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© 1999-2010 by Francis Edward Su

From the Fun Fact files, here is a Fun Fact at the Medium level:

# Mug Trick

Take a mug that has a handle, and fill it with coffee. Hold the mug in your hand. Now, without letting go of the mug (or changing the position of your mug relative to your hand), and without spilling the coffee, see if you can rotate the mug TWO FULL TURNS and return your hand, arm, and cup to their original positions.

If you can do that, can do it the same trick with only ONE full turn? (No!)

Presentation Suggestions:
Do it! Just don't fill the mug with hot coffee! You may wish to point the handle north at the beginning of the trick, so that students can track how many times the mug turns. Now gradually rotate the cup towards your chest, then under your elbow... this will bring the cup through one rotation, but your arm will be twisted. Keeping the cup upright and rotating in the same direction will now force you to rotate the cup over your head. Then bring the cup down and you will have given it two full turns and brought your arm back to its starting position.

Students will enjoy knowing that this trick is evidence of some deep mathematics, even if they don't understand it, so be sure to tell them about the math. Also, even if students only pick up the "buzzwords" (like "fundamental group"), it is still useful, because it all helps to popularize mathematics. (How many of us know what "string theory" really is? But hasn't that given physics a lot of glamor?)

The Math Behind the Fact:
See the Fun Fact on the Fundamental Group. Continuously rotating a mug is equivalent to following a path in "SO(3)", the space of rotations in 3-space, and if you start and end the mug in the same orientation, you have traced a "loop" in SO(3).

The reason this trick works for 2 twists but not 1 twist is because the fundamental group of SO(3) is "Z/2", the cyclic group of order 2. In layman's terms, this means that of all the ways you can twist a mug and return it to its original position, in only half of them will you be able to return your whole arm to its original position, and these are precisely the ones in which the mug rotates an even number of times.

Su, Francis E., et al. "Mug Trick." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

References:
Kauffman's On Knots, or
Bredon's Geometry and Topology.

Keywords:    fundamental group, topology
Subjects:    algebra, geometry, topology
Level:    Medium
Fun Fact suggested by:   Joshua Sabloff
Suggestions? Use this form.
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