The Inner Tube Problem

• But this is too simple.....

1: The answer to the first part of your question is yes....

2. Now coming to the second part of the question::

It is not possible to turn the tube inside-out...

Just visualise... you have two concentric circles (of different radii)...

Now when we warp the space in-between, what we form is the tube....

When we turn the tube inside out, as you say... or try to ...

What we are actually doing is trying to make the two circles interchange their places.... That is not practically posible.....

In the process if one tries too hard... he/she can end up tearing the whole tube......

But to turn the tube inside out is not possible... by just passing the tube through the hole......

When we wish to turn the tube inside-out... We are in fact trying to resize the dimensions... not possible... by passing it through the hole...

It is possible if you cut the seam... open it uo and turn it inside out... as out forth by Electroman... but that would not be correct.... But why is something, so trivial... being discussed.... Does it have a more profound impact on our thinking... something beyond the obviously visible.. Some thing that Bob would like to share with us... just musing....

1. YES.

2, YES. It can be done. It isn't easy. It will not look the way you might think. This was discussed at length in "Scientific American" magazine IIRC years ago.

Hi Stan...

with regards to the answer to the second part of the question...

I am really puzzled coz I thought it was not possible unless you tear the tube along its seam or any one side... As you say... I may be wrrong..

But can you please elaborate on your answer or try and explain...or give the link to the document that you are mentioning...

Hi Vish, I did indeed read the question very carefully and the first part of the question was can I do it in my head...

The answer is YES!

The second part was can a tube with a cut in it be pulled through to turn it inside out?

As I said the shape and size of the hole isn't defined so a cut along the circumference is classified as a hole... Think of a Mobius loop?

Pulling the inner tube through the 'hole' is now a simple case of forming the now flat strip of thin rubber into a loop in either direction...

So YES! it can be done! ....... John.

I still dispute that this is cutting a hole, but I suppose that depends on the definition of a hole. I would define it as the result of a closed cut removing a portion of the material thus leaving a hole. Also, intuitively, a hole is surely something one could put something into. Your cut doesn't conform to that one.

Also, if you make such a cut as you describe in a Mobius loop you end up with a Mobius loop (with an extra twist) - not a loop with a hole in it.

So what's the 'definative' definition of a hole?

Well that's why the question isn't precise enough to disallow a slit to be a hole...

As one poster said if you slit a sphere and pull the inside out you would still have a sphere...

I'm just taking the slit to be a larger version... A hole doesn't have to remove material... In fact I'd say that a complete slit in a sphere would result in two halves and that wouldn't be a hole... But in this case the torus would remain as a single piece so it would still just be a slit / hole... Pulling the inside out is simplified by the large slit!!

John.

Hello electroman... but if the cut/hole/inscision/slit changes the property of the toroid (tube.. precisely refering to a car or cycle tube representing a hollow toroid)... makes it into a flap of rubber or a hollow cylinder then... then the question and the answer consecuently become irrelevent...

Or would the question still hold true....

Why do you need a slit all round? I think that a cut right through the tube is suitable, and creates what is apparently a simple hole (a tube that is open at both ends).

But no longer a torus. I think that the idea is to maintain the torus as a 3D figure, rather than a curved 2d surface. Do what you want. Apparently there are no rules to be concerned with.

Rich

Sorry Rich - I was being a little facetious, because the two cuts are topologically equivalent - Electroman's cut opens the interior hole to leave a simple tube, and the proposed 'alternative' is a simpler cut that opens the exterior hole to give the same result.

BTW, I went away before reading the proper solutions and worked my way up so I could visualise the process. Your and Stirling Stan's pictures confirmed to what I had visualised – but I suspect you may already need to be able to visualise the situation before the (inevitably 2D) pictures made sense.

I think most people will be able to visualise the solution if they follow something similar to the route that I took - so here is my description of how I worked my way to a solution (it may take a bit of working at, and it's by no means the only way). My explanation may not be too good, so suggested improvements would be welcome:

First, consider a bicycle tyre (because the shape contrasts are easier to describe)

Visualisation 1 (the crucial first stage)
Using a specially-shaped hole:
Stretch the basic tube with long edges vertical, so it consists of four (long-and thin) nearly flat rectangles (connected at the edges),
Bisect the closest rectangle with a vertical cut, and cut horizontals at top and bottom to create a capital letter I.
The bend back the sides of the rectangle so they come around the back, rejoin the cuts. A closed continuous line can be drawn around the new front and (rejoined) back in a horizontal plane (originally, such a line would have have been in a vertical plane).
You now have a torus with a very long hole vertical hole external to the tyre, and a very short fat horizontal hole inside the tyre.

Visualisation 2
Forget the horizontals, and use the slot to distort the front face around the rest of the tyre.

Visualisation 3
Imagine that a small slit or hole could be stretched around the whole structure to invert the front surface. Unlike the other visualisations, this cannot be done with a physical tyre - but it's preparation for:

Visualisation 4
Replace stretching the hole by distorting the tyre through the hole.

