8 ms·

Why can you turn clothing right-side-out?

145 points by ivoflipse 14y ago

sakai 14y ago
And this is why math is... awesome.
- verganileonardo 14y ago
  And this is why kids are awesome!
acheron 14y ago
Love this question and all the answers. I find topology fascinating even though I understand maybe 5% of it. I took a decent amount of math as part of my CS degree, but beyond basic calculus it was concentrated in probability, stats, and linear algebra; never came near topology. In hindsight I wish I had taken more math, but as a 19-20 year old student at the time, I was happy to be done with it. If I invented a time machine, sometimes I think my second use of it would be to give my college self class choice and scheduling advice.
- jacobolus 14y ago
  Topology is fun, and not especially mysterious if you give it the time. One of those subjects that is at the same time highly abstract and exercises the visual–spatial–kinetic thinking part of your brain. You can learn it whenever you like! (For a motivated student, I think self study of mathematics topics is better than a course with fixed problems anyhow, because you can move at your own pace, and take any path you prefer. Requires some focus though.) I don’t have enough experience with all the various textbooks to compare them, but I thought Munkres was alright.
  - xyzzyz 14y ago
    It depends, modern algebraic topology is frequently considered to be one of the most mysterious and abstract fields of math, along with things like algebraic geometry.
    - grn 14y ago
      I wouldn't compare algebraic topology to algebraic geometry. In my opinion the later is much more difficult both one the conceptual and technical side.
      - xyzzyz 14y ago
        Really? I don't find conceptual and technical difficulty of considering (co)homology theories determined by spectras/infinite loop spaces like K-theories and cobordism theories, study of stable homotopy theories, spectral sequences, triangulated categories etc. to be much smaller that the difficulty of algebraic geometry. As I said, I find them both to be really abstract and mysterious.
        grn 14y ago
        Many concepts from algebraic topology can be applied to algebraic geometry after appropriate redefinitions. These new tools seem harder to use than originals. For example I find it technically more difficult to work with étale fundamental group than with a topological fundamental group. Similarly cohomology of a topological space looks simpler than cohomology of a scheme with coefficients in a sheaf. There are many similarities but working with schemes and sheaves is more troubling (at least for me). This of course can follow not from intrinsic difficulty of the subject but from my ignorance and inappropriate intuitions.
- mturmon 14y ago
  Nice observation, and 100% agree. Here's how one of the most painstaking explicators put it in the thread: "I'm summarizing about 200 years of mathematics, almost none of which is standardly taught to undergraduates at almost any university." which made me feel a little better.
  - oggy 14y ago
    I majored in math with a minor in CS (undergrad). Of his entire technical explanation, I'm only familiar with the notions "automorphism" and "topology". To add insult to injury, I did have a course on topology, but it covered only so-called "point-set" topology, and not the algebraic variety thereof.
ctdonath 14y ago
It's a matter of people imputing notions of "special" where they don't belong. Hence the importance of "proof" in math. ETA: People are surprised that they can turn a sweater inside-out thru a sleeve or neck hole ONLY because they've imputed a "special" ability to the largest hole in the garment. The mathematical concept of "proof" strips away such imputations, leaving surprisingly unsurprising results - in this case, you can reverse a garment by pulling it thru one of its holes, be it the largest, smallest, or even a tear, because they're all just holes with nothing inherently topologically special about them. In a larger social concept: people tend to impute special attributes to various things where such attribution is not warranted; people who understand the concept of mathematical proof are less likely to get caught up in such incorrect imputation.
- xyzzyz 14y ago
  I didn't quite get your comment (I'm a non-native speaker). Could you please be more elaborate?
  - karamazov 14y ago
    It's not because you're a non-native speaker - that comment doesn't make sense.
    - wging 14y ago
      The comment is basically saying that the size of a hole in a surface is irrelevant, but people incorrectly assume it matters.
      - zedadex 14y ago
        That's what I got from it as well; however it did seem needlessly verbose.
        ctdonath 14y ago
        karamazov's comment was about the first line, which wasn't. His prompted me to add the ETA, which is verbose to be clear and to address more than just clothing reversal.
pfortuny 14y ago
The real problem is: what would the world look like if the result were different? That looks more interesting to me.
- karamazov 14y ago
  For the result to be different, the real world would need to have a different sort of topology. For example, if we lived in a non-Hausdorff space, the result would potentially be different. A Hausdorff space is one where for any two different points, you can take a small ball around each point, and the two balls won't intersect. A non-Hausdorff space is just a space where this isn't true for at least one pair of points. So, for example, if wormholes are real, the space we live in might not be Hausdorff. And if you had a wormhole sewn into your shirt in the right way, you wouldn't be able to turn it inside out.
- adavies42 14y ago
  thank you! i love asking that question of math results.
josefonseca 14y ago
Is this relevant? https://upload.wikimedia.org/wikipedia/commons/5/55/Tesseract.gif https://upload.wikimedia.org/wikipedia/commons/5/55/Tesserac...
- eru 14y ago
  Not really. That looks like a 2-dimensional projection of a rotating 4-dimensional hypercube. The StackExchange question concerns 3-dimensional topology.
- adavies42 14y ago
  owww, my head....
jes5199 14y ago
a fun introduction to topology is this article "Using Asteroids [the game] to explain the topological classification of 2-manifolds" http://everything2.com/user/sockpuppet/writeups/Using+Asteroids+to+explain+the+topological+classification+of+2-manifolds http://everything2.com/user/sockpuppet/writeups/Using+Astero...
ars 14y ago
When I think of topology the first I do is remember that size means nothing in topology. So the long sleeve of the shirt? Shrink it right down - you are left with just a hole, and no sleeve. Then flatten out the curvature of the neck and other parts, and you are left with a flat sheet of clothing, with two holes in it. There is no inside or outside to this, meaning the two sides are interchangeable, and that's why you can turn the real shirt inside out.

