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        riddles >> hard >> Topological Rings
        
(Message started by: william wu on Feb 3rd, 2003, 1:34pm)
Title: Topological Rings
Post by william wu on Feb 3rd, 2003, 1:34pm
TOPOLOGICAL RINGS
A real nice puzzle. My favorite kind: people of all ages can understand that it seems totally impossible :)

http://www.ocf.berkeley.edu/~wwu/images/riddles/topologicalRings_linked.jpg   http://www.ocf.berkeley.edu/~wwu/images/riddles/topologicalRings_unlinked.jpg

Imagine the object above in the figure to the left made from perfectly elastic material. Can you transform it so as to unlink the two rings as in the figure on the right? One possible way is to cut one ring, move the other ring through the gap, and rejoin the the first ring exactly as it was. That would be a legitimate topological transformation. However, it is also possible to transform the first shape into the second without any cutting, simply by manipulating the objects in the appropriate manner (stretching, bending, but not breaking). Can you see how to do it?

Note: Source: Brandon McPhail. Play his spiffy set applet at http://www.reed.edu/~mcphailb/applets/set
Title: Re: Topological Rings
Post by Chronos on Feb 3rd, 2003, 4:57pm
OK, I think I've figured out, but I'ven't a clue how one might try to express the answer, without some sort of 3D drawing tools.
Title: Re: Topological Rings
Post by Icarus on Feb 3rd, 2003, 7:34pm
One solution came to mind almost immediately, but I'm not sure it is the one you are looking for. To find out, I will ask this hidden question:
[hide]Is the inverse transformation also continuous? The one I thought of isn't.[/hide]
Title: Re: Topological Rings
Post by SWF on Feb 3rd, 2003, 8:22pm
This is interesting. It looked impossible at first, but after finding a good way to visualize, it seems almost obvious. I can give a fairly brief and clear written description that I believe would make it pretty clear to most people. This problem is too good give away so soon, so I will generously let everyone enjoy the problem.
Title: Re: Topological Rings
Post by Icarus on Feb 3rd, 2003, 8:34pm
I have it! Just push a section of one of the rings in the direction of a 4th spacial dimension! You can then bring the other ring out in that area, and pull the first ring back into 3 dimensions again. So easy, anyone can do it! 8)
Title: Re: Topological Rings
Post by Kozo Morimoto on Feb 5th, 2003, 4:37pm
Are you allowed to join?  (like playdoe)
Title: Re: Topological Rings
Post by aero_guy on Feb 5th, 2003, 10:28pm
I have a solution, but it isn't a 'eureka' solution.  So I will ask:

[hide] is perspective key? [/hide]
Title: Re: Topological Rings
Post by Chronos on Feb 6th, 2003, 11:39am
aero_guy, to (hopefully) answer your question:  The solution which I found does not depend on any feature of the objects which is not immediately apparent.  They're just like how they look.  My solution also does not rely on any more than the usual three spatial dimensions, and I actually found the inverse transformation first.  I'm not sure how one transformation could be continuous without the other being so.

A minor hint, by the way:[hide]The solution is easier to see if the center section (the part not on any loop) is shorter.  Since we're allowed to stretch and squish as much as we'd like, start by squishing the middle[/hide].
Title: Re: Topological Rings
Post by Phil on Feb 6th, 2003, 12:27pm
I got it from your hint, I think.
[hide]Shrink the connecting bar down to nothing, and you simply have two rings joined at one point, not intertwined, just connected. Just turn the inner ring out before restretching the connecting bar. [/hide]
Title: Re: Topological Rings
Post by Icarus on Feb 6th, 2003, 7:03pm

on 02/06/03 at 11:39:17, Chronos wrote:
I'm not sure how one transformation could be continuous without the other being so.


overlap the two intertwined loops and merge them into a single loop (like play-doe). The two loops on the left form a single loop on the right. The mid-section on the left forms the other loop on the right. The mid section on the right is formed by stretching out the intersection. This is a continuous transformation, but it does not have a continuous inverse, since the inverse would involve tearing apart what was joined together, which would be heresy! :o

Matthew 19:6 - "What God has joined together, let no man separate."