Regards

Fyz

Visually/spatially, tried to push it through the hole, and the center wall would not let the donut hole pass. Vish_al's radius problem. Looking for the inner and outer walls maintaining their size znd shape, and adding that what's inside should become outside ( The inner tube) and what's outside should become inside (within the tube) The hole is outside of the torus, and to move inside, it would no longer be the donut hole, but a hole of a different description. The tube. The resulting tube therefor would have the normal interior becoming the donut hole and the normal interior of the tube becoming the rest ('the universe not previously inside) including the donut hole. This was the only solution that did not deform nor resize the final surface, maintained the same properties as the original torus, but certainly looked different. The C shape maintained the interior radius sections with their radius, and exterior also. You start with a tube around a hole, and you end with a tube around a hole.

Then from theory to lab, and the added discovery that talcom powder is a very critical part of the eversion.

After the reversion for the pictures, I gave it to my son-in-law, about as wall trained in math or science as my mother-in-law, and in 10 minutes he had re-everted it without a clue.

My first image was the mirror eversion, but I could not convert to that in my mind, so watching the C form in my mind, it did appear to be an ugly half-way point. But continueing was reversion, and then the new torus became obvious, even if the original axis of revolution became a semicircular centerline of revolution.

RichH

For what it is worth, I see the end point as being with one half of the tube left in its original position, and the other half of the tube (the new outer) wrapped inside-out around it to make a new doughnut. As the new outer tube had the same original dimensions as the inner, it is hardly surprising that you need good lubrication to manipulate it into the new shape.
If the wall of the tube is very thin, the new toroid should look like half the original. Any wall thickness will tend to cause the section to straighten out somewhat.
Of course, this comment applies only to a uniform tube that is unstressed in its original shape.

The square handkerchief returns to the same shape as it was originally - but in both cases the 'natural' shape has a zero-thickness interior.

I think there may also be a family of tubes with non-zero interior that transform into the original shape - these would be a lot easier to manipulate. If I'm right, both versions would have non-uniform cross section and the central hole of the doughnut would be somewhat curved.

This was literally a mind-bending topic - thanks to all.

Fyz

Physicist:

Check out the photos in Posts $41 & 42.

It is most instructive to actually make the handkerchief sized piece of cloth into a topological toroid. When you turn it inside out it looks as it did originally.

Then make one similarly from a length of thin cloth twice as long as it is wide. Once sewn together, turn it inside out and the difference is immediately obvious.

CAUTION: Do not sew too close to the corners, leave about 1" in order to pull the far corner thru. Foceps do a great job. If turning a auto inner tube use plenty of lubricant
before starting the pull. Here pliers are a big help with this one.

I referred to ##41,42 in post #60, which is in the direct line to this posting - this last comment was intended to address the basis for the observed shapes of transformed inner tubes. As I said, the square handkerchief is the only version so far presented that transforms into itself, and it has a zero-volume interior in both inversions. What I was trying to say was that there should be some shapes with non-zero internal volume that transform into themselves; the description above suggests how those I have been able to identify would look. (N.B. that I was quite specific about the handkerchief for the "inversion-invariant" toroid being square).

Fyz

Reply to post #60

Thanks Fyz, Brilliant! I used (about 30 years ago) to be able to nearly turn a bulbous, stretchy torus inside out in my head; I can't get anywhere near it now. Your flattening, cutting and folding approach is much better.

Thanks. I'm flattered: something I wrote actually turned out to be comprehensible.

As is usual, a concept is far harder to communicate than are any factual information. The rules are defined by facts, and sometimes the why of the rule is forsaken for the simpler task of following the rules.

RichH

Maintain the torus? I've seen this repeated in this thread. What do you think happens during the process of pulling it through the hole? The torus is destroyed in this step and recreated once the step is completed. What is the difference if we destroy the torus one step sooner, and recreate it later. You are right that apparently there are no rules. I created a unique solution that took advantage of that fact. I'm a little disappointed that it drew no comments.

I wonder why people apply artificial constraints. This is an example of refusing to think 'out of the box'. I hear that phrase often. Everyone thinks they know what it means. But when given an assignment that specifically asks you to consider strange possibilities, some people not only constrain themselves (unconsciously), but they try to impose those constraints on others. While the 'out of the box' solution may not necessarily achieve desired results, it is still important to consider it. Why? Because it shows up holes in the specifications. Asking a better question gets better answers. Consider the guy that asks "what is a good pump?" (Don't be that guy) Suggest to him that a Breast Pump is a good type of pump. This forces him to reconsider his requirements and craft a better specification. Now you both understand the problem better and he is more likely to get what he really needs as opposed to what the engineer thinks he might need.

That may be the root of the problem. We know that people don't always say what they mean. We fill in the blanks with what we know. The person asking the question is building a small cabin in the woods. As yet he has not drilled a well but is planning to soon. Based on your experience and education you suggest a Jet pump. You send him to a supplier you know. He stands there looking at the jet pump you recommended with his wife standing next to him. They both wonder how it gets connected to her boob.

PS. Referring back to the first paragraph, I suppose it could be argued that the torus still exists while it is being turned inside out as in the photos. It is just severely distorted. But it is hardly functionally equivalent.

Call it what you will, TORUS or TOROID. Donut shape of some distortion thereof it MUST conform to being a topological torus. It is and remains a topological torus throughout any and all manipulations as long as you don't destroy it by cutting it around any circumference and making a tubular cylinder out of it.

