# A strange twist of fate

April 28, 2018 9:29 AM Subscribe

I was doing the laundry this morning and noticed a pair of underwear got twisted in a weird way. I spent a few minutes trying to fix them--turning them inside out, flipping them around--but nothing worked! Metafilter, please help me untwist these knickers.

This reminds me a bit of those disentanglement puzzles where you have to think about the topography of the pieces...it almost seems like a mobius strip, but there are definitely two sides and not one. What am I not seeing here?

This reminds me a bit of those disentanglement puzzles where you have to think about the topography of the pieces...it almost seems like a mobius strip, but there are definitely two sides and not one. What am I not seeing here?

As you're facing them (like in the picture), reach your right hand in the leg hole on the right. Take the waistband at upper left through the leg hole. They should be inside out and untwisted. Using words to do this was challenging!

posted by XtineHutch at 9:44 AM on April 28, 2018 [1 favorite]

posted by XtineHutch at 9:44 AM on April 28, 2018 [1 favorite]

Maybe something happened in the crotch gusset - can you put your hand between the two layers of fabric at the crotch?

posted by mskyle at 9:53 AM on April 28, 2018 [1 favorite]

posted by mskyle at 9:53 AM on April 28, 2018 [1 favorite]

Yes if it’s not obvious, and I am giving you some credit for good faith reasonable efforts, then the

Also btw you basically started a Mexican plaited belt, which were tratitionally made by passing the belt through itself in clever ways. See here for a mathematical treatment of the topic, with pics, it might be helpful.

I have those undies and an interest in this kind of puzzle so if you don’t get it sorted by tonight I can try to put some into that configuration and back out.

posted by SaltySalticid at 10:04 AM on April 28, 2018 [5 favorites]

**fly**will likely need to be engaged. Try passing the waistband through the fly hole.Also btw you basically started a Mexican plaited belt, which were tratitionally made by passing the belt through itself in clever ways. See here for a mathematical treatment of the topic, with pics, it might be helpful.

I have those undies and an interest in this kind of puzzle so if you don’t get it sorted by tonight I can try to put some into that configuration and back out.

posted by SaltySalticid at 10:04 AM on April 28, 2018 [5 favorites]

The photo makes it look like the seams on the left and right leg holes are now forming a pair of linked rings. I don't understand how they could have done that without help from a sewing machine. Could these be pranked underpants?

posted by flabdablet at 10:17 AM on April 28, 2018

posted by flabdablet at 10:17 AM on April 28, 2018

Reminds me of the "impossible braid". Just do this (youtube link), but in reverse.

posted by yohko at 11:36 AM on April 28, 2018

posted by yohko at 11:36 AM on April 28, 2018

SaltySalticid, I tried to pass the waistband through the fly hole several times in a few different ways, but it always comes out with the twist still there. Let me know if you figure it out!

posted by Lurch at 3:50 PM on April 28, 2018

posted by Lurch at 3:50 PM on April 28, 2018

The topological equivalent in leatherwork for these underpants would be a piece cut as if in preparation for an impossible braid, but with

metafilter: the topological equivalent in leatherwork for these underpants

posted by flabdablet at 8:52 PM on April 28, 2018 [2 favorites]

*no*braids and a full twist in the centre strip. The more I think about this, the harder I'm leaning towards "fiendish prank".metafilter: the topological equivalent in leatherwork for these underpants

posted by flabdablet at 8:52 PM on April 28, 2018 [2 favorites]

Yes, one way to think of a pair of underpants like this mathematically/topologically is as a triple torus (two leg holes would make a double torus but the additional fly hole makes it a triple torus). And your situation kind of reminds me of this video.

However, for a few reasons this type of scheme will not work with your underpants--the fabric of the pants can't morph and bend in the ways the clay can.

Yohko's idea of an "impossible braid" is a clever one and I spent some quality time with a pair of my underwear this afternoon* convincing myself that it could be done. And it can--but the result is braided underwear that looks a lot like the impossible-braided leather bracelet shown on the video (except, you know, more white and underwear-y) but nothing at all like what you have.

(Also, flabdablet makes the correct observation that your current underwear situation is the equivalent of "no braids and a full twist in the centre strip" and not anything braided.)

On further thought, a better topological model for the underpants is the "pair of pants surface". (And yes, believe it or not, this is a real thing that is actually researched in mathematics, and least now and then--this paper by Hatcher and Thurston appears to be the original source of the idea.)

The pair of pants surface is just the surface of a sphere with three round holes cut in it.

Your underpants are equivalent to the surface of a sphere with FOUR round holes cut in it (because of the extra fly hole).

