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My nephew was folding laundry, and turning the occasional shirt right-side-out. I showed him a "trick" where I turned it right-side-out by pulling the whole thing through a sleeve instead of the bottom or collar of the shirt. He thought it was really cool (kids are easily amused, and so am I).

So he learned that you can turn a shirt or pants right-side-out by pulling the material through any hole, not just certain ones. I told him that even if there was a rip in the shirt, you could use that to turn it inside-out or right-side-out, and he was fascinated by this and asked "why?"

I don't really know the answer to this. Why is this the case? What if the sleeves of a long-sleeve shirt were sewn together at the cuff, creating a continuous tube from one sleeve to the other? Would you still be able to turn it right-side-out? Why? What properties must a garment have so that it can be turned inside-out and right-side-out?

Sorry if this is a lame question, but I've always wondered. I wouldn't even know what to google for, so that is why I am asking here.

If you know the answer to this, could you please put it into layman's terms?

Update: Wow, I really appreciate all the participation. This is a really pleasant community and I have learned a lot here. It seems that the answer is that you need at least one puncture in the garment through which to push or pull the fabric. It appears that you can have certain handles, although it's not usually practical with clothing due to necessary stretching.

Accepted (a while ago actually -- sorry for not updating sooner) Dan's answer because among the answers that I understand, it is the highest ranked by this community.

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@Qiaochu: Maybe I'm missing something, but in the video, the sleeves seem to pass through eachother (which is allowed for eversions, but not for actual clothing). –  Jason DeVito Aug 19 '10 at 0:26
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@Jason: Maybe my geometric intuition is failing me, but if the operation described in the video is done to the part of the garment where the holes are (the head hole and whatever you would call the other one) I think you can pass through the holes. –  Qiaochu Yuan Aug 19 '10 at 0:31
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@Qiaochu: The material passes through itself at :17 in that video. I will comment further under your question. –  Chris Aug 19 '10 at 1:13
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What I really like in the SE Universe is that people can ask such great questions and get answers to them. –  sharptooth Nov 1 '10 at 15:29
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HA!!! you math geeks are awesome!!! –  Keng Nov 1 '10 at 16:54
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6 Answers

up vote 116 down vote accepted

Everything I'm wearing is a topological sphere with holes (t-shirts have 4, pants 3, shoes and socks 1) in which cases any hole works. Instead of a long-sleeve shirt with the arms sewn together, consider a pair of pants with legs sewn together to form a topological torus with a hole (so if you were to wear them your feet would be touching and it would be impossible to put your shoes on). This pair of pants has two parameters which are roughly constant, the circumference of the leg and the total length of the two legs. When it is turned inside out through the waist, these parameters swap roles, so you will have a tube about the length of a pant leg with an opening on each end, the same as if you had turned one leg inside-out and pushed the other leg through it prior to sewing. I think this would be possible to do with real toroidal clothing as long as it is thin enough, because the process doesn't require any stretching.

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Because the parameters of the torus have interchanged, it will look somewhat different. You won't be able to completely unfold it, but try putting your arm all the way through through the tube to demonstrate that it is still a torus. –  Dan Brumleve Aug 19 '10 at 2:35
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To clarify, it won't look the same as if you had turned the article inside-out and then sewn the holes together. In fact you will not be able to reach one of these states from the other without cutting. This is a problem in homotopy. It is similar to the parity of a knot. –  Dan Brumleve Aug 19 '10 at 3:10
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Ok, I tried it. If the legs are connected, creating a punctured torus, you cannot turn the pants inside-out such that they appear the same way that they would if you had turned them inside-out before sewing. If you attempt to turn the pants inside-out after sewing, you get one leg stuck inside the other, and the pants are not wearable. They are "tangled". So, it seems that in order for a garment to be inside-outable, it needs at least one puncture and ZERO handles. Does that make sense? Am I correct? –  Chris Aug 19 '10 at 14:34
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Chris, consider that if you had a stretchy pair of toroidal pants with vertical stripes, turned it inside out through the waist, and then stretched it to resemble the original pair of right-side-out toroidal pants, it would now have horizontal stripes. –  Dan Brumleve Aug 19 '10 at 20:43
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It is not topologically important whether or not the material can be stretched as long as it is thin and flexible enough. Either we can stretch the hole to accommodate the passage of a surface, else we can fold the surface to accommodate passage through a hole, with the same effect. I don't know the mathematical terminology for this distinction but I would like to. –  Dan Brumleve Aug 20 '10 at 5:32
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A shirt with no hole, or all holes sealed is homeomorphic to a sphere.

