/minor hyperplane.
320-tiger ((III)(II)) is a nonbisecting rotation of a torisphere around a horizontal hyperplane or a bisecting rotation of a tiger around a hyperplane.
Torus tiger (((II)I)(II)) is a nonbisecting rotation of a ditorus around a major/major/medium hyperplane or a nonbisecting rotation of a tiger around a hyperplane.
32-torus ((III)II) is a bisecting rotation of a torisphere around a horizontal hyperplane or a bisecting rotation of a spheritorus around a major/minor/minor hyperplane.
212-ditorus (((II)I)II) is a bisecting rotation of a ditorus around a major/major/medium hyperplane or a nonbisecting rotation of a spheritorus around a major/minor/minor hyperplane.
221-tiger ((II)(II)I) is a nonbisecting rotation of a spheritorus around a major/major/minor hyperplane.
23-torus ((II)III) is a nonbisecting rotation of a glome around a hyperplane or a bisecting rotation of a spheritorus around a major/major/minor hyperplane.

[PWrong]

I figured it was probably obvious, but I don't remember having seen it explicitly stated anywhere.

Tiger ((II)(II)) is a nonbisecting rotation of a torus around a horizontal plane.

So tiger coordinates are probably a nonbisecting rotation of toroidal coordinates around the horizontal plane. I'm not sure how exactly that will work, but here's another guess.

[PWrong]

I've been reading up on what makes a PDE separable. It's something to do with Stackel determinants and the Robertson condition, but it seems quite difficult. The idea seems easily extended to higher dimensions in principle, but it doesn't seem like it's ever been done.

[PWrong]

I found something! It's from 1978.

http://projecteuclid.org/download/pdf_1/euclid.cmp/1103901664

Abstract. We present a complete list of all separable coordinate systems for the equations (Helmholtz equation and Hamilton-Jacobi equation) with special emphasis on nonorthogonal coordinates. Applications to general relativity theory are indicated.

So this paper is using nontrivial coordinate systems, which likely includes some toratopic coordinates, to solve important problem that has applications to general relativity. It's not clear to me how they're describing the coordinate systems, and they're certainly not talking about the geometry of them. I'm now going through all the articles that have cited this one.

[Marek14]

General theory of relativity is a bit over my head once the math starts, but the fact that it's so old is encouraging -- the authors certainly didn't have access to the computing equipment we have today!

Vaguely reminds me of the connection between special theory of relativity and hypercomplex numbers...

[PWrong]

The whole article is currently over my head, but I have enough familiarity with PDEs that I think I could probably understand it given a few solid weeks of work.

The point is, this is exactly the sort of thing I needed to justify publishing an article on toratopes.

[Marek14]

The whole article is currently over my head, but I have enough familiarity with PDEs that I think I could probably understand it given a few solid weeks of work.

The point is, this is exactly the sort of thing I needed to justify publishing an article on toratopes.

It should be really interesting!

[PWrong]

I just told my supervisor and he says this will put the idea on a more solid foundation, and that I should go ahead and work on both projects at once, at least until I can get funding for my dynamical billiards project.

So I'll go ahead and start writing an article on toratopes

[Marek14]

I just told my supervisor and he says this will put the idea on a more solid foundation, and that I should go ahead and work on both projects at once, at least until I can get funding for my dynamical billiards project.

So I'll go ahead and start writing an article on toratopes

Good for you

[ICN5D]

Sweet, awesome. This sounds interesting. There had to be something! You asked earlier if anyone could write LaTeX. I'm starting to get the hang of it with the implicit equations, if that's any help.