Torus Rotation

can you verify if my visualization of this behavior in post 18 is correct?
 
krizh said:
using this as reference: View attachment 33986

would this be correct visualization for a torus? or would angular velocity follow the yellow line shown by Cubist ?
View attachment 33987
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If you rotate around the Z axis (marked by [imath]\Omega[/imath]) then your diagram is right. The rotation shown by @Cubist is not possible in [imath]\mathbb R^3[/imath] without deformation of the surface.
 
blamocur said:
The rotation shown by @Cubist is not possible in [imath]\mathbb R^3[/imath] without deformation of the surface.
Click to expand...
Surface deformations are possible.

EDIT: Shouldn't @krizh be telling us which way the torus is rotating? That was the purpose of my diagram, containing 4 possibilities in order to obtain the relevant info. Maybe I've missed a relevant post. Perhaps we're talking in circles :LOL:
 
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blamocur said:
If you rotate around the Z axis (marked by [imath]\Omega[/imath]) then your diagram is right. The rotation shown by @Cubist is not possible in [imath]\mathbb R^3[/imath] without deformation of the surface.
Click to expand...
I've only noticed the rotation marked by the yellow arrow in the @Cubist's diagram. On the other hand, if we only consider solid tori then the yellow arrow does not correspond to a real rotation of a solid since it would cause deformations.
 
blamocur said:
I've only noticed the rotation marked by the yellow arrow in the @Cubist's diagram.
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I've drawn thick black arrows to indicate where the other three rotations are drawn in my image (around the three axes x,y and z)...

torus.png

blamocur said:
On the other hand, if we only consider solid tori then the yellow arrow does not correspond to a real rotation of a solid since it would cause deformations.
Click to expand...
Obviously. And on the other hand, if we consider a torus that can be deformed then the yellow arrow corresponds to a real rotation ?‍♂️

It's OP's question and it's up to them to tell us about it. I don't think that we should be guessing what they need (Also see post#19)
 
Cubist said:
I've drawn thick black arrows to indicate where the other three rotations are drawn in my image (around the three axes x,y and z)...

View attachment 34007


Obviously. And on the other hand, if we consider a torus that can be deformed then the yellow arrow corresponds to a real rotation ?‍♂️

It's OP's question and it's up to them to tell us about it. I don't think that we should be guessing what they need (Also see post#19)
Click to expand...
I have spent sometime analyzing "auto tires" and the stresses developed in use and associated failure mechanics.

That yellow twist is one of the prime reasons of "radial tire belt separation" during "cornering" of the moving vehicle around a bend in the road.
 
Cubist said:
I've drawn thick black arrows to indicate where the other three rotations are drawn in my image (around the three axes x,y and z)...

View attachment 34007


Obviously. And on the other hand, if we consider a torus that can be deformed then the yellow arrow corresponds to a real rotation ?‍♂️

It's OP's question and it's up to them to tell us about it. I don't think that we should be guessing what they need (Also see post#19)
Click to expand...

Sorry for my poor phrasing of the response. I should have written "At the time I'd only noticed..." -- or something similar but in better English :)
 
Subhotosh Khan said:
I have spent sometime analyzing "auto tires" and the stresses developed in use and associated failure mechanics.

That yellow twist is one of the prime reasons of "radial tire belt separation" during "cornering" of the moving vehicle around a bend in the road.
Click to expand...

To avoid such problems we should all relocate to [imath]\mathbb R^4[/imath] where a torus can be defined as [imath]x_1^2 + x_2^2 = x_3^2 + x_4^2 = 1[/imath] and both the toroidal and the poloidal rotations are deformation-free.
 
blamocur said:
Sorry for my poor phrasing of the response. I should have written "At the time I'd only noticed..." -- or something similar but in better English :)
Click to expand...
Thanks, your apology is accepted :thumbup::). And credit where credit is due, it does seem that you somehow found the interpretation that krizh required, and so well done for that.

blamocur said:
To avoid such problems we should all relocate to [imath]\mathbb R^4[/imath] where a torus can be defined as [imath]x_1^2 + x_2^2 = x_3^2 + x_4^2 = 1[/imath] and both the toroidal and the poloidal rotations are deformation-free.
Click to expand...

I didn't know this fact about [imath]\mathbb R^4[/imath] :eek: I've spent quite a lot of time thinking about higher dimensions but have never managed to get an intuitive feel for it. Sphere packing density seems to go down as the number of dimensions increases almost like they develop spikes ?‍♂️

Specifically, regarding the yellow arrow on the torus, I was thinking that a ring of gas or plasma could rotate like this in 3d. Plasma was mentioned in the op's link, "toroidally confined plasmas, as encountered in magnetic confinement fusion..."

Subhotosh Khan said:
I have spent sometime analyzing "auto tires" and the stresses developed in use and associated failure mechanics.

That yellow twist is one of the prime reasons of "radial tire belt separation" during "cornering" of the moving vehicle around a bend in the road.
Click to expand...
That sounds very interesting! Have we got you to thank for these wide rim car wheels that have tires with much reduced wall height? :D
 
thank you guys so much! yes I was thinking of the scenario where a torus is rotating around the green line - z axis.
 
krizh said:
thank you guys so much! yes I was thinking of the scenario where a torus is rotating around the green line - z axis.
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Just out of interest, are you thinking of planetary scale? I guess it's very unlikely for a torus shaped planet to form? (You don't have to answer if you're busy)

EDIT: Wikipedia page (click). I think the authors of that page must have swallowed some dictionaries :LOL:
 
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Cubist said:
Thanks, your apology is accepted :thumbup::). And credit where credit is due, it does seem that you somehow found the interpretation that krizh required, and so well done for that.



I didn't know this fact about [imath]\mathbb R^4[/imath] :eek: I've spent quite a lot of time thinking about higher dimensions but have never managed to get an intuitive feel for it. Sphere packing density seems to go down as the number of dimensions increases almost like they develop spikes ?‍♂️

Specifically, regarding the yellow arrow on the torus, I was thinking that a ring of gas or plasma could rotate like this in 3d. Plasma was mentioned in the op's link, "toroidally confined plasmas, as encountered in magnetic confinement fusion..."


That sounds very interesting! Have we got you to thank for these wide rim car wheels that have tires with much reduced wall height? :D
Click to expand...
No - that concept came before my time. However, I worked on replacing the "sidewall fabric" material - from polyester/nylon to Kevlar/aramid. Kevlar being lot stiffer and stronger than incumbent - but it had other problems (e.g. adhesion). Now we have truck-tires and racing car tires designed with Kevlar fabric (assisted by us at DuPont).
 
Subhotosh Khan said:
No - that concept came before my time. However, I worked on replacing the "sidewall fabric" material - from polyester/nylon to Kevlar/aramid. Kevlar being lot stiffer and stronger than incumbent - but it had other problems (e.g. adhesion). Now we have truck-tires and racing car tires designed with Kevlar fabric (assisted by us at DuPont).
Click to expand...
Interesting. Unlike the main driving surfaces of tires the sidewalls cannot be repaired, as I have had an unfortunate occasion to discover. Another such occasion and I might consider switching to Kevlar :)
 
blamocur said:
Interesting. Unlike the main driving surfaces of tires the sidewalls cannot be repaired, as I have had an unfortunate occasion to discover. Another such occasion and I might consider switching to Kevlar :)
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It breaks my heart to say:

Kevlar in passenger-car tire side-walls produces minor advantage at a large cost disadvantage (at this time).​
 
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