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.
In September Laura Taalman put together a page to help 3D print all of the pentagon tilings. It is a wonderful way to explore these pentagons. We did a short math project (for kids) with a few of her prints:
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For me the informal out of school math is a possible reason to get involved or stay engaged in school math, and usually worthwhile on its own merits. Not every project is going to be for every kid, but math is so vast and diverse that I really believe that there is such a context for every person.
This one has plenty of school math for which it could be the context. Angles, lengths, transformations, etc. not even getting into the practices.
When I was playing with the pentagons in the summer (with Simon Gregg [who has great pattern block connections, too] and Daniel Ruiz Aguilera) I was amazed how hard it was to sort which pentagon pattern a given tiling was.
Is it school math?
When you ask this question, are you assuming there is universal consensus on school math? I think I can safely assume you do not.
So – is the question
Is it school math?
or is the question, really,
What is school?
What a great post, and a great question too.
Here are some questions I had, when I stopped to think about your’s: How do people come to see themselves as “math people”? How fragile is that identity in the face of lousy experiences in school math? Given the nasty role math plays in doling out economic benefits, is it possible to find space for a math identity that isn’t tied to all that baggage?
I became obsessed with pentagons a couple months ago, too. I heard about this new pentagon on NPR. I actually thought this was a great thing to bring into the classroom: a current event that shows that math IS happening (present tense), a testament to the benefits of persistence, and pentagons are fairly accessible. Conveniently, the discovery happened right here in WA state! We were able to arrange to have Dr. McLoud-Mann come in and speak to middle schoolers about her experience, and they had a hands-on afternoon with tiling as well.