For our purposes we can ignore the rest of the surface and just concentrate on the boundaries of the four holes--which are four circles located in three dimensional space.

In a regulation pair of underwear, none of those four circles is interlinked with any of the other.

In your twisted-up pair of underwear, two of the circles (the leg holes) are interlinked with each other.

Here is the crux of the argument: In a three-dimensional space, there is NO WAY for two circles that are not interlinked with each other to move to a position where they are interlinked with each other.

You've either got to CUT one of the circles and then glue it back together interlinked, or MOVE ONE OF THE CIRCLES (or part of it) INTO SOME HIGHER-DIMENSIONAL SPACE, move it into position interlinking the other circle, then move it back to normal 3-space.

Within 3-space and with no cutting allowed it is just straight impossible. Explanation and proof outline here.

(Again, this is a point flabdablet correctly makes above: "the left and right leg holes are now forming a pair of linked rings. I don't understand how they could have done that without help from a sewing machine.")

So with that established, we are left with these possibilities for the situation with your underwear:

If you have worn it a few times before, maybe you were just really really drunk each and every time you wore it, and/or got dressed in the pitch dark each time, and just didn't notice? This seems unlikely, but who knows . . .

posted by flug at 9:52 PM on April 28, 2018 [109 favorites]

However, for a few reasons this type of scheme will not work with your underpants--the fabric of the pants can't morph and bend in the ways the clay can.

Yohko's idea of an "impossible braid" is a clever one and I spent some quality time with a pair of my underwear this afternoon* convincing myself that it could be done. And it can--but the result is braided underwear that looks a lot like the impossible-braided leather bracelet shown on the video (except, you know, more white and underwear-y) but nothing at all like what you have.

(Also, flabdablet makes the correct observation that your current underwear situation is the equivalent of "no braids and a full twist in the centre strip" and not anything braided.)

On further thought, a better topological model for the underpants is the "pair of pants surface". (And yes, believe it or not, this is a real thing that is actually researched in mathematics, and least now and then--this paper by Hatcher and Thurston appears to be the original source of the idea.)

The pair of pants surface is just the surface of a sphere with three round holes cut in it.

Your underpants are equivalent to the surface of a sphere with FOUR round holes cut in it (because of the extra fly hole).

For our purposes we can ignore the rest of the surface and just concentrate on the boundaries of the four holes--which are four circles located in three dimensional space.

In a regulation pair of underwear, none of those four circles is interlinked with any of the other.

In your twisted-up pair of underwear, two of the circles (the leg holes) are interlinked with each other.

Here is the crux of the argument: In a three-dimensional space, there is NO WAY for two circles that are not interlinked with each other to move to a position where they are interlinked with each other.

You've either got to CUT one of the circles and then glue it back together interlinked, or MOVE ONE OF THE CIRCLES (or part of it) INTO SOME HIGHER-DIMENSIONAL SPACE, move it into position interlinking the other circle, then move it back to normal 3-space.

Within 3-space and with no cutting allowed it is just straight impossible. Explanation and proof outline here.

(Again, this is a point flabdablet correctly makes above: "the left and right leg holes are now forming a pair of linked rings. I don't understand how they could have done that without help from a sewing machine.")

So with that established, we are left with these possibilities for the situation with your underwear:

**#1. You and/or your underwear briefly entered some twisted higher-dimensional space.**Have you flown to a black hole recently? Or perhaps you have a pesky wormhole hiding out in your washing machine?**#2. Someone has pranked us by taking apart the underwear & sewing it back together the wrong way.**Inspecting the underwear, this can be done by taking out one seam, twisting the fabric through 360 degrees, and re-sewing in place.**#3. (My preferred/most likely solution) This is a new-ish bit of underwear and came from the factory this way.**It would be really easy for a pair to be sewn in the factory this way--as noted above, it just requires a 360-degree twist in that section of fabric before it is sewn. That is the sort of thing that could happen any number of ways at the factory.If you have worn it a few times before, maybe you were just really really drunk each and every time you wore it, and/or got dressed in the pitch dark each time, and just didn't notice? This seems unlikely, but who knows . . .

** Metafilter: A few quality hours twisting my tighty-whities in mathematically interesting ways.*posted by flug at 9:52 PM on April 28, 2018 [109 favorites]

Trying to reproduce this at my end. I can't. Mysterious!

posted by carter at 5:35 AM on April 29, 2018

posted by carter at 5:35 AM on April 29, 2018

can you reach in through the fly hole, grab the waistband where your bellybutton would be, and pull the undies inside out through the fly hole?

posted by pseudostrabismus at 2:38 PM on April 29, 2018

posted by pseudostrabismus at 2:38 PM on April 29, 2018

I think

In the video, the craftsperson starts with an oblong piece of leather several times as long as it is wide, and cuts two slits in it parallel to the long sides, but which do not reach either end, effectively partitioning the piece of leather into three parallel strips over most of its length.

