Make a Real Projective Plane (Boy’s Surface) out of Paper

I am teaching an undergraduate course in topology. We are now looking at what we get if we take a square and glue the sides together. (These are called identification spaces.) We are assuming that our spaces are made out of very stretchy rubber. So, if the space begins as a square, we could, for instance,…

The Magnificent Möbius Band

As I write this blog post, we are all either struggling with the impact of the COVID-19 virus or waiting nervously as cases start to rise in our area. I am currently teaching remotely. My college students are scattered around the globe, and we are interacting through various online methods. This semester I am teaching…

BraidTiles—A Mathematical Braid Puzzle

We can view braids mathematically as n strings hanging from a horizontal bar.  Each piece of string runs downward and can cross neighboring strings. In the 1920s Emil Artin observed that braids of n strings form an algebraic group. To “multiply” two braids, we append the bottom of one braid with the top of another braid. The identity element in this group…

Zip-Apart Möbius Bands

I’ve taught topology many times. One of the highlights for the students (and for me) is the investigation of the Möbius band—the one sided, one edged, non-orientable surface with boundary. On the day we introduce the Möbius band I bring many strips of paper, clear tape, and scissors and have the students make conjectures about what…

How I teach topology: an inquiry-based learning approach

Recently I’ve had a number of people ask for more information about how I teach topology. I’ve taught it five times using a “modified Moore method” or “inquiry-based learning” approach. I’ve modified it each time, trying to work out the bugs. I think it is pretty successful now. Context. At our college all math majors…

Cantor set applet

I made this Cantor set applet for my Real Analysis class. It is nothing fancy, but it saves me from drawing it on the board.

Beautiful theorems about dynamical systems on the plane

I was reading through some papers written by my Ph.D. advisor (John Franks) from the early 1990’s and was reminded of a few beautiful results about the dynamics of planar homeomorphisms. So I thought I’d share them here. For those of you who are not familiar with the terminology, a planar homeomorphism is a bijective…

Furstenberg’s topological proof of the infinitude of primes

I just returned from a 10-day trip to India. It was my first visit there. I gave a talk at the ICM Satellite Conference: Various Aspects of Dynamical Systems. The conference was hosted by the Department of Mathematics at the M. S. University, Baroda, which is in Vadodara (formerly Baroda) in the Indian state of…

Mathematical magic tricks for kids

My six-year-old son loves the website ActivityTV.com, especially their science, origami, cooking, and magic videos. I watched a few of the magic how-to videos with him and was pleasantly surprised to see that some of them had a distinctly mathematical feel to them. For example: Jumping rubber bands: topological properties of circles and linked circles…

Turing’s topological proof that every written alphabet is finite

Recently one of my colleagues was reading Alan Turing‘s groundbreaking 1936 article “On Computable Numbers with an Application to the Entscheidungsproblem.” This is the article in which Turing introduced the turing machine, solved Hilbert’s Entscheidungsproblem (`decision problem’), and proved that the halting problem is undecidable. It is viewed by many as the foundation of computer…