Is the Belt Trick Possible with Continuous Deformation in 3D Rotation Space?

Hi, in the following video at 15:15 the twist of ##4\pi## along the ##x## red axis is "untwisted" through a continuous deformation of the path on the sphere 3D rotations space.

My concern is the following: keeping fixed the orientation in space of the start and the end of the belt, it seems there exist a continuous deformation of the belt taking place only in the ##x,y## plane that allows to untwist completely the belt.

Is it actually the case ? In other words my concern is whether the belt actually supports this kind of continuous deformation.

Ps. See also at 12:25 why the rotation path along the twisted belt takes place only in ##x,y## plane of sphere rotations space.

cianfa72 said:

Is it actually the case ? In other words my concern is whether the belt actually supports this kind of continuous deformation.

I'm not sure what you're asking. Do you suspect that the physical demonstration shown near the start of the video is a fake? It's real -- you can test this yourself in the privacy of your own home. (Btw, it's also possible to untwist the ##4\pi## twist without raising the book like in the video, iirc.)

strangerep said:

Do you suspect that the physical demonstration shown near the start of the video is a fake? It's real -- you can test this yourself in the privacy of your own home.

Yes, it'real (I tested it).

strangerep said:

it's also possible to untwist the ##4\pi## twist without raising the book like in the video, iirc.

Yes, however I've not a clear understanding why in the video the untwisting process may take place involving only points in the ##x,y## plane of the representative sphere of 3D spatial rotations (i.e. not involving any point off ##x,y## plane).

strangerep said:

(Btw, it's also possible to untwist the ##4\pi## twist without raising the book like in the video, iirc.)

I was not able to do this. Is there a trick or a reference that I can look at?

FactChecker said:

Is there a trick or a reference that I can look at?

Take a look to the c) picture from Penrose's book


Simply loop the belt over the book !

cianfa72 said:

Take a look to the c) picture from Penrose's book

View attachment 338604
Simply loop the belt over the book !

Right, but I interpret "loop the belt over the book" as requiring the book be up so that the belt can go under it. I can't se how to get it to work otherwise.

FactChecker said:

Right, but I interpret "loop the belt over the book" as requiring the book be up so that the belt can go under it. I can't se how to get it to work otherwise.

I see in this video (not the one in the OP) that it is not necessary to circle anything around the stationary end (the book or her body in the video). It is only necessary to circle the rotated end around the belt (or hair in the video). So I will experiment more.
UPDATE: Sure enough. It works.

cianfa72 said:

Take a look to the c) picture from Penrose's book
...
Simply loop the belt over the book !

This whole thing seems similar to a rubber-band around some object.
A rubber band going once around an object can lay flat, without twists, similar to the first picture.
If the band goes around twice, there's a full twist to it (2π), and it cannot lay flat (2nd picture)
If it goes around thrice (4π twist), it can be made to lay flat or not (3rd picture).

Ditto for any even/odd number of times around.

The physics of this similarly applies to folding up one of those mesh laundry bags with stiff wire hoops encircling 4 sides. They sold in the stable state of being folded in triplicate. It doesn't work if you try to go around only twice.

We can calculate the composition of 3D rotations using composition of quaternions. According to the ##4\pi## untwisting path in OP video, the belt must support rotations along axes that involve a component along the ##z## axis.

Or, said in other words, which is the condition on the belt such that the twisting process at 12:25 is described by a curve that lies entirely in the ##x,y## plane of the representative sphere of the 3D rotation space (and it is not an arbitrary closed curve through the origin) ?