The rest is braiding, but a state topologically equivalent to your twisted briefs is reached after the first iteration of the braiding process.

To see that the flat starting point of the braid, after the slits have been cut but before any braiding has taken place, is equivalent to your briefs in the untwisted state, imagine that the elastic waistband is super-elastic and is stretched into an oblong shape such that the leg holes have become parallel slits, and the stretched waistband has become the outer perimeter of the oblong.

This is the exact shape of the starting point for the braid.

In terms of

As

posted by jamjam at 5:50 PM on April 30, 2018

**yokho**is correct that this is a version of the "impossible braid" they link to a video of (and which the Mexican plaited belt mentioned by**SaltySalticid**is a more complex version of).In the video, the craftsperson starts with an oblong piece of leather several times as long as it is wide, and cuts two slits in it parallel to the long sides, but which do not reach either end, effectively partitioning the piece of leather into three parallel strips over most of its length.

The rest is braiding, but a state topologically equivalent to your twisted briefs is reached after the first iteration of the braiding process.

To see that the flat starting point of the braid, after the slits have been cut but before any braiding has taken place, is equivalent to your briefs in the untwisted state, imagine that the elastic waistband is super-elastic and is stretched into an oblong shape such that the leg holes have become parallel slits, and the stretched waistband has become the outer perimeter of the oblong.

This is the exact shape of the starting point for the braid.

In terms of

**flug**'s "pair of pants surface" the hole corresponding to the waistband has become the outer perimeter of the oblong.As

**yokho**says, watch what she does and do the reverse -- to describe all that in words is beyond me, I'm afraid, but the leftmost strip of leather corresponds to the left edge of the waistband of your stretched into an oblong briefs; the middle strip of leather is the part of your briefs that goes between your legs; and the rightmost strip of leather is the right edge of the waistband of your stretched briefs.posted by jamjam at 5:50 PM on April 30, 2018

Hmmm, it's actually not braiding, it's generating a 360 degree twist in the middle strip, which involves a similar but distinct way of passing parts through the slits.

posted by jamjam at 6:24 PM on April 30, 2018 [1 favorite]

posted by jamjam at 6:24 PM on April 30, 2018 [1 favorite]

*a state topologically equivalent to your twisted briefs is reached after the first iteration of the braiding process.*

It really isn't.

The underlying reason for that is because, as flug correctly says, linking unlinked rings is impossible in 3-space; but since that's just incomprehensible math for most people, the best way to get a gut feel for it is simply to stop and

*check*the condition of the boundaries as you make each braiding move. What you will

*always*find is that as you braid, what you're doing to the

*edges*of your braided strips is topologically equivalent to pushing the end of one rubber band through the middle of one or more others.

If you pay close attention to the

*boundaries*of the surfaces in the impossible-braid starting leather - one boundary around the outside and one created by each slit, none of them linked with any other - you will see that

*no*braiding move causes two of those boundaries to link.

It's important to understand that the "linking" we're talking about here is the kind that you'd find in an ordinary steel chain. It is indeed possible to "link" two rubber bands so they don't come apart when stretched, but what you end up with where they join will always be some variation on a figure-8 knot - a knot which you

*can*take apart just by pulling on its tight loops until they loosen. In fact, the knot that ties the rubber bands together is just a little braid, and as such it can always be unbraided; the tied rubber bands are wound around each other, but they're not actually

*linked.*

But that's not what we're seeing with these underpants. If you trace the seams around the leg holes, you will find that those two seams are indeed properly linked, just like links in a chain, and not merely braided together. And that means there is

*no*sequence of braid-equivalent moves you can make to unlink them.

posted by flabdablet at 10:10 PM on April 30, 2018 [6 favorites]

This backgrounder on 4-space includes instructions for unlinking linked rings by "lifting" one of them along an axis orthogonal to the 3-space in which the linked rings currently reside, and some discussion about how that process works.

