This is probably a bad thing to post on a thread about visualizing inverting a surface, but since I've been making my clothes for over 2 decades, I had to check this out. So I took off my trousers and pinned the cuffs together along the circumference (it just so happens that I'm making a new winter coat, so I had pins handy), matching the seams to each other. I grabbed the cuffs and pulled one leg up through the other. The result is that the pinned cuffs sit right at the crotch seam -- there is no way to completely invert the trousers any more than this, and I don't think it would be possible even if the pants had a higher lycra content, as there is no way to get the cuffs THROUGH the crotch to complete the inversion. However, if I grab the pinned cuffs through the waist band and flick the trousers away from me, they turn right side out with a lot less effort than it takes to turn them inside out.
I was able to do it with my jeans, so I expect it's possible with any trousers. You should note that the inversion is not symmetrical. You are basically pulling one pant leg into the other one. In the end you are left with one pant leg which is inside-out and straight and the other pant leg is inside of the first.
The inversion of the parameters is now noticeable. You started with basically a donut with a very wide hole (formed by the pant legs -- I'm ignoring the hole where your waist goes). After the inversion, the pant legs form a donut which is very narrow (just the width of the inside of the pant leg) and is very tall (the height of the pant leg).
I stared at the animated example for nearly two hours last night. I did get the one leg inside of the other donut, but it didn't /look/ like the animated example, so I disregarded it. Thank you -- "not symmetrical" did not occur to me -- it was bugging me. There's a lot of this same sort of visual thought that goes into designing various garments, but the surface is not a torus.
An impressive layman indeed, knowing to post a question to StackExchange under Topology.
And I mean that sincerely - that page has some pretty awesome 'layperson'-level (as much as it can get, I suppose) information on topology and manifolds in particular, for a an interested onlooker.
If you click on the edited date on Stack Exchange sites you can see the revision history. That one shows that Qiaochu Yuan changed the tag from "geometry" to "topology".
That was an impressive answer from Ryan Budney! ... Then I clicked to his homepage and discovered he is a math professor with interest in knots. I love the internet.
I took 3 terms of discrete math from him at the University of Oregon a few years back. He was a great instructor and really smart guy that knows more about knots than I realized could be known at the time.
I made a small search on youtube to see some videos related to turning a sphere inside out.
Please have a look at this gem : http://www.youtube.com/watch?v=R_w4HYXuo9M#
Especially the snelpiller's comment
I wish HN would show the sub-domain (math.stackexchange.com) because at first I thought there was a new "fashion" SE and I was starting to feel sick. Glad I clicked through though, very interesting stuff.
point in case, I got chided for posting a tex question on se instead of tex.se. Silly but that one irked me, tex looks a lot like code to me. On the other hand I did get prompt replies.
Not true—in the case of a punctured torus (like the shirt with joined sleeves), it's not like a sack at all!
It can still be turned inside-out, but it's a little trickier to visualize. You can imagine the hole stretching until you have a pair of identical rings glued together, then it's easy to see that you can shrink the hole the same way except in the other direction. But there's no way to make it look like a sack with holes.
Not necessarily. For all reasonable articles of clothing, yes, it's true. That's because clothing is designed to enclose an interior which is isomorphic to a sphere [1]. Thus, clothing is simply a sphere with holes.
But there are plenty of hypothetical clothes for which this isn't true. Consider a mobius strip - like a sack, there is only one hole, but a mobius strip isn't a sack (and you can't patch it shut).
[1] I don't want to think about clothing for which this isn't the case, since it would involve the digestive tract.
A real world example of a diffrent topology is is a dress whose straps intercect with one wrapping around the other so they from an X on someones back.
That picture is a "sack with a couple more holes" however if rather than cloth connecting the straps they tied together in a knot. In that case the knot is a topologically distant feature which is maintained regardless how you deform the garment.
I can't find a picture of it. However while this is still a "sack with a couple more holes" http://s7.kmart.com/is/image/Sears/049B018579550001?hei=500&... if you picture having that strap be wrapped around at the point of intersection it can still be unwound. However, if you form a slip knot at the intersection it can not be removed by deforming the garment.
It's a rather neat installer for Ubuntu that installs Linux on top of a working Windows system - allowing you to dual boot. The Windows install remains intact.
Also, all of the cash fell out of my pockets.