Tether a Spinning Ball

[Mark]

Judging from the recent thread “Where are you?” some people here are interested in spherical geometry, so you might be interested in a curious fact about space discovered by the English theoretical physicist P.A.M. Dirac (1902-1984) – specifically the “space” of rotations in three dimensional space.

At the bottom of the following webpage you’ll find a link to a computer program (Windows) that generates movies illustrating his discovery in various ways:
How a Spinning Object Can Remain Connected to a Stationary One

It can also show a movie that illustrates the principle behind the spinning jenny used to twist fiber into thread. 

 

[BaalChatzaf]

13 hours ago, Mark said:

Judging from the recent thread “Where are you?” some people here are interested in spherical geometry, so you might be interested in a curious fact about space discovered by the English theoretical physicist P.A.M. Dirac (1902-1984) – specifically the “space” of rotations in three dimensional space.

At the bottom of the following webpage you’ll find a link to a computer program (Windows) that generates movies illustrating his discovery in various ways:
How a Spinning Object Can Remain Connected to a Stationary One

It can also show a movie that illustrates the principle behind the spinning jenny used to twist fiber into thread. 

 

A disk mounted on a mathematically thin axle with a bearing one point thick at the center. 

[Darrell Hougen]

On 1/3/2019 at 7:27 PM, Mark said:

Judging from the recent thread “Where are you?” some people here are interested in spherical geometry, so you might be interested in a curious fact about space discovered by the English theoretical physicist P.A.M. Dirac (1902-1984) – specifically the “space” of rotations in three dimensional space.

At the bottom of the following webpage you’ll find a link to a computer program (Windows) that generates movies illustrating his discovery in various ways:
How a Spinning Object Can Remain Connected to a Stationary One

It can also show a movie that illustrates the principle behind the spinning jenny used to twist fiber into thread. 

 

Wow! Mind blown!

Darrell

[Jon Letendre]

On 1/3/2019 at 8:27 PM, Mark said:

Judging from the recent thread “Where are you?” some people here are interested in spherical geometry, so you might be interested in a curious fact about space discovered by the English theoretical physicist P.A.M. Dirac (1902-1984) – specifically the “space” of rotations in three dimensional space.

At the bottom of the following webpage you’ll find a link to a computer program (Windows) that generates movies illustrating his discovery in various ways:
How a Spinning Object Can Remain Connected to a Stationary One

It can also show a movie that illustrates the principle behind the spinning jenny used to twist fiber into thread. 

 

I don’t have a Windows machine handy so haven’t downloaded the software.

I have clicked the website and read the page and viewed the tethered ball pic.

That  ball can spin without twisting the ropes?

[Mark]

Glad you liked it, Darrell.

Jon,

Yes, it can spin forever without twisting the ropes to the breaking point.  The ropes must stretch and twist but only by a finite amount.  

Since the webpage gives a rather abbreviated account of what the program does here’s a more leisurely description.

Imagine a ball in the middle of the room suspended in the air by six elastic cords.  I’ve gotten used to calling them ropes.  Each rope is glued to the ball at one rope-end and to a wall, ceiling, or floor at the other rope-end.  The ropes are all straight and untwisted.  The following schematic diagram gives some indication of what I mean:
......................................................... | /
...................................................... —O—
....................................................... / |

Then the ball starts turning around on its vertical axis.  The rope-ends attached to the ball’s surface go around with it while the rope-ends attached to the room remain stationary.  The ropes become tangled and twisted.

The ball turns around exactly once and pauses.  During this pause you can try to untangle the ropes but you won’t succeed.  

Then the ball continues turning around, in the same direction, and stops after a second turn.  Surprise, now you can untangle the ropes so they are all straight and untwisted, as they were at the beginning.

The manipulation can be done while the ball turns, then the ball can spin without pause while the ropes never tangle and twist beyond a certain amount.

You can do this with any number of ropes, I said six just to be definite.

 

[Jon Letendre]

10 minutes ago, Mark said:

Glad you liked it, Darrell.

Jon,

Yes, it can spin forever without twisting the ropes to the breaking point.  The ropes must stretch and twist but only by a finite amount.  

Since the webpage gives a rather abbreviated account of what the program does here’s a more leisurely description.