Why hide your identity. Stand up and be counted.

s/SS

In reply to Stirling Stan, my name is Dustin Maki registered here as sloco. I wrote posts 65, 28, and 31 in this thread. Sorry if I caused any confusion. I often forget to log in, I apologize. I am nobody. 34, unemployed, barely a high school graduate so I never studied topology. I must confess I still do not fully understand what topology is but I found this on Wikipedia "Intuitively, two spaces are topologically equivalent if one can be deformed into the other without cutting or gluing." In order to pull the torus through a hole in itself, you NEED to cut a hole. Doesn't that change it?

On a personal note, I have a disability that keeps me from getting out to see other people much. It also affects my mind, so on occasion I may ramble on making little or no sense. For that I apologize. I read these forums to keep the wheels turning and reply to help maintain some type of social skills. For that opportunity I thank you all very much. The last few months have been great.

sloco aka Dustin Maki;

My apologies, there are all too many posts that I suspect are traps or test or something of the sort by the anonymus "Guest's."

Exercising the mind is great exercise when there are limitations on other activities. Let me suggest that you Google for "Kleins Bottle." I think it was mentioned in another post.

In general you are allowed by rules of topology to cut holes but not any cuts that would otherwise destroy the integrity of the original surface (s).

Cutting a multitude of holes in a topological entity would make for some very interesting possibilities. Like a tangled fish net in appearance.

s/SS

Looks to me as if Dustin's idea at post #31 and Koshy's at post #45 are the same:-

The thick line is the hole in the tube; the hole through the doughnut is horizontal behind the "wall/frame" in the first picture, and, vertical in front of the "wall/frame" in the second. The inside of the tube is fairly obvious in the first picture: in the second, imagine that the picture shows a recessed handle: someone grabbing the handle from the "other side" of the wall has their hand around the inside of the tube.

Hi sloco

If it's any reassurance, once on this site, most of us are 'nobody'.

Looking at post 31:

It's an excellent summary of the possibilities.

Version 3 represents the inversions using the "classical" hole, and I think is fairly well covered by now.

Versions 1 and 2 are in a sense equivalent - in that each of them cuts one of the two circumferential connections. They are actually slightly more interesting than has been recognised, because each of them can be reformed into four different simple toroidal inversions - compared to just two for the classical version. The downside is that the toroidal surfaces are no longer self-connected. We could also create knotted tori using these inversions (though it would be physically quite difficult to knot an inner tube with a circumferential cut)

Version 4 was mildly amusing, but didn't truly turn the tube* inside-out, and the hole in the wall was (strictly speaking) irrelevant to the process.

*The problem here is that you need a boundary to define the difference between inside and outside. With an open tube, the linear edges form this boundary, which is between surfaces. The torus doesn't intrinsically have edges, so this solution is not properly defined. In any event, at least in 3-dimensions, separation by a surface is more definite than separation by an edge.

A final note - one may question whether thinking outside the box has intrinsic merit. What matters is the quality of the solution (whoops - I see value judgements coming on). Sometimes, original and unconstrained thought results in a solution that is clearly superior. Other-times, as I believe in this case, linear development has the edge.

Regards, and good luck

Fyz

Re: The inner tube problem

Thank you Fyz. You've made my point for me perfectly. I am referring specifically to post 72 the text following "version 4". Contrary to what you claim, version 4 does "truly turn the tube* inside-out" and "the hole in the wall was (strictly speaking)" highly relevant "to the process".

The tube* refers to the physical object known as "inner tube" not any geometrical form. I submit that absent any clarification by the author of the original question, my definition of "inside out" is as valid as yours. You say "*The problem here is that you need a boundary to define the difference between inside and outside." I submit that If you hand the tube to any 5 year old kid and tell them to put their hand "inside" the tube, they place their hand through the hole in the center. Ask them to place their hand "outside" and they place it near the outermost surface. I refer to the illustration in post 7, not the text. Anything inside circle B is inside, anything outside circle A is outside. The space between IS the boundary. Its fairly clear. If I can make the points along circles A and B swap places I have turned the tube inside out.

I hold in my hand right now a tube sock with a hole in the toe. This particular sock is constructed of a double layer of material. A soft fabric facing the foot and more durable patterned fabric facing the world. They are stitched together at various points, but topologically it is a torus. Most similar to the "classically" everted inner tube. Now turning socks inside out is a pretty universally understood process. Do you see where I'm going here?

Anyway, I'm not saying that any solution is wrong, just that all 4 versions I described(including yours) are equally valid given the requirements of the question as stated. And choosing 1 version over the other is irrelavent. Can we agree on that?

Regarding the hole in the wall. While it is quite possible to turn the tube inside out by the version 4 method without using a hole in the wall, doing so would not conform to the requirements of the question. The hole not only needs to be present, it needs to be an integral useful part of the process. In version 4, it is.

The artificial constraints of maintaining a topological toroid, limiting the hole configurations, and even the form of the end result ARE IRRELAVANT to the question at hand.