However, since the manoeuvre concerned involves taking advantage of the extra spatial direction in order to move one of the rings

As a result, I'm assigning the black-hole and wormhole-in-the-washing-machine options even lower likelihoods than flug did.

posted by flabdablet at 11:18 PM on April 30, 2018 [2 favorites]

However, since the manoeuvre concerned involves taking advantage of the extra spatial direction in order to move one of the rings

*completely clear*of the other, I'm not sure that it would actually be feasible if the rings in question were constrained to remain physically attached to each other via other connections, as is the case for the leg seams on a pair of ordinary 3-space underpants. If you could indeed untwist a gusset by this method, I suspect that some portion of it would end up constrained to remaining outside normal space.As a result, I'm assigning the black-hole and wormhole-in-the-washing-machine options even lower likelihoods than flug did.

posted by flabdablet at 11:18 PM on April 30, 2018 [2 favorites]

*>However, since the manoeuvre concerned involves taking advantage of the extra spatial direction in order to move one of the rings completely clear of the other, I'm not sure that it would actually be feasible if the rings in question were constrained to remain physically attached to each other via other connections, as is the case for the leg seams on a pair of ordinary 3-space underpants.*

Yes, this is a slightly tricky question.

**First fact we need is to realize that our twisted underwear is equivalent topologically to a doubly-twisted ribbon with the ends glued together.**

Think of a mobius strip: A mobius strip is a ribbon with a SINGLE twist and the ends glued together.

Instead of a single twist, our twisted-underwear is like a ribbon with a double twist, then ends glued.

(To be painfully accurate, our underwear shape is a double-twisted ribbon which then has two holes cut into the ribbon. These two holes correspond to the waistband hole and the fly hole. However, the waist hole and the fly hole don't help us at all, or hinder either, so I'm just going to ignore them in the discussion below, and imagine they are filled in. You can reinsert them any time you like by just snipping two holes into our double-twisted ribbon.)

**The second fact we need to know is that our double-twisted ribbon is the same essential shape (topologically speaking) as a ribbon with NO twists.**

Taking this idea a step further:

- If you take a ribbon and twist it 1, 3, 5, 7 or any odd number of times, then glue the ends together, those are all equivalent to a mobius strip.
- If you take the ribbon and twist it 0, 2, 4, 6, or any even number of times before gluing the ends back together, it is equivalent to a cylinder.

So that got a bit technical, but the essential fact we need to know is that, in terms of basic topological shapes, our double-twisted ribbon is the same as a ribbon with NO twists.

Since they are the same shape, this raises the question: How can we transform the double twisted ribbon to a non-twisted ribbon? Can we do it in 3-dimensional space? 4-dimensional space? Or perhaps some higher dimension?

**This brings us to our third fact--the one we established in our discussion above in regards to the twisted underwear: You cannot transform from a doubly-twisted ribbon to a non-twisted ribbon in 3 dimensional space (R**(Less technical discussion here - little more technical discussion here.)

^{3}).Two helpful bits of information about this "ribbon with no twists" that will help further our discussion as we move to higher dimensions:

- This
*ribbon with no twists*is the same shape as an*ordinary cylinder*--thinking of the cylinder surface only and no top or bottom caps: Take a sheet of paper, roll it up, glue opposite sides to one another. - The
*cylinder*is the same shape as a*sphere with two holes cut into it*(remember our "pair of pants surface" mentioned above? That has 3 holes, the cylinder has just 2).

**Our fourth fact--and the point of this comment--is that you CAN untwist the double-twisted ribbon if you work in four dimensions (R**

^{4}).Here is a hand-wavy explanation:

Think of two circles in 3-space, one just above the other. You can 'tip' the uppermost circle in 4-space so that only two points of this upper circle remain in 3-space. Then turn this circle 90 degrees on its side and shift it down a bit so that it interlinks with the lower circle. Now tip the circle back out of 4-space and you have two interlinked circles.More detailed explanation & nice animation here. More technical approach (that arrives at the same answer) here.

This same transformation works just fine if you just fill in between the two circles to create a cylinder and then follow the same moves to transform it to a double-twisted cylinder.

**A fifth (bonus!) fact is that you can't untwist a mobius strip to a no-twist ribbon--no way, now how, whether in 4, 5, 6, or any number of dimensions.**

The no-twist ribbon (or "cylinder") and the one-twist ribbon ("mobius strip") are essentially different shapes that can never be transformed one to the other.

As mentioned above, you can add or subtract TWO twists from either type of shape by manipulating it in 4th dimensional space. But you can never add or subtract just one twist.

And now we have totally overthought a pair of twisted underpants . . .

posted by flug at 4:58 PM on May 5, 2018 [4 favorites]

Looks like what you have there is some monster underpants.

posted by The Underpants Monster at 12:34 PM on May 6, 2018 [8 favorites]

posted by The Underpants Monster at 12:34 PM on May 6, 2018 [8 favorites]

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posted by EndsOfInvention at 9:36 AM on April 28, 2018 [4 favorites]