Archives For Euler’s characteristic

On Friday, November 2, I gave a 30-minute talk titled “Euler’s characteristic, soccer balls, and golf balls” during NAU’s Friday Afternoon Undergraduate Mathematics Seminar (FAMUS). This was my second FAMUS talk of the semester. You can read more about my first talk by going here. As the name of the seminar suggests, the target audience for the FAMUS talks are undergraduates. My last talk was well-received, but I wanted to discuss something a little “lighter.” Here is the abstract for my talk.

A typical soccer ball consists of 12 regular pentagons and 20 regular hexagons. There are also several golf balls on the market that have a mixture of pentagonal and hexagonal dimples. Both situations are examples of convex polyhedra. Loosely speaking, a polyhedron is a geometric solid in three dimensions with flat faces and straight edges. In this case, the faces are pentagons and hexagons. The adjective convex refers to the fact that a line segment joining any two points of the solid lies entirely inside or on the surface of the solid. For mathematicians, a natural question arises. Namely, what sorts of convex polyhedra can we build using only regular pentagons and regular hexagons? For example, is it possible to build a convex polyhedron using only regular pentagons? How about just hexagons? If we allow both, how many of each are possible? In this talk, we will explore these types questions by utilizing Euler’s characteristic formula for polyhedra, which establishes a relationship between the number of vertices, edges, and faces of a polyhedron.

And here are the slides.

I was inspired to start thinking about this topic while I was coaching my 4-year old’s soccer team this fall. As those of you with knowledge of the subject know, there’s so much more I could have said. However, 30 minutes isn’t a lot of time and I wanted to make sure that I discussed the proof of Euler’s characteristic slowly enough so that most of the audience could follow it.

I am currently overloaded with work, so I was planning to do a chalk talk and skip making any slides. However, the morning of the talk, I decided to make a couple slides that included pictures. Of course, as soon as I started dropping in images, I found myself adding text and before I knew it, I had slides for most of my talk. Only a few things didn’t make it in the slides. In particular, when I discussed the proof of Euler’s characteristic formula, I drew lots of pictures on the chalk board.

As a special treat, my mom, my mom’s husband, and both my sons were in the audience. I’m pretty sure this is the first time that my mom had ever seen me give a talk before and definitely the first time my kids had been to a math talk of any kind. It is customary for the FAMUS host, Jeff Rushall, to interview a faculty member after the talk. Typically, the speaker and the faculty member to be interviewed are not the same person, but this time, I did both. After Jeff asked his usual list of questions, the audience was allowed to ask me questions. It was a lot of fun.

Note: It just occurred to me that I cheated a little bit in a couple spots during my talk. See if you can figure out where.