There is another aspect which I believe is of PRIMARY relavance to the original question. It has been at the heart of all my posts in this thread. That is the cognitive process by which we craft our solutions to problems. I find it interesting given the obvious ambiguity in the original question that few asked for clarification of details, none asked for the purpose or end use of the procedure, and the original author provided neither. Had this been a real, non hypothetical question, all those things would have been quickly asked and answered resulting in a much shorter discussion.

The intrinsic merit of 'thinking outside the box' is as a foil for linear development. A challenger to the status quo. The Rebels to the Imperial Empire. Snub fighters to the Death Star. End result? New improved Death Star. That metaphor didn't end where I thought it would when I started but you get the point. The word rut comes to mind. It is the music and open window that keeps you awake and out of the ditches. The fact that occasionally it actually produces a workable, even superior solution is an unexpected side benefit.

Ok, I've definitely begun to ramble and this thread is quite long enough. I thank you and leave with just 2 questions. What do Klein bottles have to do with anything. I've looked them up and fail to see the connection. Finally, how do you spell relavant. relevant. relavent. relevent. I've found each of these spellings in various online dictionaries including 2 different spellings within the same entry.

Hi Sloco

Most of the following is pure pedantry, so not to be taken too seriously:

We can agree that the first three meet the question as stated, but not the fourth. The failing of the fourth is that it turns the ring inside out, but not the tube. The five-year-old would indeed do as you say, but only (I think) because of social habit - because he/she assumes you would not ask the impossible. If you asked the question "can you place your hand inside the tube without cutting it", most kids would either answer "no" or that it depended what you mean by the question.

I would add that the intended question is clear, even though the literal interpretation would have it ambiguous. This makes answer three the most appropriate answer, but answers 1 and 2 are acceptable as supplements on the basis that they allow a solution whose shape might be preferred.

Now to the Klein bottle. No relevance at all, so far as I can see. Which is why I didn't bother mentioning the following before:
Contrary to some statements in this thread, a Klein bottle can no more exist in three dimensions than can a Moebius strip exist on the surface of a flat sheet of paper. In both cases the best that you can do is produce a projected representation of the shape.

Penultimately: the standard (and semantically correct) spelling is "relevant". It is the present participle of the Latin verb "relevare" = to raise up. I have failed to find a definition on-line for "relavant", but I believe it to be an obsolete word with two irrelevant meanings - one relating to volcanic activity, the other to cleaning.

Regarding on-line dictionaries: depending national on preferences, Webster, Oxford and Chambers are reliable sources.

Regards

Fyz

The hole, being not descirbed, should mean that it can be reduced in size to miniscule to a pure point. At least for the inversion in your head. The hole as cut represents the point as is a commonly used concept in topography/topology. This is very similar in that respect to the representation of the klien bottle. The klien bottle is missing one dimension, as much as a drawn mobius figure, though a 2 dimensional figure, requires a third dimension to make it's properties work. The same goes for the klien bottle.

The relationship of the klien bottle to the mobius strip is demonstrated by the reduction of each to a 2D figure in any written discussion of them. The property of the mobius strip as a 2d figure in a 3d universe is rather similar to a 3d figure residing in a 4D universe. Anything about the concept of a klien bottle is invisible from only three.

Everting the torus is a similar exercise in working spaces and surfaces without distortion.

It appears to be more of an exercise of visually working in added dimensions. The classical "hole" would work equally well as a a minimal size slit, with the exception of the valve stem, the removal of which leaves a hole. An inflatable pool tube float has a small enough valve to pass through a very small slit without removing material. from there, the concept of a zero thickness wall passing through a slit of zero size. The solution to the problem leads to further conceptual exercise and possibilities. The cross cut through or circumfrential cut and restoration are dead ends, not able to vary in size, and producing the easy to visualise result. You are welcome to your solutions, some folks prefer mine. Happy New Year.

Please identify this unique contribution by number. It could be that we didn't recognise your meaning.

Fyz

vish_al210: If I had had the article(s) mentioned I would have included that in the post.

Unfortunately I disposed of a large accumulation of S.A. some years ago to help the library fill in gaps in their back issues collection.

It is a topological problem which generates a lot of discussion by those not having encountered it previously. I remember the problem because it cautioned of the distinct possibility of losing or damaging some finger nails if you attempted to turn a normal auto inner tube inside out. This was in the days before tubeless tires. It would be properly described as "How or Can you turn a torus inside out?"

Post 17 by Randall, re: Handkerchief

"Now take a square handkerchief, or, other square of material: sew it along two opposite sides; then the other two so that you end up with a torus with a hole where the four corners come together. Clearly mark the inside and outside; and pull the material through the hole. You've turned it inside out!" (I'll have to try that demo!)

Google for: "turn torus inside out" and get these and more links.

Article: <http://www.sciencenews.org/pages/sn_arc98/10_10_98/bob1.htm>

Images: <http://new.math.uiuc.edu/optiverse/images.html>

Google for: "turn innertube inside out" and get these and more links.

Such as: <http://www.fortunecity.com/emachines/e11/86/klein2.html> which takes another tack on the visualizations, etc.

Now everyone should have a better understanding of what the problem is and the solution!

s/SS

PS Comment: Using search engines is more of an art than a science.

Another way of thinking about this:

Take a rather deflated beach-ball. Cut a hole in it, big enough to push the rest of the ball through - this works! You can turn something connected like a sphere inside out.

