Sunday, May 20, 2018

Tangles and Symmetry

This is a description of Dr. David Pengelley's talk "All Tangled up and Searching for the Beauty of Symmetry" which I just attended. This makes an excellent Math Circle topic and I think I might use it next year.

Materials



1. 3 different colors of dental floss or some equivalent string cut into 3 foot lengths.
2. A 6-8 inch wooden dowel
3. A note card with 3 holes for the strings and with an X on one side and an O on the other.

Its helpful to tape the string in bundles of 3 to keep them from tangling before handing them out.

Notecard Explorations

1. First hand out the note cards. Have the kids explore how many different states of the note card there are with same orientation.

There are 4 (2 with the x showing on top and bottom and 2 with o showing on top and bottom.) Note: the rectangle must stay with the longer side vertical.





2. Next explore how many ways there are to transition between states.


  • Flip, Rotate and Spin and most importantly None. 


Make a an operation chart to explore the transitions:




Things to look for: closure, identity element, commutativity. i.e. this is an Abelian Group.

Full Space (Quarternions)

3. Next add the strings. Each string should be threaded through the prepunched holes and tied to the dowel.





4. Next explore whether anything has changed. Key question are the strings the same after a flip, rotate etc?

Introduce notion of clockwise/counterclockwise transitions. Also Full rotation = 2 in one direction.

5. Can we get back to the original state w.r.t to the strings and card after 2 and 4 rotations? You are allowed to move the strings but not the dowel or card and the strings may only move around the card. Split group in half and have each piece work on part of the problem. Remember to pre-practice play with the transitions before hand so they are very familiar.




(Only 4 rotations works. Several ways to move the strings to prove it. The easiest is to take all of them and move them around the card.) This takes some time.

6. What state does a flip / rotate end up in. (Either a forward or backwards flip.

7. Build the new operations table:







8 states: None, Full Turn, Rotate Clockwise, Rotate Counter Clockwise, Forward Flip, Backwards Flip, Clockwise Spin, Counter Clockwise Spin.

Note new patterns: This is not commutative for example. How do both tables relate to addition and subtraction?

Wrap up: Tie to Algebraic Group and Quarternion I like bringing in a bit of math history about Hamilton and the Broome Bridge. See: https://en.wikipedia.org/wiki/History_of_quaternions





















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