Euclid in The School of Athens

This is a very special fresco painting by Italian Renaissance artist Raphael in the Vatican Museum called The School of Athens. It depicts the great philosophers of ancient Greece, famously with Plato pointing up and Aristotle pointing down.

The Mathematics Community on Google+ had our second ever Hangout On Air last week. I was joined by Luis Guzman, Jason Davison, and Amy Robinson in a conversation that ranged from fluid dynamics to applying the mathematics of networks to map our ideas to JPEG image compression.

While the target audience of this article is my fantastic calculus students, other math teachers might enjoy it as well.

Sneaky Continuous Functions

When students in first semester calculus first start learning about limits, they are often asked to determine limits using the graph of a function, which we will call the graphical method, and also by constructing a table of values of the function, which we will call the numerical method. Students should be warned that these methods, while perfectly legitimate and often quite useful, are really just fancy ways of guessing the value of the limit, that is, the graphical and numerical methods do not supply us with mathematical certainty regarding the value of the limit. After all, what if your function is very sneaky and merely looks like it’s approaching a value $L$ as $x$ approaches $c$ when in fact it ultimately approaches a different value $K$?

It’s over. Getting my internet fix in the hotel lobby at midnight, walking 18 miles a day through the convention center, sitting in uncomfortable chairs for hours, getting dinner in a local restaurant and realizing halfway through the meal that every single person dining there is also a mathematician, randomly bumping into an old friend or mentor or student–it ended today at around lunchtime when I made the trek back to the hotel lobby one last time.

I love the exhibit hall. I love books, and the exhibit hall is full of some of my favorite kinds of books. I spend hours picking through the texts, flipping through their pages. The AMS sold me on a great book directed at undergraduates on Fourier analysis and an advanced text on Riemannian manifolds. In fact, they managed to photograph me mid-purchase! I also learned they are selling their first ever children’s book this spring.

The best part of the Joint Meetings is networking with like-minded people. I ran into many old friends and colleagues. Today (Thursday) several math bloggers and Google+‘ers organized an impromptu meeting for lunch. I stuck a sign on the message board inviting other math bloggers to join us, and a cohort of tumblr bloggers discovered it and joined us. Such is the power of the JMM message board.

Today I learned about connections between musical rhythm and knot theory, abstract algebra and dance, a Navy ship from the mid 1800s called the U.S.S. Constellation, and some great undergraduate real analysis pedagogy. In today’s post, I’ll share with you some of the most interesting mathematical ideas I heard today.