Turn it back right-side out. Mark two diametrically-opposite points, make small holes, flatten the ball so that the holes line up, and fit a pop rivet. Ther resulting shape is connected like an inner tube.

Now try turning it inside-out through the hole - you can't! Even if you imagine a great big flexible 'pop rivet' - it still can't be done.

Errrrmmmmm the original question didn't state the size or shape of the hole that is cut did it?

If a cut is made along the outer edge of the tube all the way round the circumference...

Then the tube can be folded inside out with the cut edge facing the inner diameter...

Wouldn't it??? - John?

Topological skulduggery! I'm not so sure that such a cut constitutes a 'hole', either!

Happy Christmas !

Hi Electroman...You need to read the problem properly....

There are two questions asked...

...and as John says... It is a hole we are talking about and not a full fledged cut along the circumfrence of the tube....that would not be a hole now.. would it??

And by "tube" specificly meaning.. a Doughnut shaped tube.... (as mentioned car or bi-cycle tube)....

And a merry and joyous X-mass to you all....

Vish see my answer #18

I think reading the question shows up the 'holes' in the question...

Instead of thinking about the question I suggest you think about an answer before stating its not possible...

Your comment about re-sizing the two inner and outer diameters is nonsense, if you read in the question that we are dealing with a thin elastic skin and not a rigid thick tube. There is no mention that the reformed tube should be able to hold its new shape for any length of time either...

So use a bit of thought..... John.

Hi Bob..

I guess you are waiting for more responses....

So what was it that yu wanted to surprise us with??? I am sorry, the startling fact.. that waits to be exposed....

OK.. ok... I aint pushing you for answers... I can wait my turn as John Milton did.....

Hello Vish,

Actually the result might not surprise any one but it surprised me. I have to own up. As many do, I thought that I could visualise this problem in my head but actually I had to go to the local tip, get an inner-tube and perform the experiment to find the answer. Why I wanted to know? - just curiosity.

Why would I pose this question, you might ask. I want to know if the sort of folk that contribute to this site (whom I guess fall into a particular category in which I include myself) can solve this sort of problem in their heads; is the human brain that capable? As someone said, it is a simple problem (I don't agree) but if we cannot solve even this one mentally then how do we suppose that we can solve more difficult problems.

I have learned a lot from the answers given and the way they have been expressed. This interests me. The answers are good but not correct so far.

The Engineer's Engineer

With regards to comment 7 posted by me.. I may probably be wrong with the concentric circle theory....

No. It would only be possible to "exhange" the inner and outer surfaces of the tube provided that you pulled only the inner surface through the opening. But, in order to pull the inner, inner tube through the hole (in the outer, outer tube) you must also pull the outer, outer tube (surface), along with the inner surface, through the opening.

...even if the hole was cut around the valve stem.

Can you clarify further... You also are saying that it is not possible to turn the tube inside out... through the hole.. If I have understood you right??

a hole is a hole and always will stay a hole

but what if you cut the hole in half?

The concept of "CAN'T" doesn't fit. Inverting the hollow torus of the inner tube would result in a complementary torus, with the hole running in another dimension (lengthwise through it). You would have a semicircular double walled "half donut" as the only way to keep both surfaces continuous. Then the continuous "outer" surface would become a continuous "inner" surface. They should have had the concept of manifolds when I was in school.

I love this problem. First you prove conclusively that it cannot be done: poke a blue pen through the hole and draw a circle inside, all the way round the "large circumference"; now draw a red circle round the outside around the "small circumference". If you turn the torus inside out it is clear that the circles which started off linked become unlinked; therefore it cannot be done.

Now take a square handkerchief, or, other square of material: sew it along two opposite sides; then the other two so that you end up with a torus with a hole where the four corners come together. Clearly mark the inside and outside; and pull the material through the hole. You've turned it inside out!

You must use a square (not a real inner tube) or it is obvious that the axes of the torus have swapped; as described by NoSciFi at post #16 above.

I cannot contain myself any longer. The answers are great which just goes to show that there is usually more than one correct answer to a problem

Here is one of mine; the tube can be pulled through the hole entirely (over look the pun) but it is not inside out. It is exactly as it was in the first place. I found this surprising.

I wonder if perhaps in your attempt to get the "logical" solution of the same torus, you may have actually inadvertantly 'everted" it twice, because the result of the dual eversion would be normal. Paint the entire outside of the tube white, and if you reach a point where there is no white showing, you have a single inversion. The double walled half circular construct of a single eversion would not have the completed appearance one would expect, and continueing would be a normal response to the strange result of a true singular eversion. With a painted surface, it would become apparent when you started re-everting it.

I always did like to imagine the klien bottle, an imaginary 3d mobius strip with a single surface, which could never be inverted.

The double wall half circle result of maintaining the smaller radius inside the curve and the outer radius outside of the curve, without breaking contiuity, is actaully a torus, with the curved donut hole through the "circumferential" direction, actually outside of the tube even through the new donut hole, while the normally exterior surface is inside of the resulting half-circle. The effect of a single eversion would appear to be an ugly half point of the process, but in reality would be a complete single. Once no previously exterior surface is still outside, the process is complete. The result one would expect could only be obtained by cutting and reattaching, either radially or circumfrentially, to get the same shape torus with the inside out.