A shirt with holes is homeomorphic to a sphere with punctures.

Now you intuitively see that a rubber sphere with punctures can be turned inside out by continuous motion, whereas a perfect sphere cannot.

To understand why, let's concentrate on 2-dimentional plane, taking a circle instead of a sphere. It can be proven that any continuous deformation that inverts the circle will make it cross itself. However if the circle has 'holes', or gaps, then it topologically is equivalent to one or more line segments, which can all be inverted in place without touching each other (appropriate scaling may be necessary).

Similar feature holds for a sphere too. Hence you cannot turn a shirt inside out if all the holes are sealed, but you can do it even if you have one hole (and provided that the shirt is flexible enough to go through that hole if the hole is small).

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I do not think it is a good idea to use the word "homeomorphic" if the goal is to explain surface topology to a layman. –  Qiaochu Yuan Aug 19 '10 at 0:33
    
Right... I should add homeomorphic elements are those which can be morphed into each other using "rubber sheet" transformations. –  KalEl Aug 19 '10 at 0:45
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Homeomorphism are a bit more general than that (and I know that you're purposely trying to be vague for the layman sake). For example, if you take a rectangle and glue two opposite sides together with 2 twists, the resulting space is homemorphic to an annulus (no twists), but there is no "rubber sheet" transformation which takes one to the other. –  Jason DeVito Aug 19 '10 at 2:59
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@Jason You bring up a very nice point indeed. –  KalEl Aug 21 '10 at 20:15
    
To add to Jason DeVito's remark, I'd say that your '"rubber sheet" transformations' are a better mental image for ambient isotopies than for homemorphisms. –  Omar Antolín-Camarena May 14 '12 at 22:27
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I can at least answer the question about the rip, and point you to some terms to look up. What you are looking at is not geometry as it is usually understood but topology (particularly of surfaces with boundary), which informally is the study of properties of things that are invariant under deformation. It is often said that a topologist can't tell a coffee mug from a doughnut (it is possible to deform one into the other) and this is the attitude you should take towards questions of this kind.

From this perspective, a rip is the same thing as the bottom of the shirt is the same thing as a sleeve: you can deform each of them into the other, so topologically they're the same thing. (Imagine making the rip bigger, or squashing a long sleeve into a short sleeve.) They are all just examples of what topologists call punctures.

I don't know the answer to your general question, but one property a surface must have to be able to be turned inside out is that it must be orientable; in other words, it must have an inside and an outside to begin with! Surfaces such as the Möbius strip famously don't have this property, so it doesn't even make sense to ask about whether it's possible to turn such things inside-out or not. (And some non-orientable surfaces can't even be realized in three dimensions...)

Orientable surfaces happen to have a particularly simple description: they are all topologically just spheres with some handles sewn on and with some punctures. To turn them inside out, at least one puncture is necessary, and if you can turn one handle inside-out then you can presumably turn all of them inside-out. If what I said in the comments is wrong and you can't turn handles inside-out, then a necessary and sufficient condition is that there is at least one puncture and no handles.

