
Why can you turn clothing right-side-out? - ivoflipse
http://math.stackexchange.com/questions/2755/why-can-you-turn-clothing-right-side-out
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ctdonath
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.

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xyzzyz
I didn't quite get your comment (I'm a non-native speaker). Could you please
be more elaborate?

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jacobolus
Ctdonath was claiming that many people notice that a shirt can be turned
inside-out through the main hole at the bottom, or a glove through the wrist
hole. Far fewer people try turning a shirt inside-out through a sleeve or neck
hole, or a pair of pants through a rip in the knee. They then abstract the
wrong way from this experience, inventing the idea that the size of the hole
is important (or maybe just that the “main” hole is special).

I’m not sure how prevalent these misconceptions are. It would be interesting
to do surveys of the general public to see what kinds of mental models people
have here.

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philwelch
If you actually tried to turn a pair of pants inside out through a rip in the
knee, the rip might get bigger. In fact, it almost certainly would. Likewise
for turning a sweater inside-out through the neck hole, it would stretch it
out a bit. Yes, it's mathematically possible but there are real reasons people
don't treat their clothes that way even if they know they can.

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acheron
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.

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jacobolus
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.

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xyzzyz
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.

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grn
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.

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xyzzyz
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.

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grn
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.

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jes5199
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+Astero...](http://everything2.com/user/sockpuppet/writeups/Using+Asteroids+to+explain+the+topological+classification+of+2-manifolds)

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pfortuny
The real problem is: what would the world look like if the result were
different?

That looks more interesting to me.

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karamazov
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.

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ars
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.

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sakai
And this is why math is... awesome.

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verganileonardo
And this is why kids are awesome!

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josefonseca
Is this relevant?

[https://upload.wikimedia.org/wikipedia/commons/5/55/Tesserac...](https://upload.wikimedia.org/wikipedia/commons/5/55/Tesseract.gif)

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eru
Not really. That looks like a 2-dimensional projection of a rotating
4-dimensional hypercube. The StackExchange question concerns 3-dimensional
topology.