Imagine a ball in the middle of the room suspended in the air by six elastic cords.  I’ve gotten used to calling them ropes.  Each rope is glued to the ball at one rope-end and to a wall, ceiling, or floor at the other rope-end.  The ropes are all straight and untwisted.  The following schematic diagram gives some indication of what I mean:
.......................................................... | /
...................................................... —O—
........................................................ / |

Then the ball starts turning around on its vertical axis.  The rope-ends attached to the ball’s surface go around with it while the rope-ends attached to the room remain stationary.  The ropes become tangled and twisted.

The ball turns around exactly once and pauses.  During this pause you can try to untangle the ropes but you won’t succeed.  

Then the ball continues turning around, in the same direction, and stops after a second turn.  Surprise, now you can untangle the ropes so they are all straight and untwisted, as they were at the beginning.

The manipulation can be done while the ball turns, then the ball can spin without pause while the ropes never tangle and twist beyond a certain amount.

You can do this with any number of ropes, I said six just to be definite.

 

Really. I have to think about this more.

Any axis? Or do we need to avoid having a rope exactly on a pole?

[Mark]

Any axis, but since the ropes can be attached anywhere we need -- "without loss of generality" as they say -- only show that it can be done for some one axis.

[Jon Letendre]

37 minutes ago, Mark said:

Any axis, but since the ropes can be attached anywhere we need -- "without loss of generality" as they say -- only show that it can be done for some one axis.

Ok. So the ball in the room has the 6 ropes fixed to it. I am imagining it. I am going to have it spin twice, then the ropes can be untangled, then I can keep repeating that indefinitely, right?

I am imagining the top rope that goes to the ceiling, the "north pole" of the turning ball. After the two turns it will be twisted X2 and the untangling will have to involve that rope being pulled down and around the south pole, twice.

Do I have that right? 

 

[Jon Letendre]

Not twice, once. The untangling will have to involve the ceiling rope being pulled down and around the south pole once.

(I discovered it is only once that works by taping a flat shoelace to my desk and to a baseball. The flat shoelace discloses twisting. Once around the south pole fixes the twisting from spinning the ball twice.

Now do I have that much right?

[Jon Letendre]

I do have it right! This is fascinating. I now have two flat shoelaces taped to the baseball and to my desk. Each lace just needs one trip around the south pole and that resolves both their tangling and their twisting.

[Jon Letendre]

Now three laces. It still works. The laces have to sequence their trips about the south pole in the correct order, taking their turn. To my surprise there are three sequences that work, instead of just one. There are three laces so there are six possible sequences of turn—taking: 1,2,3  1,3,2  2,1,3  2,3,1  3,1,2   3,2,1. Three of these six sequences successfully untangle the laces. (All of them successfully untwist the individual laces.)

[Jon Letendre]

I think I am starting to see a rule: if a rope is attached in the northern hemisphere then in the untangling it will need to make one trip around the South Pole, while a rope attached in the Southern Hemisphere of the ball will need to make one trip around the North Pole for its untangling.

[Max]

This is a fascinating subject, quite counterintuitive. I downloaded the program, but that doesn't do anything on my computer (at least I hope it doesn't do some hidden damage...). I'd like to play with such a system, but I don't have flat shoelaces or something similar that can be used to observe the twists clearly. Parhaps I should order aome of them on Amazon... I'd like some hands on experience.

11 hours ago, Mark said:

Then the ball starts turning around on its vertical axis.  The rope-ends attached to the ball’s surface go around with it while the rope-ends attached to the room remain stationary.  The ropes become tangled and twisted.

The ball turns around exactly once and pauses.  During this pause you can try to untangle the ropes but you won’t succeed.  

Then the ball continues turning around, in the same direction, and stops after a second turn.  Surprise, now you can untangle the ropes so they are all straight and untwisted, as they were at the beginning.

Rotating twice to get the original configuration back, that seems to be tied(!) to the SU(2) group and its difference from the SO(3) group. I should lookup those things again...

7 hours ago, Jon Letendre said:

Now three laces. It still works. The laces have to sequence their trips about the south pole in the correct order, taking their turn. To my surprise there are three sequences that work, instead of just one. There are three laces so there are six possible sequences of turn—taking: 1,2,3  1,3,2  2,1,3  2,3,1  3,1,2   3,2,1. Three of these six sequences successfully untangle the laces. (All of them successfully untwist the individual laces.)

Let me guess: if 1,2,3 works, so do 2,3,1 and 3,1,2, or just the other three?

[Mark]

3 hours ago, Max said:

 I downloaded the program, but that doesn't do anything on my computer ... at least I hope it doesn't do some hidden damage

Just what did you experience?  It has run perfectly under Windows XP, Windows Vista, Windows 7 (32 bit), and Windows 10 (64 bit). I haven't had a chance to try it on a Windows 8 computer.

