PWrong wrote: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?
Marek wrote: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 wrote: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.
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.
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.
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.
Quickfur pointed out to me how an orbital is some kind of 6D standing wave, with a real and complex component.
He talked about how they rewrote some thermo equations in an 8D context, making it closely resemble general relativity. Not sure how that applies.
PWrong wrote: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?
Tiger ((II)(II)) is a nonbisecting rotation of a torus around a horizontal plane.
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.
PWrong wrote: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.
PWrong wrote: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
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