TITLE: The harmonic (series) beetle: live illustrations of mathematical theorems
QUESTION [8 upvotes]: In my analysis class I use the following problem to illustrate the divergence
of the harmonic series (consider this as a hint for solving it).
Exercise.
A beetle creeps along a 1-meter infinitely elastic tape with constant velocity.
Every hour the tape is lengthened out by 1 meter, and the beetle remains at the
same rate of the tape it has already reached to the moment. Will the beetle ever reach
the end of the tape?
This is not a paradox but a calculation of a mathematically idealised model
"from life". Do you have some other, probably nicer examples which illustrate
some standard but deep results in analysis, algebra, probability, geometry,
and so on, and so on, and so on. Please keep in mind an average undergraduate
student as the audience for your example and allow others to use it in his/her
teaching.
Thank you!

REPLY [2 votes]: How do you hold a pizza (with your hands) so that the toppings don't slide off? That is, how do you keep the pizza "straight"? You bend it through the middle and use one of your fingers to support the bend. Why does it work? Because a pizza is inherently "flat." You would expect any reasonable definition of curvature to give a (flat) pizza slice a 0. It turns out this expression (Gaussian curvature) is the product of two numbers, $\kappa_1$ and $\kappa_2$, corresponding to the two orthogonal curvatures. Thus, one of the two curvatures must be 0 -- so if you're bending one direction, the other must necessarily be "straight" (no curvature).