Needless to say, your solution is much better, not to mention safer for our immortal souls! ;)
Title: Re: Topological Rings
Post by SWF on Feb 6th, 2003, 8:16pm
Here is how I visualize this (hidden text):
[hide]Imagine two metal hoops that are not linked. Bring them together so they almost touch at a single point. Put a drop of rubber cement at that point to join them together. This is the topological equivalent of a figure-8. If you pull the rings apart so the rubber cement stretches you get two unlinked loops joined together by a long connection as shown in the 2nd picture.

Repeat, but this time start with two linked metal hoops with a drop of rubber cement bonding them at a point. Again, it is equivalent to a figure-8, so it is same as the first case. Separate the rings as far as you can to stretch the cement, then grab the rubber cement and stretch it into a long strand. This leaves two linked rings joined by a long connection, as show in the first picture.

It may not be obvious to everyone that both cases are equivalent to a figure-8. If that is a problem, perhaps it would help to imagine the drop of cement to be a large blob of rubber cement than covers enough of the rings near the contact point so that one cannot see whether the rings are linked or unlinked.[/hide]
Title: Re: Topological Rings
Post by James Fingas on Feb 7th, 2003, 11:35am

Quote:
overlap the two intertwined loops and merge them into a single loop ... the inverse would involve tearing apart what was joined together, which would be heresy!  

Matthew 19:6 - "What God has joined together, let no man separate."


Heresy? Pronouncing that you are God--now that's heresy!
Title: Re: Topological Rings
Post by Icarus on Feb 7th, 2003, 9:18pm
Aye - but that heresy is an old tradition in my family! (Google for "sinclair Templar Jesus" if you want to know - but you'd be wise not to! ;))
Title: Re: Topological Rings
Post by Guest on Nov 22nd, 2003, 4:23am
    Does this mean there is no actual solution to this problem? That is, if you have a stretchable rubber pipe which has been formed into the configuration in the first figure, is there no way you can manipulate it to unlink the rings?
Title: Re: Topological Rings
Post by Icarus on Nov 22nd, 2003, 12:39pm
No - it has a solution. Chronos and Phil have described one way of looking at it (the descriptions are hidden - highlight their posts to read them).

Note though that describing the material a being like rubber is not quite the right idea. The rings are to be considered as made of "topological clay", which can be deformed freely (streched, compressed, twisted, pulled along itself, etc) without any resistance, but can never be cut or torn.

An alternative - but equivalent - description to the one offered by Chronos and Phil is: [hide]Think of the loops as being made from a "rope" of the material, with the ends of the rope looping back to attach to points in the middle. Because of the deformation properties, we can move these attachment points in a continuous fashion. So, move one of the attachments up until it moves past the other attachment, so that the one loop reattaches in the middle of the other loop. Now move the attachment in a 360o trip around the diameter of the rope there - this unlinks (if you go in the right direction) the two loops. Finally you migrate your attachment point back down to where it started from.[/hide]

This is a bit hard to explain in words, but it is actually fairly simple. If this is too confusing a description, I'll try to exercise my poor artistic talents to produce recognizable pictures.
Title: Re: Topological Rings
Post by rmsgrey on Nov 22nd, 2003, 5:24pm
My visualisation is:
::[hide]
Imagine inflating the center section like a balloon to get a sphere with a couple of linked "handles" - the attachment points of which can float around on the surface of the sphere, but can't touch each other. It's then much easier (IMO) to visualise moving one end of one handle round an end of the other handle to unwind them. If you then let the sphere deflate back down to the original center section you get the second object
[/hide]::

And a quick quibble: making the object of perfectly elastic material would be counterproductive since, as soon as you let go of it, it would return to its original form. Making it of perfectly plastic material would allow you to exhibit the object in the terminal states as well as in any intermediate state you wish.
Title: Re: Topological Rings
Post by Icarus on Nov 22nd, 2003, 5:57pm
Yes - that is a much more easily described solution than mine. (It is not a better solution than mine or Chronos or Phil's since all the solutions are equivalent, but it is certainly easier to see from your description than mine is.)