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This is blowing my mind. I am carefully reading and looking things up. –  Chris Aug 19 '10 at 1:00
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I went and fastened the sleeves of a shirt together :). I was NOT able to get the shirt inside-out while the sleeves were connected (and it was not a limitation of the flexibility of the fabric, as far as I know). Maybe I did it wrong, but I could not make it happen. I basically tried to push the whole shirt through the sleeve-tube. I was able to do this, and I ended up with an inside-out shirt BUT one sleeve was inside the other! I did not have a means to get the sleeves back to the tube shape. So maybe the requirements are: 1) at least one puncture 2) no handles? –  Chris Aug 19 '10 at 1:19
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@BBischof: Is a shirt with its sleeves sewn together orientable? It seems that it is, because the sleeve-tube still has an inside and and outside, right? –  Chris Aug 19 '10 at 1:27
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@Chris: yes. A shirt with its sleeves sewn together is a sphere with one handle and two punctures. –  Qiaochu Yuan Aug 19 '10 at 1:39
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Oh, last night in bed I got it. The handle => donut hole –  bobobobo Dec 5 '10 at 13:54
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First, a warning. I suspect this response is likely not going to be immediately comprehensible. There is a formal set-up for your question, there are tools available to understand what's going on. They're not particularly light tools, but they exist and they're worthy of being mentioned. Before I write down the main theorem, let me set-up some terminology. The tools belong to a subject called manifold theory and algebraic topology. The names of the tools I'm going to use are called things like: the isotopy extension theorem, fibre-bundles, fibrations and homotopy-groups.

You have a surface $\Sigma$, it's your shirt or whatever else you're interested in, some surface in 3-dimensional space. Surfaces have automorphism groups, let me call it $Aut(\Sigma)$. These are, say, all the self-homeomorphisms or diffeomorphisms of the surface. And surfaces can sit in space. A way of putting a surface in space is called an embedding. Let's call all the embeddings of the surface $Emb(\Sigma, \mathbb R^3)$. $Emb(\Sigma, \mathbb R^3)$ is a set, but in the subject of topology these sets have a natural topology as well. We think of them as a space where "nearby" embeddings are almost the same, except for maybe a little wiggle here or there. The topology on the set of embeddings is called the compact-open topology (see Wikipedia, for details on most of these definitions).

Okay, so now there's some formal nonsense. Look at the quotient space $Emb(\Sigma, \mathbb R^3)/Aut(\Sigma)$. You can think of this as all ways $\Sigma$ can sit in space, but without any labelling -- the surface has no parametrization. So it's the space of all subspaces of $\mathbb R^3$ that just happen to be homeomorphic to your surface.

Richard Palais has a really nice theorem that puts this all into a pleasant context. The preamble is we need to think of everything as living in the world of smooth manifolds -- smooth embeddings, $Aut(\Sigma)$ is the diffeomorphism group of the surface, etc.

There are two locally-trivial fibre bundles (or something more easy to prove -- Serre fibrations), this is the "global" isotopy-extension theorem:

$$Diff(\mathbb R^3, \Sigma) \to Diff(\mathbb R^3) \to Emb(\Sigma, \mathbb R^3)/Aut(\Sigma)$$

$$Diff(\mathbb R^3 fix \Sigma) \to Diff(\mathbb R^3, \Sigma) \to Aut(\Sigma)$$

here $Diff(\mathbb R^3)$ indicates diffeomorphisms of $\mathbb R^3$ that are the identity outside of a sufficiently large ball, say.

So the Palais theorem, together with the homotopy long exact sequence of a fibration, is giving you a language that allows you to translate between automorphisms of your surface, and motions of the surface in space.

It's a theorem of Jean Cerf's that $Diff(\mathbb R^3)$ is connected. A little diagram chase says that an automorphism of a surface can be realized by a motion of that surface in 3-space if and only if that automorphism of the surface extends to an automorphism of 3-space. For closed surfaces, the Jordan-Brouwer separation theorem gives you an obstruction to turning your surface inside-out. But for non-closed surfaces you're out of tools.

To figure out if you can realize an automorphism as a motion, you literally have to try to extend it "by hands". This is a very general phenomena -- you have one manifold sitting in another, but rarely does an automorphism of the submanifold extend to the ambient manifold. You see this phenomena happening in various other branches of mathematics as well -- an automorphism of a subgroup does not always extend to the ambient group, etc.

So you try your luck and try to build the extension yourself. In some vague sense that's a formal analogy between the visceral mystery of turning the surface inside-out and a kind of formalized mathematical problem, but of a fundamentally analogous feel.