145 points by ivoflipse 14y ago

sakai 14y ago

And this is why math is... awesome.

verganileonardo 14y ago
And this is why kids are awesome!

acheron 14y ago

Love this question and all the answers. I find topology fascinating even though I understand maybe 5% of it. I took a decent amount of math as part of my CS degree, but beyond basic calculus it was concentrated in probability, stats, and linear algebra; never came near topology. In hindsight I wish I had taken more math, but as a 19-20 year old student at the time, I was happy to be done with it. If I invented a time machine, sometimes I think my second use of it would be to give my college self class choice and scheduling advice.

jacobolus 14y ago
Topology is fun, and not especially mysterious if you give it the time. One of those subjects that is at the same time highly abstract and exercises the visual–spatial–kinetic thinking part of your brain. You can learn it whenever you like! (For a motivated student, I think self study of mathematics topics is better than a course with fixed problems anyhow, because you can move at your own pace, and take any path you prefer. Requires some focus though.) I don’t have enough experience with all the various textbooks to compare them, but I thought Munkres was alright.
- xyzzyz 14y ago
  It depends, modern algebraic topology is frequently considered to be one of the most mysterious and abstract fields of math, along with things like algebraic geometry.
  - grn 14y ago
    I wouldn't compare algebraic topology to algebraic geometry. In my opinion the later is much more difficult both one the conceptual and technical side.
    - xyzzyz 14y ago
      Really? I don't find conceptual and technical difficulty of considering (co)homology theories determined by spectras/infinite loop spaces like K-theories and cobordism theories, study of stable homotopy theories, spectral sequences, triangulated categories etc. to be much smaller that the difficulty of algebraic geometry. As I said, I find them both to be really abstract and mysterious.
      - grn 14y ago
        Many concepts from algebraic topology can be applied to algebraic geometry after appropriate redefinitions. These new tools seem harder to use than originals. For example I find it technically more difficult to work with étale fundamental group than with a topological fundamental group. Similarly cohomology of a topological space looks simpler than cohomology of a scheme with coefficients in a sheaf. There are many similarities but working with schemes and sheaves is more troubling (at least for me). This of course can follow not from intrinsic difficulty of the subject but from my ignorance and inappropriate intuitions.
mturmon 14y ago
Nice observation, and 100% agree. Here's how one of the most painstaking explicators put it in the thread: "I'm summarizing about 200 years of mathematics, almost none of which is standardly taught to undergraduates at almost any university." which made me feel a little better.
- oggy 14y ago
  I majored in math with a minor in CS (undergrad). Of his entire technical explanation, I'm only familiar with the notions "automorphism" and "topology". To add insult to injury, I did have a course on topology, but it covered only so-called "point-set" topology, and not the algebraic variety thereof.

