## Inside-out torus

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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?

### Re: Inside-out torus

I think I have it.

**Spoiler:**

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### Re: Inside-out torus

JonMW wrote:I think I have it.Spoiler:

Yes.

However,

**Spoiler:**

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### Re: Inside-out torus

Token wrote:However,Spoiler:

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.

### Re: Inside-out torus

DrStalker wrote:Token wrote:However,Spoiler:

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:**

All posts are works in progress. If I posted something within the last hour, chances are I'm still editing it.

- VectorZero
**Posts:**471**Joined:**Fri Nov 02, 2007 7:22 am UTC**Location:**Kensington

### 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.

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.

Van wrote:Fireballs don't lie.

### 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.

[edit] Spoilerizing since I turned out to be right.

**Spoiler:**

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.

### Re: Inside-out torus

@DrStalker: that is indeed the right answer.

Edit: spoilerised to be on the safe side

**Spoiler:**

Edit: spoilerised to be on the safe side

### 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:**

### Re: Inside-out torus

Zomg that is so cool!

### 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?

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