Damn, now I'll have to go buy a tube. And there cannot be more than one correct answer for either question. But we all know what opinions are like. Mine included.

Rich

Rich:

Klien bottles are available for purchase on the web. I have one in front of me at the moment with a pair of the tiny magnets from Radio Shack to demonstrate that the inside is outside and the outside is inside.

There is a LARGE Klien Bottle on the grounds of a management training institute in eastern Canada. It is shown in several stages of construction on the same web site as above.

s/SS

Then the answer to your question is, No. You said it.

Correct. You cannot turn it inside out. So is there a rule? can you think of any other shape that cannot be turned inside out?

Bob, send me a PO box number and I will send you the everted innertube, or I will email a pic . It not only can be done, the video on the one site showing eversion resulting in the same torus does not demonstrate the change in the axis that results. What it appears to show is the inner hole passing through the wall as it passes through the hole. What actually happens is the axis of revolution (of the torus donut) changes from the perpendicular to the one half of circular arc of the center of the torus body, and the resulting torus bears no resemblence to the original. It's not a mirror image. It is still something of a torus based figure, but far different from the original.

Rich H

Actually, I can turn the tube inside out using a simple hole in the wall of the tube, as can Stirling Stan, NoSciFi, sloco, and several other contributors. Hopefully, you too can do this by now. Maybe you can even visualise it by adapting the explanation in post 60. As it was your problem initially (which I believe to be the reason the answer given to both questions in post 36 was "no"), please tell us where you stand now.

because you are pulling it out of the hole as you are pushing it into the hole?

Rich,

Your answer and opinions are spot on, but, don't go buy a tube: use a square of material with a printed and plain side (see my post #17), it's much more baffleing for the people who don't realise that the "long" and "short" (= for a square) axes have swapped.

I did stop and buy a tube this morning. For simplicity, my hole was where the valve stem was. I strongly recommend some talc for the process. Everting the tube means that after you push the first piece of inner tube (not innertube) through, the rest of the inside can only be extracted by pulling the inside of the innertube out through the tube forming with the inside out. Once you reach the end of material to feed through the hole, and it everts itself, you do end up with a semi circular double walled tube. The interesting thought is that, even though you have pulled all of the tube through the hole, the inner tube of the resultant double wall tube has somehow maintained the outside(large diameter) toward the outside, and the inner (small diameter) toward the would be center. It sounds like it's only half way everted, but you have to realize that the axis of the resulting torus has changed, from a perpendiclar to a circumfrential. The new torus is the cavity between the two walls, a very long oblong rotated about an arc. The new axis is the middle of the semicircular result. Thus, the inner wall of the semicircular tube is now the ouside face of the torus facing the torus axis.

The result is a semicircular double tube, with the inside tube of the construct appearing as originated, but the new axis being lengthwise through the tube means that it really has everted, and lies outside of the new configuration, just as food you eat is still outside of your body, residing in a tube passing totally through your body.

Remember the talc.

$2.69 and 10 minutes, for a fat bicycle tube. If you can find a larger onelike a wheelbarrow or lawn tractor, or perhaps a vinyl pool tube, it might be easier.

Happy Holidays

Rich

NoSciFi:

CONGRATULATIONS !

THEY SAID IT COULDN'T BE DONE.

BUT YOU DID IT !

There is nothing like someone who will take the bull by the horns and give it the old college try!

It took me a bit of work to be able to visualize it, but I believe I have it down correctly. But many of the other posts don't seem to agree. So I've re-thought the puzzle and still come to the same conclusion.

The first step -- the mental warm-up -- is to visualize the tube with one part of the circle severed (not just a hole cut into it). Now imagine reaching into the hole on one side of where it is severed and pull the rest of the tube through the hole until the tube is completely inside out. Now you have the inner-tube inside out, with the one part severed.

The second step is more difficult. Now imagine repeating the first step, but without completely cutting through -- leave a small strip joining the inside circumference of the donut. If you can visualize this, I believe you still come up with the inner-tube turned inside out.

If you follow the second step as it seems to me, the third step is to apply this principle and conclusion to any size of hole in any size of torus. This then leads to a torus with the normally exterior surface hidden on the inside, and the normally unseen, interior surface being on the outside.

If I am wrong in this, somebody please let me know where.

OK. I finally see where I went wrong. Step two requires the "final end" of the cut torus to pass completely through the "front end," and all the way through the rest of the tube, before it inverts. But it can't do this while the "final end" is still attached to the "front end". So what is described in post #31, solution 3, is what I should have pictured at step 2.

I apologize for any confusion I may have added to the discussion.

Answer to first question. Yes I can work it out in my head. At least a couple of solutions. Possibly there are others which I have not worked out.