[Max]

20 minutes ago, Mark said:

Just what did you experience?  It has run perfectly under Windows XP, Windows Vista, Windows 7 (32 bit), and Windows 10 (64 bit). I haven't had a chance to try it on a Windows 8 computer.

Nothing happened, at least I didn't see anything happen,  I downloaded it again, but with the samer result. This was on my PC with Windows 10 pro. I just tried it on my laptop with Windows home, and now it worked! Strange. I'll try it later again on the other computer and compare the files.

[Max]

I copied the file from my laptop to my PC, but there it still doesn't do anything...

[Jon Letendre]

5 hours ago, Max said:

This is a fascinating subject, quite counterintuitive. I downloaded the program, but that doesn't do anything on my computer (at least I hope it doesn't do some hidden damage...). I'd like to play with such a system, but I don't have flat shoelaces or something similar that can be used to observe the twists clearly. Parhaps I should order aome of them on Amazon... I'd like some hands on experience.

Rotating twice to get the original configuration back, that seems to be tied(!) to the SU(2) group and its difference from the SO(3) group. I should lookup those things again...

Let me guess: if 1,2,3 works, so do 2,3,1 and 3,1,2, or just the other three?

Maybe elastic bands or a ribbon or some sort. Lamp cord.

The other three. You’ve probably gathered already that all three were attached in the northern hemisphere.

[Jon Letendre]

It’s fun and quite counter–intuitive.

Now I have a lace attached to each pole of the baseball and to the ceiling and my desk. I twist the ball twice and there is no entanglement, just both laces are now twisted. The top one makes a pass around the South Pole which untwists it, but now the laces are entangled. Then they become unentangled when the lower lace makes its untwisting pass around the North Pole.

[Max]

18 minutes ago, Jon Letendre said:

Maybe elastic bands or a ribbon or some sort. Lamp cord.

I've no such things that could be used, but Amazon is your friend: I ordered a set of laces, flat and 160 cm long and a set of 10 cm styropor balls. 

[Mark]

Max,

Probably your browser just copied it to your “Downloads” folder.

To run it open Windows Explorer.  On the left pane of that window very near the top you see “Desktop” and underneath it “Downloads.”  Click on “Downloads.”  Then on the right panel you see a list of files.  Double-click the exe at the top (if the list is sorted by Date most recent first).

Then you see a scary message saying “Unknown Publisher-,” then click “Run.”

[Max]

As I already wrote, the program runs fine on my laptop (Windows 10 home). I copied that file to my PC (Windows 10 Pro), but there it just doesn't do anything, not even saying "Unknown Publisher". I tried changing protection settings, run as Administrator, compatibility mode, all to no avail.

[Mark]

Max,

On your Windows 10 Pro machine, what exactly happens when you double-click the exe file?  It just doesn’t do anything, OK, but what exactly do you see?  Pretend I’m from Mars.

 

[Max]

14 hours ago, Mark said:

Max,

On your Windows 10 Pro machine, what exactly happens when you double-click the exe file?  It just doesn’t do anything, OK, but what exactly do you see?  Pretend I’m from Mars.

I saw just for a second or so the Windows hourglass, and then everything was the same as before, filename etc. No message or other symbols appeared. But since today it suddenly works! As far as I know, I didn't change anything since the last time it wouldn't work, so it's a great mystery. The only thing I can think of is that one of those attempts to remedy the problem I mentioned in my previous post, did in fact have the desired effect, but only after restarting the computer, although there was no message to that effect. But I'm glad that I now can explore the program on my big screen. It looks great!

[Mark]

It’s a mystery to me as well.  I don’t see how any of the three things you mentioned could have affected the program.

Compatibility mode:  The program should run without this setting and if it were required, restarting your PC wouldn’t affect the setting.

Administrator mode:  Ditto.

Protection setting:  I don’t know much about this but judging from what I read on various Internet references the only such setting called protection has to do with turning on or off the Windows “restore points” feature.  This wouldn’t affect how any program runs.

Anyway, glad you got it working.

 

[Mark]

Two more motions were recently added to those the Antitwister program can display and, if memory serves, two more before those but after the date of the last post above.  (The new motions aren’t listed on the linked-to webpage; I didn’t want the advertising to be over-detailed.)

This program teaches you to visualize 3D rotations.  I imagine it as a “plaything of Krell children” – a reference to the old science fiction movie Forbidden Planet (1956).