Exploring the Relationship between Spinors and Mobius Strips in Rotations

In summary, the conversation is about the relationship between spinors and rotations, and whether there are closed paths that can rotate someone back to their original orientation. The speaker also mentions the Spin groups and their connection to the rotation groups of Euclidean space.
  • #1
jfy4
649
3
Hi,

I was pondering a bit about the mobius strips and I was wondering if there is a relationship between spinors and there transformation under rotations and that, in a manner of speaking, one must go around a mobius strip twice to return to the original position. To me it seems there would be some underlying relationship here.

I was thinking, abstractly, some sort of closed path, [itex]\gamma[/itex], with non-zero torsion; such that after traversing the loop once, one has been spun around half way, and then after a subsequent traversal, another half to return to the original orientation.

Are there such closed paths such that, say, after one traversal the torsion of the path returns one to their original orientation (a full rotation) and other paths like the one mentioned above.

Thanks in advance,
 
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  • #2
Not sure about your question but the Spin groups are two fold covers of the rotation groups of Euclidean space. I don't know much about these groups but SO(3) is covered by the 3 sphere and is homeomorphic to the 3 dimensional real projective space. Projective space has fundamental group Z/2Z that arises from a cross cap around its equator. A cross cap is a Moebius band with a disk attached to its bounding circle.
 

1. What is a Mobius strip and how is it different from a regular strip of paper?

A Mobius strip is a mathematical object that has only one side and one edge, while a regular strip of paper has two sides and two edges. This unique property of the Mobius strip is due to its twisted shape.

2. How is a Mobius strip related to spinors?

Spinors are mathematical objects that are used to describe the rotation of particles in quantum mechanics. The shape of a Mobius strip is closely related to the concept of spin, which is a fundamental property of particles that determines how they behave when they are rotated.

3. Can a Mobius strip be made in real life?

Yes, a Mobius strip can be made in real life by taking a strip of paper, giving it a half twist, and then connecting the two ends together. This results in a shape that has only one side and one edge, just like the mathematical object.

4. What are some real-life applications of Mobius strips?

Mobius strips have been used in various fields such as engineering, biology, and art. Some examples include conveyor belts, DNA molecules, and Möbius bridges. They can also be found in everyday objects like pasta, paperclips, and circuit boards.

5. Are there any other variations of the Mobius strip?

Yes, there are several variations of the Mobius strip, such as the double Mobius strip, the triple Mobius strip, and the Klein bottle. These variations have multiple twists and can result in even more unique properties, such as having multiple sides and edges.

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