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!)
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.
You can learn more about the fundamental group in a topology course.
How to Cite this Page:
Su, Francis E., et al. "Mug Trick."
Math Fun Facts.