Lesson 33: The Geometry of Braids—Curious Applications
The Geometry of Braids—Curious Applications
Lesson 33Topics
x Braids with no free ends.
x Dirac’s string trick.
x The waiter’s trick.
x The national mathematics salute.
Result
x If strings that are attached to an object and that object is rotated two full turns, then the tangle of strings
that results is physically equivalent to the beginning state of untangled strings. This is not the case if
the object were given only one full rotation.
Summary
This highly unusual lesson demonstrates that the physical effect of rotating an object through one full turn is
fundamentally different from rotating that object through two full turns. This topic usually does not appear in
any school or undergraduate curriculum, but it is richly engaging, completely accessible, and shows a side of
mathematical thinking that will astound.
Example 1
In the lesson, we proved that it is impossible to untangle three strings attached to a teacup if that teacup has
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underwent one full turn?
Solution
If it were possible to untangle four strings, then it would be possible to untangle three: Simply imagine a fourth
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