On Friday, September 20, I gave a 30-minute talk titled “Proofs without Words” during NAU’s Friday Afternoon Undergraduate Mathematics Seminar (FAMUS). As the name of the seminar suggests, the target audience for FAMUS is undergraduates. I usually give a couple talks at FAMUS each semester and this was my first of the semester. Here is the abstract with words for my talk.

In this FAMUS talk, we’ll explore several cool mathematical theorems from a visual perspective.

The talk basically went like this. I displayed a figure or drawing and then the goal was for the audience to come up with the corresponding theorem. I had a ton of fun and the audience seemed to enjoy it. The initial idea for the talk came from the book Charming Proofs: A Journey into Elegant Mathematics by Claudi Alsina and Roger B. Nelsen. This is a wonderful book that incorporates lots of visual proofs. If you don’t have a copy, I highly recommend it. I borrowed lots of ideas from it when I taught a class titled “Introduction to Formal Mathematics” while I was at Plymouth State University.

My original plan was to recreate a lot of the figures I had in mind using TikZ, but I should have known that I wouldn’t have time for that. When I was brainstorming the talk a couple days before, I decided to do a Google search in the hope that I could find some figures to borrow that others had already made. In my search, I stumbled on several references to “proof without words”, which is what I ultimately named my talk. In fact, there is a Wikipedia entry and Roger B. Nelsen also wrote a book called Proofs without Words: Exercises in Visual Thinking. Moreover, I was thrilled to find lots of cool figures on the Internet. For my talk, I borrowed images and content from the following sources:

In the end, my slides ended up being a sample each of these three sources. If you are ready to see some cool figures, check out the slides below. I’ve left the pauses in so that you can ponder the theorem before you see it.

As anticipated, I had more figures than I had time to discuss, but we did get through most of them.

The custom at FAMUS is to interview a faculty member after the 30 minute talk. The usual FAMUS host, Jeff Rushall, was out of town, so I was filling in for him. I had the honor of interviewing Michael Falk. This was fun for me because Mike was my masters thesis advisor.


Dana C. Ernst

Mathematics & Teaching

  Northern Arizona University
  Flagstaff, AZ
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Land Acknowledgement

  Flagstaff and NAU sit at the base of the San Francisco Peaks, on homelands sacred to Native Americans throughout the region. The Peaks, which includes Humphreys Peak (12,633 feet), the highest point in Arizona, have religious significance to several Native American tribes. In particular, the Peaks form the Diné (Navajo) sacred mountain of the west, called Dook'o'oosłííd, which means "the summit that never melts". The Hopi name for the Peaks is Nuva'tukya'ovi, which translates to "place-of-snow-on-the-very-top". The land in the the area surrounding Flagstaff is the ancestral homeland of the Hopi, Ndee/Nnēē (Western Apache), Yavapai, A:shiwi (Zuni Pueblo), and Diné (Navajo). We honor their past, present, and future generations, who have lived here for millennia and will forever call this place home.