We're looking for automorphisms that reverse orientation. For an arbitrary surface with boundary in 3-space, it's not clear if you can turn the surface inside out. This is because the surface might be knotted. Unknotted surfaces are examples like your t-shirt. Let's try to cook up something that can't be turned inside-out.

The automorphism group of a 3-times punctured sphere has 12 path-components (12 elements up to isotopy). There are 6 elements that preserve orientation, and 6 that reverse. In particular the orientation-reversing automorphisms reverse the orientation of all the boundary circles. So if you could come up with a knotted pair-of-pants (3-times punctured surface) so that its boundary circles did not admit a symmetry that reversed the orientations of all three circles simultaneously, you'd be done.

Maybe this doesn't seem like a reduction to you, but it is.

For example, there are things called non-invertible knots:

http://en.wikipedia.org/wiki/Invertible_knot

So how do we cook up a knotted pair-of-pants from that?

Here's the idea. The non-invertible knot in the link above is sometimes called $8_{17}$. Here is another picture of it:

http://katlas.org/wiki/8_17

Here is a variant on that.

Interpret this image as a ribbon of paper that has three boundary circles. One boundary circle is unknotted. One is $8_{17}$. The other is some other knot.

It turns out that other knot isn't trivial, nor is it $8_{17}$.

So why can't this knotted pair of pants be turned inside-out? Well, the three knots are distinct, and $8_{17}$ can't be reversed.

The reason why I know the other knot isn't $8_{17}$? It's a hyperbolic knot and it has a different ($4.40083...$) hyperbolic volume than $8_{17}$ ($10.9859...$).

FYI: in some sense this is one of the simplest surfaces with non-trivial boundary that can't be turned inside-out. All discs can be turned inside-out. Similarly, all annuli (regardless of how they're knotted) can be turned inside-out. So for genus zero surfaces, 3 boundary components is the least you can have if you're looking for a surface that can't be turned inside-out.

edited to correct for Jason's comment.

comment added later: I suggest if you purchase a garment of this form you return it to the manufacturer.

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You're awesome... –  BBischof Aug 19 '10 at 5:14
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I just can't make heads or tails of this. By the upvotes, it is clear that this answer is the most useful to the community, but I think it's beyond the grasp of a layman. –  Chris Aug 20 '10 at 4:29
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Could I request a "for dummies" version of this? If you had to explain this directly to my nephew (he's 10), what would you say? You've already put a LOT of effort into this and clearly many people have benefited from your answer, so if you're not up for this, I completely understand. –  Chris Aug 20 '10 at 4:32
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@Chris: I think that the issue is not so much that Ryan is not up to translating his answer for 10 year olds, but that the answer is essentially untranslatable to 10 year olds. The statements (but not the proofs) that Ryan mentions should be comprehensible to an undergrad math student with good intentions and some knowledge of topology, and that is the closest to 10-year-olds you are going to get. –  Mariano Suárez-Alvarez Aug 20 '10 at 4:52
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The wikipedia page for the torus has a very nice animation of a punctured torus turning inside out. In just the same way, if you stitch together the two cuffs of a pair of pants together you can turn the result inside out. You have to attach the cuffs in the simplest way, without knotting, to get a torus with a hole, not a Klein bottle with a hole.

As Ryan says, it is not possible to turn every piece of clothing inside out, However such "bad" pieces of clothing have to be knotted in some fashion. Here is an example, perhaps a bit simpler than Ryan's, that avoids hyperbolic geometry :). Take a pair of pants, tie the legs in an overhand knot, and stitch together the cuffs to get a knotted punctured torus. The knot in this case is a trefoil knot.

I'll call this piece of clothing $S$. The proof I know that $S$ cannot be turned inside out uses some ideas from low-dimensional topology. Notice that the curve of stitches compresses (bounds a disk $D$) in the inner direction. If you could turn $S$ inside out then that motion of $S$ would also give a motion of the disk $D$. Thus the motion sends the curve of stitches to some curve on $S$ (still not parallel to the boundary!) that compresses in the outer direction. But since the trefoil is not the unknot there is no such curve.