ctdonath 14y ago

It's a matter of people imputing notions of "special" where they don't belong. Hence the importance of "proof" in math. ETA: People are surprised that they can turn a sweater inside-out thru a sleeve or neck hole ONLY because they've imputed a "special" ability to the largest hole in the garment. The mathematical concept of "proof" strips away such imputations, leaving surprisingly unsurprising results - in this case, you can reverse a garment by pulling it thru one of its holes, be it the largest, smallest, or even a tear, because they're all just holes with nothing inherently topologically special about them. In a larger social concept: people tend to impute special attributes to various things where such attribution is not warranted; people who understand the concept of mathematical proof are less likely to get caught up in such incorrect imputation.

xyzzyz 14y ago
I didn't quite get your comment (I'm a non-native speaker). Could you please be more elaborate?
- karamazov 14y ago
  It's not because you're a non-native speaker - that comment doesn't make sense.
  - wging 14y ago
    The comment is basically saying that the size of a hole in a surface is irrelevant, but people incorrectly assume it matters.
    - zedadex 14y ago
      That's what I got from it as well; however it did seem needlessly verbose.
      - ctdonath 14y ago
        karamazov's comment was about the first line, which wasn't. His prompted me to add the ETA, which is verbose to be clear and to address more than just clothing reversal.

pfortuny 14y ago

The real problem is: what would the world look like if the result were different? That looks more interesting to me.

karamazov 14y ago
For the result to be different, the real world would need to have a different sort of topology. For example, if we lived in a non-Hausdorff space, the result would potentially be different. A Hausdorff space is one where for any two different points, you can take a small ball around each point, and the two balls won't intersect. A non-Hausdorff space is just a space where this isn't true for at least one pair of points. So, for example, if wormholes are real, the space we live in might not be Hausdorff. And if you had a wormhole sewn into your shirt in the right way, you wouldn't be able to turn it inside out.
adavies42 14y ago
thank you! i love asking that question of math results.

josefonseca 14y ago

Is this relevant? https://upload.wikimedia.org/wikipedia/commons/5/55/Tesseract.gif https://upload.wikimedia.org/wikipedia/commons/5/55/Tesserac...

eru 14y ago
Not really. That looks like a 2-dimensional projection of a rotating 4-dimensional hypercube. The StackExchange question concerns 3-dimensional topology.
adavies42 14y ago
owww, my head....

jes5199 14y ago

a fun introduction to topology is this article "Using Asteroids [the game] to explain the topological classification of 2-manifolds" http://everything2.com/user/sockpuppet/writeups/Using+Asteroids+to+explain+the+topological+classification+of+2-manifolds http://everything2.com/user/sockpuppet/writeups/Using+Astero...

ars 14y ago

When I think of topology the first I do is remember that size means nothing in topology. So the long sleeve of the shirt? Shrink it right down - you are left with just a hole, and no sleeve. Then flatten out the curvature of the neck and other parts, and you are left with a flat sheet of clothing, with two holes in it. There is no inside or outside to this, meaning the two sides are interchangeable, and that's why you can turn the real shirt inside out.