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Can you give examples for mechanical system that has a Mobius strip as their configuration space?

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the configuration space of two unordered points on a circle $C$ is a Möbius strip --- not what you're looking for? –  Carlo Beenakker Jan 4 at 22:16
    
There are also applications of Mobius strips to electrical systems, see the Mobius resistor: en.wikipedia.org/wiki/M%C3%B6bius_resistor –  Mark Lewko Jan 4 at 22:44
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up vote 8 down vote accepted

Here is an example in which the Mobius strip is also physically visible. Namely, the old eight-track tape system, on which I listened to the Cars and Led Zeppelin as a teenager, has an endless tape with one twist, giving the basic tape-position configuration space the nature of a Mobius strip. (The configuration of the head position keeps track of an additional four positions across the tape.)

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am I right that this is actually the double cover of the Möbius strip? –  Carlo Beenakker Jan 4 at 22:26
    
Carlo, I'm not sure what you mean. What I had in mind is that the basic tape position (ignoring the four tracks across) is determined by a length marker on the the tape and an indication of which side is facing out, which would seem to be the Mobius strip. –  Joel David Hamkins Jan 4 at 22:38
    
I was just wondering about the fact that the tape has a certain thickness, while the mathematical Möbius strip has no thickness. –  Carlo Beenakker Jan 4 at 22:50
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Oh, I see. I was just thinking about the configuration space of the system, as it is used by the music player. The four tracks across (called eight I think because it switches to the next with one revolution of the tape ignoring sides), however, make the full configuration space more complicated, which I am hoping someone will describe succinctly. But meanwhile, the picture seems to show a Mobius strip inside the cartrage. –  Joel David Hamkins Jan 4 at 22:53
    
Thnks!!!!!!!!!! –  Mari Jan 8 at 16:27
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Imagine a rod of length d confined inside a spherical shell of diameter d and free to rotate within it. Provided the two ends of the rod are indistinguishable and the rod is cylindrically symmetric, the configuration space of the rod is the real projective plane (for an application of this sort of picture in nature, see this answer). If you glue a block to the inside surface of the sphere, this confines the rod further and now the configuration space is the Mobius strip.

picture

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Thnks!!!!!!!!!! –  Mari Jan 8 at 16:27
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