## Inside-out torus

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antonfire
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### Inside-out torus

I have a rubber torus, of some finite thickness. On the outside, I paint a meridian. On the inside, I paint a parallel. These make two linked circles. Now, I poke a hole in the torus (not through either of the circles), and turn the torus inside-out. The meridian is now on the inside and the parallel is on the outside. But then the two are unlinked. Somehow, by turning a torus inside out, I've unlinked two linked circles without breaking either of them! What's going on?
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

JonMW
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### Re: Inside-out torus

I think I have it.
Spoiler:
Is it even possible to turn a torus inside-out by poking a hole in it?
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Token
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Location: London

### Re: Inside-out torus

JonMW wrote:I think I have it.
Spoiler:
Is it even possible to turn a torus inside-out by poking a hole in it?

Yes.

However,
Spoiler:
The process of doing so does not cause the two circles to become unlinked.
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DrStalker
Posts: 271
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Location: Sydney

### Re: Inside-out torus

Token wrote:However,
Spoiler:
The process of doing so does not cause the two circles to become unlinked.

Can you elaborate on that? It seems to me they would be unlinked, I just can't visualize how that happens topologically when the torus is inverted.
There are two types of people in the world: 1) those that can extrapolate from incomplete data.

Token
Posts: 1481
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Location: London

### Re: Inside-out torus

DrStalker wrote:
Token wrote:However,
Spoiler:
The process of doing so does not cause the two circles to become unlinked.

Can you elaborate on that? It seems to me they would be unlinked, I just can't visualize how that happens topologically when the torus is inverted.

Spoiler:
Visualise the process of turning the torus inside out, then follow the location of the rings.
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VectorZero
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### Re: Inside-out torus

For those having difficulty visualising the scenario, remember a torus can be assembled by taking a rectangle and fold it such that opposite edges meet (i.e. make a cylinder then a torus.)

So, draw one line on each face, from the midpoint of one edge to the midpoint of the opposite edge, the two lines at right angles. Create a cylinder such that the circle formed by the drawn line is on the outside. Then fold the cylinder into a torus.
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DrStalker
Posts: 271
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Location: Sydney

### Re: Inside-out torus

VectorZero wrote:For those having difficulty visualising the scenario, remember a torus can be assembled by taking a rectangle and fold it such that opposite edges meet (i.e. make a cylinder then a torus.)

So, draw one line on each face, from the midpoint of one edge to the midpoint of the opposite edge, the two lines at right angles. Create a cylinder such that the circle formed by the drawn line is on the outside. Then fold the cylinder into a torus.

 Spoilerizing since I turned out to be right.

Spoiler:
Does refolding the same paper in the opposite direction on step 1 (so the cylinder is inside out compared to the way it was previously) result in a model of the torus after it has been turned inside out? Or do you need to roll the paper perpendicularly instead? (roll top-to-bottom instead of left to right)

If the later I can then make a second torus configuration with linked rings, but the meridian ring is now the parallel ring and vice-versa, so the answer would be "they are still linked, we were visualizing it wrong in teh original puzzle." Except I don't think thats right.
Last edited by DrStalker on Sat Nov 03, 2007 11:04 pm UTC, edited 1 time in total.
There are two types of people in the world: 1) those that can extrapolate from incomplete data.

rhino
Posts: 123
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Location: Cambridge, UK

### Re: Inside-out torus

@DrStalker: that is indeed the right answer.

Spoiler:
The meridian on the outside becomes a parallel on the inside, and the parallel on the outside becomes a meridian on the inside.

You can imagine/draw this (which I did to convince myself) or conclude that since it's topologically impossible to separate the circles, that has to be what happens. The "reason", if you like, is that the tunnel round the inside of the donut becomes the hole of the inverted donut, and vice versa.

Edit: spoilerised to be on the safe side

Ralp
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### Re: Inside-out torus

Let's say our inner-tube with a hole is colored black on the outside and white on the inside. Let's consider it to be "inside out" if we can transform it so that it's the other way around, white on the outside and black on the inside.

Spoiler:
It turns out this is possible; the result is indeed a torus that's white on the outside and black on the inside, but it doesn't look like an inner-tube anymore. Since it's made out of real rubber instead of imaginary topologystuff, it will be roughly a C shape, with the rubber material stuffing back in on itself at both tips of the C. The meridian and parallel that we painted will still be linked circles.

It's not possible to get the object that we all imagine from the description "a torus turned inside-out" from just a hole in the side -- we need to either cut all the way along a meridian to break the "links" of all the parallels, or vice versa, and then seal our cut back up afterwards.

Blatm
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### Re: Inside-out torus

Zomg that is so cool!

antonfire
Posts: 1772
Joined: Thu Apr 05, 2007 7:31 pm UTC

### Re: Inside-out torus

By the way, I got this from Martin Gardner's "Mathematical Puzzles and Diversions". It has tons of other cool stuff as well.
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?