And "plastic" is indeed the correct word. The only problem with it is that people are not nearly as aquainted with the meaning of it as used here, as they are with using it to refer to a family of materials (which are given the name because most of them exhibit a high plasticity).
Title: Re: Topological Rings
Post by Icarus on Nov 22nd, 2003, 9:28pm
They say a picture is worth a thousand words. Everyone knows that I am a fan of the thousand words, but here is a picture of what I was trying to describe anyway. As you can see, I am also a great artistic talent.
Title: Re: Topological Rings
Post by Guest on Nov 23rd, 2003, 2:54pm
   No no, that is okay. I do understand the two solutions. I wanted to know if it is possible to do it without going into the stretching and compressing stuff. I think there are some configurations which seem equally impossible but can be solved ( not using the ideal compression and stretching used in this problem). I was wondering if such a solution exists for this problem.
Title: Re: Topological Rings
Post by Icarus on Nov 23rd, 2003, 6:17pm
I'm not even sure what sort of a puzzle you have without the "stretching and compressing stuff". This is a topology problem from the study of knots and links, which is itself a part of homotopy theory. The stretching and compressing are just a "layman" description of homotopy.

Without stretching or compressing or cutting or joining, then no - you cannot unlink them in 3D space. (You can in 4D.)
Title: Re: Topological Rings
Post by SWF on Nov 23rd, 2003, 10:30pm
Guest, you could try and reproduce these manipulations with an equivalent shape made of flexible material like a pair of shorts, but you are not going to get the leg holes to link together.

The shape in the puzzle is a single shape with two holes in it, so I would not say there are two rings that become linked and unlinked.  If you paint two unlinked circles anywhere on the surface of the object, those circles will remain unlinked.

The figure below shows what I previously described. On the left is an 8 shape. It is all one object, but is colored for descriptive purposes. The green and blue remain rigid and are almost complete rings.  They join at a blob of stretchy material which I made the same color as Silly Putty. If you pull apart as the picture on the lower right it looks like two unlinked rings joined by a bridge of Silly Putty. If you twist the rings relative to one another and push together as shown on the top right, they look like they are linked.  

Also shown is what happens if you paint a red circle around the inside of the blue ring.  When deformed to form the "linked" configuration the new shape of the red paint is shown (with the dotted lines being behind the putty). From that it is seen that the red line never links with the green circle, but goes around the outside of it. It is more natural to assume the red line goes straight across the putty to close the blue ring, which causes one to think rings have become linked.

If you tried this with a non-stretchy but flexible object, the deformation required of the red line (and the comparable line on the green ring) would become apparent. For example, with something made of fabric, the perimeters of the holes would never link, but you could fold them in the path of the red line.
Title: Re: Topological Rings
Post by Icarus on Nov 24th, 2003, 3:23pm
Okay, I see what you meant now. I didn't get the right picture in my mind from reading your original post. I've removed the offending sentence! :-X
Title: Re: Topological Rings
Post by Guest on Nov 25th, 2003, 12:56pm
    Thanks a lot. Further, does this topological equivalence of the two shapes have any significance somewhere in physics or other sciences? Or is it just a beautiful mathematical result?
Title: Re: Topological Rings
Post by Icarus on Nov 27th, 2003, 9:53pm
This particular equivalence is rather doubtful. But more generally, homotopy has application in particle theory, and more specifically, the theory of links has applications in string theory (don't ask what they are - I never got that far into it).

