Wednesday, October 26, 2016

Math Circle: Mobius Strip and its Hexa-Flexagon Connection

From our teacher:


Today we worked on three experiments for Mobius strip (band).
 
1) Showing it has only one side: We compared regular strip and mobius strip by drawing a continuous line on (one of) the surface(s) for each strip. 
2) Cut a mobius strip in half to see what happens.
3) Cut a mobius strip at 1/3 distance from the right edge to see what happens.

We connected the math involved in this to three sided hexa-flexagon by discussing mobius strip like surface obtained by rotating one end of the paper strip three times (540 degrees) instead of one time (180 degrees).

We looked at two dimensional projections ( pictures) of twisted bands made with 180 and 540 degrees twists. I am attaching the pictures.

Then we made three sided flexagons. The goal here was not only should kids be able to make a 3 sided hexa flexagon without directions, but should grasp the 540 degree twisting that leads to the flexagon. 


Those who want to explore this further should see what happens when they make strips involving 2 or 3 or 4 or 5 twists. When do they get two surfaces? When one? Why so?

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