Solution 1: Cut the tube along the major circumference. All the way around, with your final incision ending where you started. Peel the 2 cut edges in opposite directions till they are reunited toward the center. Doing so fulfills the requirement "pulling the rest of the tube through the hole". Even though a better description would be pulling the hole over the rest of the tube, they are equivalent. At this point the strict requirements of the challenge are met. Interior surfaces are now on the exterior. But to make it a better demonstration would involve reconnecting the cut edges. This results in a distorted torus shape with the former minor circumference now being the major circumference, a larger average cross sectional area of the volume contained within the ring, and axial pleats all around the inner circumference. For those that don't believe this is really cutting a hole, imagine the steps in cutting a hole around the valvestem. Insert blade, create a circular incision. That is what we did. What if you just cut a 2" slit? That is a hole. I just extended the slit to make a larger hole.

Solution 2: Variations on the first. Make the cut at any section around the ring of the tube. Same result as taking a single good whack at it with a cleaver. You end up with a hole alright. The tube is now a true tube open at both ends, not a torus. Its now a fairly simple matter to turn it inside out. Get a long rod with a hook on the end. Stick it inside the tube till it comes out the other side. Hook the far end of the tube and pull it through itself. The former interior surface is now on the exterior and vice versa. The basic shape is the same as we started with.

Solution 3: The probable default assumptions. Cut a relatively small hole or slit at any point in the surface of the tube. Around the valvestem is as good a place as any. Push and pull the material through the hole. This results in a C shaped double walled tube. Essentially a very elongated torus. Half of the former interior surface is now the outer surface. The other half of the former interior surface is now the very innermost surface. That is, the inner lining of the tube. The former outer surface is now completely enclosed, forming the facing surfaces of the double walled tube. The remaining hole can be on either the inner or outer wall and can be moved from one to the other by sliding the inner and outer walls in relation to each other.

Solution 4: Takes great liberties but follows the letter of the challenge. Requires the material of the tube to be flexible enough to stretch the minor circumference to the size of the major circumference. Requires a liberal definition of "inside out". If you can live with those conditions, you may find my solution interesting.

Cut a hole in the wall. Not the wall of the tube, but an actual wall. About chest height, a little smaller in diameter than the largest diameter of the tube. With you on one side of the wall and the tube on the other put your arms through and hold the inner part of the tube. Center the tube on the hole and try to pull it through. Friction between wall and rubber will keep the contact surfaces in place. The rubber will stretch and it will 'roll' itself inside out. The part of the tube that originally faced inward toward the center now faces outward toward the wall and vice versa.

Dustin Maki

Doesn't this either give or make you wonder over test writers? Reminds me of the son of the engineer who wanted a swing from a tree. You've all seen it, I'm sure.

Hi...

just one question...

Is the description of a hole the same as making an incission throughout the circumfrence.....

My hole was a discrete opening, roughly circular, on the minor radius face of the tube, diameter roughly 1/2 the tube width. Total perimeter of the hole 2". A quarter sized hole in the wall, without modifying the integrity of the torus, rather like a slighly larger, though flat valve area. The "C" shape is a better description of the eversion product.

Rich H

Making a cut around the outer periphery of the torus would alter the topology.

But if the cut did not join itself then it would be a BIG hole in the outer surface or iow a strand left intact would still be topologically the same as a small hole. You are in effect just stretching the hole all the way around the outer circumference of the torus.

Cutting the remaining strand, in effect converts the torus to an open ended cylinder.

Please note the following pictures of the reversion of the everted torus, as described:

Everted torus, on pvc pipe, demonstrates the tall, narrow torus with it's circumfrential axis of rotation (the donut hole)

The natural "C" shape with the axis along the arc. Arrow showing hole.

Begining the reversion. Pulling the outside back out.

More outside out

Bunching result of drawing outside out through inside.

Starting to form the innertube from reversion

Nearly complete

For some reason, I'll have toput the final 2 pictures in another post.

Rich

Completed reversion to original torus shape, commonly refered to as an inner tube, or I believe Air Bladder, withaxis of revolution restored to perpendicular to the plane descirbed by the tube.

The hole for eversion.

Sorry, Bob. If you can find or demonstrate any other solution, or lack thereof, please let me know.

Rich H

NoSciFi:

Went by H.F. today to get diamond grinding wheels and picked up a 4" inner tube for a 12" truck/dolly tire.

It took a bit of effort but the final result is above.

Being totally impossible results in an early termination of oft fruitfull discussion. Being exremely difficult sometimes brings out the worst some have available.

And the answer to the first half of the question is, and was, yes

Nice torus around an arc. I think that result is due to the fact that the torus is radial rather than Cartesian figure. And your hole looks much nicer than mine.

Rich H

The easy way. Use some topology.

My logic is as follows.

I will treat all continuos sufaces can be shrunk/ or stretched without violating topology.

Now imagine the hole as constant and all the surface around is something which can be shrunk or stretched. Well now imaging you choose to shrink it such that the whole tube now becomes a diaphragm with the hole outer as the outer periphery of the diaphragm. Now you can push the diaphgram in out on either side. (Well when you invert the valve will be insde the tube).

By the way the four color theorm of colouring in maps also can be shown to true using the same method.

Your comments to:

koshy@tek1.com.au

Can you supply a sketch of the last stages and final appearance of this wonderful solution? It should be most enlightening?

Koshy:

It can NOT be done as you describe. No cuts other than to make a hole in the outer surface, then push, pull, distort, stretch, and shrink as much as you like.

