Wikipedia:Reference desk/Archives/Mathematics/2017 November 21

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November 21

Turning things inside out

Take a watch with a metal watchband, like the ones pictured here, and you can easily turn it inside out. Take a bottomless and topless paper tube (e.g. a cereal box with flaps opened) and you can't. What's the difference? Is it related to the fact that the diameter of the watchband circle is a good deal greater than the width (i.e. your arm's circumference is many cm, while the band is 1cm or 2cm wide) while the cereal box's interior diameter is a good deal larger than its circumference? I came here because this question reminds me of chatting with a former roommate about his topology-related Ph.D. dissertation, where he was talking about turning mugs inside out, or something like that. Nyttend backup (talk) 16:29, 21 November 2017 (UTC)

I think it has more to do with the stiffness of the material. For example, if you take an empty bag of potato chips (made of paper or plastic) and open the bottom, it's fairly easy to turn it inside out, even though the dimensions are similar to those of the cereal box. On the other hand if the watchband were not flexible, but something like a rigid steel ring, it would not be possible to turn it inside out. This is not really a topological issue of course, because in topology the objects under consideration are assumed to be infinitely flexible and stretchable. CodeTalker (talk) 02:05, 22 November 2017 (UTC)

[Edit conflict.] This is more a function of material property than geometry. A tube with a similar aspect ratio to you paper tube, but made out of more pliant material such as thin rubber (a quarter of a bicycle's innertube) or fabric (a sleeve cut from a long sleeve shirt or a sock with the toe cut open), could easily be everted. -- ToE 02:06, 22 November 2017 (UTC)

Only tangentially related, but you may be interested in reading our sphere eversion article and watching the first video in § External links. Also, if you ever have an old bicycle tube to be discarded, first cut the valve out and then evert the punctured torus. (Some talcum powder may help the process.) This is described in § Topology, but it is very cool to see first hand, particular given the aspect ratio of an innertube. The end result is still a punctured torus, but of a very different appearance. -- ToE 02:06, 22 November 2017 (UTC)

See [1] about turning a tube inside out. This is the tallest that has been done with rigid plates. Dmcq (talk) 12:17, 22 November 2017 (UTC)