Discrete Math - Day 55

Students presented their cryptology research papers to the class. The presentations are to highlight the topic they wrote about in their report.

Most of the presentations were well structured and indicated an appropriate level of research into the topic. One group tackled women in cryptology and drew comparisons of the participation and growth in this field, especially during and after World War II with growth in women's rights and equality. Several groups looked at Bletchley Park and the enigma machines used during World War II. A couple of groups focused on the Navajo code talkers used by the United States Marine Corps during World War II. Students made some solid connections to the topics we studied in class and asked additional questions about different topics.

I then used some time to show more of the video The Music of the Primes. Specifically, I started to show segment 3, which deals with Alan Turing, his work at Bletchley Park, and his development of an electronic calculating machine, the fore-runner of the modern computer. Besides connecting directly to many of the presentations made it also made more explicit connections to topics we studied and to our world.

I pointed out to students that one aspect of this story that struck me was that Alan Turing conceived of his idea of a computational machine as a tool in helping to prove or disprove the Riemann Hypothesis, a purely theoretical undertaking with no practical value at the time. As it turned out the computational machine became the foundation for modern computing. When students ask why they are studying a topic and when will it ever be used, I think about a situation like this where it may not be used for decades or centuries, but it can eventually be of great value. This is an aspect of human growth, the pursuit of knowledge and insight that pushes us, collectively, to greater things. The ancient Greeks looked at many purely theoretical topics in mathematics that had no practical purpose, yet two millennium later so of these purely theoretical topics are so pervasive in our daily lives that we do not even take notice. Sometimes we examine problems and dive into topics for the pure adventure of where they might take us, challenging our thinking and pushing our experience, exploring uncharted landscapes to make discoveries not thought possible.

To wrap up class, we plunged back into graph theory. I started by having students share what they had drawn for the problem of six vertices that contained four loops and two multiple edges. There was quite a variety of drawings and students could see and appreciate that there were many different graphs that could represent the same situation.

We then tackled a graph with five vertices all of which had degree four and without any loops or multiple edges. Students struggled with coming up with this graph. Many wanted to draw segments coming out of a vertex but not connecting to another vertex. This was an opportunity to revisit the definition of an edge, which states that it is a segment connecting two vertices. Finally, I drew a pentagram and connected each vertex to every other vertex. I asked the class if this met all the criteria. After a little consideration the class agreed that it did meet the criteria.

I then asked if there was another graph that could also work. They jumped into attempting to create another version. Some people made elongated pentagons but then realized they had just stretched the figure. As class wound down, I drew a second figure, in essence folding one of the pentagon vertices inside the pentagon. The class agreed that this worked, but only after arguing that the figure contained multiple edges. Once they realized the edges were not multiple ones they realized that this figure worked as well.

We'll continue working through some of these. My goal is to build comfort in the class with the definitions and terms used in graph theory.

Visit the class summary for a student's perspective and to view the lesson slide.