Exotic Tori in R^5?

Discussion of shapes with curves and holes in various dimensions.

Exotic Tori in R^5?

Postby ICN5D » Sat May 02, 2015 10:05 pm

I came across this interesting question, asking about homeomorphic versions of a 4-torus. From what I gather, there would be two of them: (((II)(II))I) and (((II)I)(II)). Apparently, this is still an open question, as toratopes aren't known beyond this forum. So, is this correct? In an attempt to answer the question, what would someone be looking for, in terms of proof? Would homology groups and implicit surface equations be enough? How about the "exotic 5-tori", which has 5 homeomorphic variants to T^5 in R^6?
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Re: Exotic Tori in R^5?

Postby Marek14 » Sun May 03, 2015 5:23 am

These are probably not the only ones -- both of them are nonbisecting rotations of ditorus around different hyperplanes, as is normal tritorus, so presumably nonbisecting rotation of ditorus (or tiger, or anything between) around ANY hyperplane should suffice.
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Re: Exotic Tori in R^5?

Postby ICN5D » Mon May 04, 2015 6:22 am

What other types are you thinking of? Like a polygonal ditorus? Something outside toratope notation, for sure. How would you even classify the mantis, in that context?
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Re: Exotic Tori in R^5?

Postby Marek14 » Mon May 04, 2015 11:45 am

ICN5D wrote:What other types are you thinking of? Like a polygonal ditorus? Something outside toratope notation, for sure. How would you even classify the mantis, in that context?


Well, it starts with torus. Both ditorus and tiger are rotations of torus around a nonbisecting coordinate plane.

But, you could also rotate the torus around a GENERAL nonbisecting plane, not just coordinate ones. This should give something with the same topology as ditorus/tiger. Unfortunately, I never managed to derive an equation for this general case which would contain both ditorus and tiger as special cases.
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Re: Exotic Tori in R^5?

Postby ICN5D » Mon May 04, 2015 5:16 pm

I haven't heard of anything like that before. So, if a torus rotates around a 2-plane, and coordinate planes will be xy, xz, xw, etc, then a general plane will be an oblique angle one? Something like xy/xz?
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Re: Exotic Tori in R^5?

Postby Marek14 » Mon May 04, 2015 5:46 pm

ICN5D wrote:I haven't heard of anything like that before. So, if a torus rotates around a 2-plane, and coordinate planes will be xy, xz, xw, etc, then a general plane will be an oblique angle one? Something like xy/xz?


Yes, something like that.
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Re: Exotic Tori in R^5?

Postby ICN5D » Mon May 04, 2015 5:54 pm

What types of equations have you used? Also, I'm trying to picture what that would look like. Wouldn't it just be a normal ditorus/tiger, but tilted with respect to the coordinate planes? Or, would it be something elliptical?
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Re: Exotic Tori in R^5?

Postby Marek14 » Mon May 04, 2015 6:31 pm

Well, I tried to combine equations from ditorus and tiger in some common form, but it didn't quite work.

As how it would look like: ditorus has a cut looking like two toruses next to each other. Tiger has a cut looking like two toruses above each other. This would have a cut looking like two tilted toruses (they need to be tilted in opposite directions like mirror images).
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Re: Exotic Tori in R^5?

Postby ICN5D » Tue May 05, 2015 5:39 am

Marek! I got it! Remember that function that was supposed to morph a tiger into a ditorus, that you wrote here?

Well..... I just experimented (and self-discovered what you did), and came up with the proper equation. This time, the two tori are perfectly formed, and maintain their shape under rotation, between the vertical stack and side by side cuts. It's also the non-bisecting general plane equation that does what you are looking for. The culprit all this time was a squaring of terms that didn't need to be.

What you had before was:

(sqrt(x^2 + ((sqrt(y^2 + b^2) - 4)^2*cos(a) + z*sin(a))^2) - 2)^2 + ((sqrt(y^2 + b^2) - 4)^2*sin(a) - z*cos(a))^2 = 1^2


What ends up working is:

(sqrt((x*sin(a) + (sqrt(z^2+b^2)-5)*cos(a))^2 + y^2)-3)^2 + (x*cos(a) - (sqrt(z^2+b^2)-5)*sin(a))^2 = 1

where the (sqrt(z^2+b^2)-5) term doesn't need to be squared, since it already is. So, how about that?


