Maths and music

Task 3c: Group theory and dancing

1. Take a minute or two to convince yourselves that these two are the same: that A B C D A C B D is the same as b, the reflection of the original square in the diagonal from top left to bottom right. Then convince yourselves that doing two inner twiddles is the same as doing b twice, and that it gets you back where you started. (This is the dance move dos-e-dos, from the French for back-to-back).



2. Now see if you can work out how the line of dancers A B C D should be reorganised to correspond to c, the reflection about the vertical axis (see below). We’ll call this dance move an 'outer twiddle'. Be prepared to demonstrate this when you come back on camera!

   c is the operation:

   

   c corresponds to the dance step:

A  B  C  D B  A  D  C

so the two outer dancers change places with their neighbours.

3. Combining the inner and outer twiddles gives us a much better dance. Try this out as a dance routine, and be prepared to show us all what you think it looks like when you return to the videoconference!



4. Satisfy yourselves that bcbcbcbc = aaaa = e (this comes from the work you did in Task 3b).



5. Now do the dance move which corresponds to bc.
   • What mathematical operation is this on the square?
   • What happens if you repeat this four times, both on the square and on the dance floor? Do you get back where you started?
     It might help if you write down a list of where each dancer is after each move, and you might find it easier to keep track of things if all your dancers hold up a sheet of paper with the letter corresponding to their starting position on it. (Remember that e means ‘do nothing’, or equivalently, ‘get back to where you started’).
   • Again, be prepared to show everyone your dance routine when you return to the videoconference!

Answers to Task 3c
Now return to Chris' talk

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