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I was wondering, when I want to change shirts in a public space, I can put another shirt on, then take my arms out, put them inside the other shirt, and take off the first shirt through my head hole. But I can’t do this with pants, or in this case underwear.

Are there any topological reasons or explanations why this is not possible?

I’m a first year math undergraduate.

J. W. Tanner
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Baltazar Perez
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    You can in fact take off your underpants without removing your trousers. You need to stretch them a little, but I've seen it done. – TonyK Dec 20 '25 at 18:07
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    @TonyK you move in sophisticated circles... – peter a g Dec 20 '25 at 18:07
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    The analogue of taking your arms out of the first shirt without taking it off would be taking your legs out of your pant legs without taking the pants off. Topologically the same, but geometrically makes less sense. – Karl Dec 20 '25 at 18:09
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    I think you can, topologically, the problem is the stretching required isn't feasible with normal cloth. – Thomas Andrews Dec 20 '25 at 18:34
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    Welcome to Math SE. Regarding doing basically the converse, i.e., putting on a swimming suit while wearing pants, you may wish to view the humourous, about $4$ minutes long, epsisode of Mr. Bean Has A Mishap At The Beach, starting at about the $14$:$30$ point. – John Omielan Dec 20 '25 at 19:09
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    This guy shows how to do it: https://www.facebook.com/watch/?v=742936172061940&vanity=Ericsurf6Channel – Gonçalo Dec 20 '25 at 20:19
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    Topologically, you could, but physiologically, you might dislike the experience: retracting extremities, you'd be a ball, and could easily remove any garment with at least one hole in any order you please. – Professor Vector Dec 20 '25 at 20:47
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    Retract your extremities until you're a sphere, then the answer is trivial: of course you can. – Ivan Neretin Dec 21 '25 at 12:25
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    Congratulations, your post has became a meme. – Brightsun Dec 22 '25 at 22:55
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    The late Prof. Robert E. Greenwood at the University of Texas at Austin used to amuse his students (I was one) by taking off his vest without removing his jacket. – John Wayland Bales Dec 25 '25 at 23:43

1 Answers1

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Topologically it is possible, you can even do it without ever lifting your left foot off the ground.

Problems like these are easier to visualize when you can deformation retract 3D objects into 1D: pants/underwear can be retracted to figure-eight spaces, while your body can be retracted to a parabolic arc for your legs (with everything above your waist squished into a single point at the apex).

Since the arc is contractible, any such clothing you put on could be removed in any order (assuming they are not linked of course, in which case it would be hard to take one off independently from another).

For an example of a similar, but topologically nontrivial problem, consider this: You can’t go from wearing “pants over underwear” to “underwear over pants” without either taking a foot off the ground, or stretching a garment around your head or the entire earth.

Ben
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    Doing it without ever lifting your left foot requires you to take the underwear off through your head (and other body parts). – Lab Dec 23 '25 at 16:54
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    @Lab Not necessarily - first remove the right side by stretching over your right foot, then transfer down your left pant leg. Now that your underwear is on the ground, only looped around your left ankle, you can stretch it around the entire earth to remove it. – Ben Dec 24 '25 at 02:08
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    How would one prove the latter assertion, though? – cody Dec 25 '25 at 22:05
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    @cody I posted as a separate question here https://math.stackexchange.com/questions/5116674/prove-there-is-no-isotopy-interchanging-a-pair-of-figure-eight-spaces-linked-wit – Ben Dec 25 '25 at 23:58
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    @Ben Touché, sir. I will personally try that next time. – Lab Dec 30 '25 at 04:06