Pentagons are taking over my life.
You may have heard the announcement this summer that mathematicians found a new tiling pentagon. Previously, there were 14 known classes of convex pentagons that tile the plane. Now there are 15. Maybe that’s all there is; maybe there is another class, or even infinitely many classes, remaining. No one knows.
My Normandale colleague Kevin Lee brought some samples of this new pentagon to Math On-A-Stick this summer, mere days after the announcement. This led to discussing the nature of sameness of the pentagons with my father, which led to further reading, and so on…
I am now drawing an example of each of pentagon type using Geometer’s Sketchpad and Adobe Illustrator, cutting them out of wood on a laser cutter, and then figuring out how they go together. No phase of this project is simple.
I consider a pentagon “solved” if I have at least once figured out how it tiles.
I have successfully drawn and cut pentagons 1 through 11. I have solved all of these but number 9.
The project is making me think a lot about learning.
For example, tonight I was working on pentagon number 8. I solved it.
These sets of four can continue to go together in a way I see and can describe.
But that’s not the only way to view the solution. Maybe someone else solves it using sets of three.
This is the exact same arrangement—the same solution—organized differently. The threes are meaningful here, whereas the fours were meaningful in the first solution. Which is better? Which is right?
Another solution uses sixes.
With that set of six pentagons, you can tessellate by translation only. The three pentagons at lower right are the beginning of the next set of six. Each of these has the same orientation as its corresponding pentagon above it. Does that make it a better solution?
I’m thinking a lot these days about the kinds of questions I’ve posed here. I’m trying to sort out my answers to a larger question:
What is (or should be) the relationship between informal outside-of-school math, and school math?
I have given a couple versions of a talk that asks four basic questions about people’s mathematical activity that occurs outside of school:
- Is this math?
- Is it school math?
- Do we value it?
- Why or why not?
I invite you to join me on this journey.
I’ll keep you posted on the pentagon project.
Overthinking my teaching 