I suspect that other applications exist as well.
Title: Re: Topological Rings
Post by LZJ on Nov 27th, 2003, 11:07pm
For applications in string theory, I seem to recall the theory of links coming in when considering the possible ways in which superstrings join together...can't be sure though.
Title: Re: Topological Rings
Post by pjay on Dec 4th, 2003, 9:05am
The normal rule in knot theory is that these things have to be
ambient isotopies which i'm pretty sure they are not (i'll spare the definition since it requires a few page of explanation, but you can look it up in any knot theory or graduate topology book).
the basic idea though is that at some point you have to pass through a shape which has a point connected to 4 "line segments".  At this point, there is no bi-continuous map between this shape and either of the shapes drawn.  Maybe the wording of the problem should be changed o specify exactly what kind of moves can be made...
Title: Re: Topological Rings
Post by Icarus on Dec 4th, 2003, 4:44pm
I admit that I never studied links other than Martin Gardner - level recreational mathematics, but I was under the impression that the sort of maneuvers described here was exactly what the theory allowed. I've seen a large number of these recreational level problems, and the solutions always worked like this.
Title: Re: Topological Rings
Post by rmsgrey on Dec 5th, 2003, 5:14am
I believe (with no research into the matter) that the knot theory assumes line-segments with branching at points. The problem we're dealing with works with a 2-dimensional surface embedded in 3-dimensional space. Among other things, there are infinitely many lines meeting at every point on the surface, and picking any representative line segments or line loops, the transformations we've described transform them continuously without introducing or removing any intersections. If you draw a set of lines "0---0" on the shape in the obvious way, then the two circles are linked both before and after the transformation, and at no point does the connecting line disappear. After the transformation, depending on the details, one of the circles goes round most of one loop, along the connecting rope and then crosses the other loop somewhere before going back along the connecting rope again.
Title: Re: Topological Rings
Post by pjay on Dec 25th, 2003, 11:59am
I was under the impression that we were to think of these objects as one dimensional embedded in 3-space (by one-dimensional, I mean a one dimensional manifold for example, any curve).  In normal knot theory, knots are always homeomophic to a circle (meaning the circle can be wiggled around and looped through itself to form "knots"). when we are dealing with more than one of these, they are calle braids.  But there is no general study of one dimensional objects that branch out- in other words a point where more than 2 segments meet up.  But if you are trying to extend the allowable moves from knot theory, a good requirement would be the following:

at any given fixed time t, there is a homeomorphism from the shape at time t to the shape at time 0.  This cannot be done in this puzzle since inevitably we must pass through a point at which time the shape will include one point which has 4 line segments coming out of it...

Anyways, I still think this is a good exercise in visualization, i just think we should be more precise about what types of moves are allowed.  I for one cannot think of a good definition of allowable moves, so maybe the puzzle as is, is the best it can be...
Title: Re: Topological Rings
Post by Icarus on Dec 26th, 2003, 6:52am

on 12/25/03 at 11:59:23, pjay wrote:
I was under the impression that we were to think of these objects as one dimensional embedded in 3-space (by one-dimensional, I mean a one dimensional manifold for example, any curve).


No - it is common in these problems to consider the object as a two-dimensional surface, just as the picture indicates. (One could also envision it as a solid in this case, but some other problems explicitly require a surface.)


Quote:
In normal knot theory, knots are always homeomophic to a circle (meaning the circle can be wiggled around and looped through itself to form "knots").


Knot theory starts there, but is more broadly applicable. Perhaps you should think of this problem as an application of knot theory (or link theory, more appropriately) rather than as being in that theory.


Quote:
But there is no general study of one dimensional objects that branch out- in other words a point where more than 2 segments meet up.


Algebraic geometry includes such objects. But in this case, the objects in question are two-dimensional and a well-known part of surface theory.
Title: Re: Topological Rings
Post by tangent on Aug 1st, 2006, 10:06pm
a slightly nicer visual representation, just remember this is really in 3D.  I threw this together in 15 minutes.

-mikehttp://personalwebs.oakland.edu/~mjpalmer/twoloops.gif


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