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That torus animation is easier to follow than the others I have seen so far. I think this proves pants with the legs joined to a tube could be turned inside out, but extremely stretchy material would be required. We're getting somewhere. –  Chris Aug 19 '10 at 18:23
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Use stretch pants. ;) –  Sam Nead Aug 19 '10 at 19:40
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Ok, I took a pair of jeans and used clothes pins to connect the two cuffs. I then managed to turn the jeans inside out by pushing everything through the waist band and rearranging a bit. However! It didn't look perfect -- even though I have fat feet, my cuff length is still much smaller than twice my inseam. When you turn the jeans inside out the cuff and the inseam have to switch places (as you can see in the Wikipedia animation). I'm off to try this on a pair of boxers. brb. –  Sam Nead Aug 19 '10 at 19:58
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Actually, I wound up not even trying the boxers, because they have the opposite problem -- twice the inseam is much smaller than the cuff length. But running shorts worked perfectly. Turn the bottom inch of one of the cuffs inside out. Tuck that inside the other cuff. Use clothes pins to attach the cuffs. Now you can invert to your heart's content. Good luck! –  Sam Nead Aug 19 '10 at 20:06
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as a layman - this is the answer that made it click for me. Particularly with Sams explaination about the cuff to inseam ratios. –  Geoff Sep 1 '10 at 4:31
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I'm going to try to give a lighter-flavoured version of my previous answer. I'd rather not edit the previous one anymore so here goes another response. I want to make clear, this response is to you, not your 10-year-old nephew. How you translate this response to any person depends more on you and that person than anything else.

Take a look at the Wikipedia page for diffeomorphism. In particular,the lead image

When I look at that image I see the standard Cartesian coordinate grid, but deformed a little.

There's a "big theorem" in a subject called Manifold Theory and it's name is the "Isotopy Extension Theorem". Moreover, it has a lot to do with these kinds of pictures.

The isotopy extension theorem is roughly this construction: say you have some rubber, and it's sitting in a medium of liquid epoxy that's near-set. Moreover, imagine the epoxy to be multi-coloured. So when you move the rubber bit around in the epoxy, the epoxy will "track" the rubber object. If your epoxy had a happy-face coloured into it originally, after you move the rubber, you'll see a deformed happy-face.

So you get images that look a lot like mixed paint. Stir various blotches of paint, and the paint gets distorted. The more you stir, the more it mixes and it gets harder and harder to see the original image. The important thing is that the mixed paint is something of a "record" of how you moved your rubber object. And if your motion of the rubber object returns it to its initial position, there is a function

$$ f : X \to X $$

where $X$ is all positions outside your rubber object. Given $x \in X$ you can ask where the particle of paint at position $x$ went after the mixing, and call that position $F(x)$.

All my talk about fibre bundles and homotopy-groups in the previous response was a "high level" encoding of the above idea. An intermediate step in the formalization of this idea is the solution of an ordinary differential equation, and that differential equation is essentially the "paint-mixing idea" above, in case you want to look at this subject in more detail later.

So what does this mean? A motion of an object from an initial position back to the initial position gives you an idea of how to "mix paint" outside the object. Or said another way, it gives you an Automorphism of the complement, in our case that's a 1-1, continuous bijective function between 3-dimensional space without the garment and itself.

You may find it odd but mathematicians have been studying "paint mixing" in all kinds of mathematical objects, including "the space outside of garments" and far more bizarre objects for well over 100 years. This is the subject of dynamical systems. "Garment complements" are a very special case, as these are subsets of 3-dimensional euclidean space and so they're 3-manifolds. Over the past 40 years our understanding of 3-manifolds has changed and seriously altered our understanding of things. To give you a sense for what this understanding is, let's start with the basics. 3-manifolds are things that on small scales look just like "standard" 3-dimensional Euclidean space. So 3-manifolds are an instance of "the flat earth problem". Think about the idea that maybe the earth is like a flat sheet of paper that goes on forever. Some people (apparently) believed this at some point. And superficially, as an idea, it's got some things going for it. The evidence that the earth isn't flat requires some build-up.