Consider a 6" torus with a 1" hole it the periphery facing you. Stretch it till is is 12" in diameter and afix it to a piece of plywood 18" square. Now shrink the remains of the original torus to something the size of a "Froot Loop," "Cherrio," or something even smaller.

You now have a large diaphragm with two tiny holes and a tiny torus behind the diapragm. There is no escape. Your are stuck with a torus. No matter how much you stretch and shrink it still retains the torus configuation, albeit a strange one.

s/SS

Hi Bob... I guess that the problem is meaningful only if we keep the property of the tube intact.... i.e.,. a hole or a cut is ok.. as long as the tube (specifically car or cycle tube.. toroid) can still be called a tube... not a hollow cylinder or a circular flap of rubber.....

If the tube is distorted then the problem and the solution fail as it no longer a tube that we are talking about....

PLease clarify if i am wrong....

Hi Bob... I guess that the problem is meaningful only if we keep the property of the tube intact.... i.e.,. a hole or a cut is ok.. as long as the tube (specifically car or cycle tube.. toroid) can still be called a tube... not a hollow cylinder or a circular flap of rubber.....

If the tube is distorted then the problem and the solution fail as it no longer a tube that we are talking about....

PLease clarify if i am wrong....

Hi Bob... I guess that the problem is meaningful only if we keep the property of the tube intact.... i.e.,. a hole or a cut is ok.. as long as the tube (specifically car or cycle tube.. toroid) can still be called a tube... not a hollow cylinder or a circular flap of rubber.....

If the tube is distorted then the problem and the solution fail as it no longer a tube that we are talking about....

PLease clarify if i am wrong....

Hi Bob... I guess that the problem is meaningful only if we keep the property of the tube intact.... i.e.,. a hole or a cut is ok.. as long as the tube (specifically car or cycle tube.. toroid) can still be called a tube... not a hollow cylinder or a circular flap of rubber.....

If the tube is distorted then the problem and the solution fail as it no longer a tube that we are talking about....

PLease clarify if i am wrong....

Hi Bob... I guess that the problem is meaningful only if we keep the property of the tube intact.... i.e.,. a hole or a cut is ok.. as long as the tube (specifically car or cycle tube.. toroid) can still be called a tube... not a hollow cylinder or a circular flap of rubber.....

If the tube is distorted then the problem and the solution fail as it no longer a tube that we are talking about....

PLease clarify if i am wrong....

Hi Bob... I guess that the problem is meaningful only if we keep the property of the tube intact.... i.e.,. a hole or a cut is ok.. as long as the tube (specifically car or cycle tube.. toroid) can still be called a tube... not a hollow cylinder or a circular flap of rubber.....

If the tube is distorted then the problem and the solution fail as it no longer a tube that we are talking about....

PLease clarify if i am wrong....

Vish, a hole will always cause distortion of the original shape...

Consider a sphere, if a slit was made round the centre the sphere would become two separate halves... if the slit is moved over to one side it will become a hole in the sphere which now makes the sphere non-spherical, it will be truncated where the hole / slit has been made.

With a torus its the same, a hole or slit will make it an incomplete torus! So sliting all round the outer or inner circumference will do the same only it makes for inverting the torus easier! Just as if you slit the torus through its section to form a tube...

The point I'm trying to make is that any hole or slit will deform the original shape from being perfect... a sphere becomes a sphere with a flattened section where a hole or complete slit has been made... If a slit was only tiny then the force required to pull the sphere inside out would distort the shape from being what it was...

So how about defining what a hole is??

John.

By the way you've picked up a nasty stutter in those last several posts of yours!!

Both curved surfaces in a torus are continuous. No boundaries. A hole may interupt a small area of the surfaces, but the basic definition of the torus surface is maintained. A longitudinal slit is a destruction of the curves in a complete surface dimension. The short radius after a slit is a planar figure, totally 2 dimensional.

If you wish to call that a hole, you are welcome to. I just don't believe that the sheet of rubber can any longer be called a torus, or innertube. Basically, a transverse cut has the same effect.

Start from a flat piece of rubber, it is the same effect. But you certainly are right. If you wish to drfine that in the rules, then you are correct. I also am still correct, but under either set of rules.

Slitting the sphere to have the result you define is most often a mercator projection, a two dimensional means of expressing a 3 dimensional figure.

The torus problem, now that I search it, has far stricter rules than you do. Merry Christmas

Rich

"With a torus its the same, a hole or slit will make it an incomplete torus! "
"So how about defining what a hole is??"

Or How Big a Hole Can be cut from a torus and still call it a Torus with a hole?

Torus, Described/Defined

Once a hole is cut in the surface it is no longer a complete torus by definition, however you can cut away almost all of the surface and still have enough to define the original torus.

frankly, i do not know the answers.

this reminds me about that old riddle: two pythons of equal length and size start eating (gulping? swallowing?)each other starting from the tail of the other. what happens in the

end?

And it was supposed to have been a yes or no question....with a simple enough explanation...

It seems the jury is still out...

As a 'head' problem it requires considerable stretching of the brain!

As a practical problem it requires considerable stretching of vulcanized rubber!

See Posts #40, #41, & #42.

As to the article in "Scientific American," it is too old to be online. Library files only.

If I can find the old inside out torus inner tube I'll provide some new photos.