A fully 4D equation ends up as:

(sqrt((x*sin(a) + (sqrt(z^2+w^2)-5)*cos(a))^2 + y^2)-3)^2 + (x*cos(a) - (sqrt(z^2+w^2)-5)*sin(a))^2 = 1

where parameter 'a' will adjust the non-bisecting plane of rotation. What's really interesting about the result of this function, is getting the two tori in the exact positions needed to express a mantis cut. We're now 2/3's of the way there. Next step is how to add another torus, and pair it with the other two. Easier said than done.
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Re: Exotic Tori in R^5?

Postby Marek14 » Tue May 05, 2015 6:09 am

Yes, I checked it and it works very well. Not yet sure how to develop this into the mantis, though...
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Re: Exotic Tori in R^5?

Postby ICN5D » Tue May 05, 2015 9:09 pm

Here's a really wild 5D equivalent. It makes several rotations and amazing morphs between ((((II)I)I)I) , (((II)(II))I) , and (((II)I)(II)). You get all three out of this function, plus the ability to rotate the specific shape a few ways. This is the wildest stuff I've seen yet with toratope functions:


(sqrt(((x*sin(d))*sin(a) + (sqrt((y*cos(b)-(sqrt((z*sin(c))^2+(x*cos(d))^2)-8)*sin(b))^2+(z*cos(c))^2)-4)*cos(a))^2 + (y*sin(b)+(sqrt((z*sin(c))^2+(x*cos(d))^2)-8)*cos(b))^2)-2.5)^2 + ((x*sin(d))*cos(a) - (sqrt((y*cos(b)-(sqrt((z*sin(c))^2+(x*cos(d))^2)-8)*sin(b))^2+(z*cos(c))^2)-4)*sin(a))^2 = 1


0 < a,b,c,d < 1.57
XYZbox = -17 , 17

a,b changes general plane of rotation
c,d rotates respective toratope
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Re: Exotic Tori in R^5?

Postby PWrong » Sun May 10, 2015 5:37 am

I'm pretty sure that all of our shapes that are homeomorphic to the 4-torus would also be diffeomorphic. Homeomorphic means there is a continuous transformation between them, that is, the transformation doesn't "break" anywhere. Diffeomorphic means there is a differentiable transformation, that is, it doesn't suddenly change direction or have sharp corners. So toratopes don't really answer this question.

I'll see if I can understand the paper on "fake tori" linked in the question.
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Re: Exotic Tori in R^5?

Postby PWrong » Sun May 10, 2015 6:03 am

Ok here's a very rough explanation of the paper. Most of it is beyond me.

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC223652/pdf/pnas00105-0067.pdf

I described "holes" earlier using homology groups. A closely related topological invariant is homotopy. Homotopy groups are simpler to explain but harder to calculate. Take a look at the table of homotopy groups for spheres and you'll see how complicated they can get.

The homotopy groups for the n-torus are actually pretty simple, being made up of just circles. The first homotopy group for the n-torus is Z^n, and the rest are all the trivial group.

A shape with the same homotopy group as a torus is called a "homotopy torus". But this doesn't mean it's actually a torus! In the paper they have a way of calculating the number of what they call "fake tori", manifolds with the same homotopy group as a torus, but nonetheless different from a torus. Unfortunately I can't figure out in what sense they're "different", or even what these shapes are supposed to be.

Take a look at this sentence though:
This is an elaboration of an idea used by J. L. Shaneson and R. Lashof to show that there is a manifold of the same homotopy type as S3 X T2 but not P.L.-equivalent.


This is the first reference I've ever seen (outside of this forum) to a toratope more complicated than an n-torus. I'm going to check this paper out and see what they say about ((II)I)(III).
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Re: Exotic Tori in R^5?

Postby PWrong » Sun May 10, 2015 6:06 am

I should mention that all of this is happening in the context of Piecewise linear manifolds, a topic I know much less about than smooth manifolds.
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