Anyhow, so 3-manifolds are the next step. Maybe all space isn't flat in some sense. That's a tricky concept to make sense of as space isn't "in" anything -- basically by definition whatever space is in we'd call space, no? Strangely, it's not this simple. A guy named Gauss discovered that there is a way to make sense of space being non-flat without space sitting in something larger. Meaning curvature is a relative thing, not something judged by some exterior absolute standard. This idea was a revelation and spawned the idea of an abstract manifold. To summarize the notion, here is a little thought experiment.

Imagine a rocket with a rope attached to its tail, the other end of the rope fixed to the earth. The rocket takes off and goes straight away from the earth. Years later, the rocket returns from some other direction, and we grab both loose ends of the rope and pull. We pull and pull, and soon the rope is tight. And the rope doesn't move, it's taught. as if it was stuck on something. But the rope isn't touching anything except your hands. Of course you can't see all the rope at one time as the rope is tracing out the (very long) path of the rocket. But if you climb along the rope, after years you can verify: it's finite length, it's not touching anything except where it's pinned-down on the earth. And it can't be pulled in.

This is what a topologist might call a hole in the universe. We have abstract conceptions of these types of objects ("holes in the universe") but by their nature they're not terribly easy to visualize -- not impossible either, but it takes practice and some training.

In the 1970's by the work of many mathematicians we started to achieve an understanding of what we expected 3-manifolds to be like. In particular we had procedures to construct them all, and a rough idea of how many varieties of them there should be. The conjectural description of them was called the geometrization conjecture. It was a revelation in its day, since it implied that many of our traditional notions of geometry from studying surfaces in 3-dimensional space translate to the description of all 3-dimensional manifolds. The geometriztion conjecture was recently proven in 2002.

The upshot of this theory is that in some sense 3-dimensional manifolds "crystalize" and break apart in certain standard ways. This forces any kind of dynamics on a 3-manifold (like "paint mixing outside of a garment") to respect that crystalization.

So how do I find a garment you can't turn inside-out? I manufacture one so that its exterior crystalizes in a way I understand. In particular I find a complement that won't allow for this kind of turning inside-out. The fact that these things exist is rather delicate and takes work to see. So it's not particularly easy to explain the proof. But that's the essential idea.

Edit: To say a tad more, there is a certain way in which this "crystalization" can be extremely beautiful. One of the simplest types of crystalizations happens when you're dealing with a finite-volume hyperbolic manifold. This happens more often than you might imagine -- and it's the key idea working in the example in my previous response. The decomposition in this case is very special as there's something called the "Epstein-Penner decomposition" which gives a canonical way to cut the complement into convex polytopes. Things like tetrahedra, octahedra, icosahedra, etc, very standard objects. So understanding the dynamics of "garments" frequently gets turned into (ie the problem "reduces to") the understanding of the geometry of convex polytopes -- the kind of things Euclid was very comfortable with. In particular there's software called "SnapPea" which allows for rather easy computations of these things.

Images taken from Morwen Thistlethwaite's webpage. These are images of the closely-related notion of a "Dirichlet domain".

Here is an image of the Dirichlet domain for the complement of $8_{17}$, the key idea in the construction of my previous post.

Technically, this in the Poincare model for hyperbolic space, which gives it the jagged/curvy appearance.

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I really appreciate the effort you put into this, but I just don't understand it at all. –  Chris Aug 24 '10 at 6:01
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I'm summarizing about 200 years of mathematics, almost none of which is standardly taught to undergraduates at almost any university. So if you want to understand what's going on, you have a pencil-sketch above and enough terms and links to start trying to assemble a picture. How much you will eventually understand depends largely on how much time and energy you're willing to put into understanding, as with pretty much anything substantial. Good luck! –  Ryan Budney Aug 24 '10 at 9:10
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+1 I don't know how good the answer is. I haven't read it as I know I won't be able to understand and I also don't have time to read it. But +1 for the efforts. –  iSid Nov 26 '10 at 14:05
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Generating two 100+ vote answers on the same question just amazes me :) –  rschwieb Feb 8 '13 at 21:52
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@rschwieb There should be a badge for that... –  gsingh2011 Nov 18 '13 at 20:11
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protected by Qiaochu Yuan May 14 '12 at 21:18

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