Patrick Honner shared an amazing post from Richard Green yesterday:
The post caught my attention for a couple of different reasons. First, the result is absolutely amazing, and following Green’s summary I clicked through and skimmed the original paper. It was really cool to see all 111 tilings. Second, the mention of the Paul Monsky result about triangles in a square was fascinating. Monsky was (and still is) a professor at Brandeis when I was in graduate school and he was always incredibly generous with his time and ideas. I found the Monsky paper with a quick google search and his proof is amazing (though pretty technical and not really something that you could share with kids).
Lastly, though, Green’s post intrigued me because I’m just finishing up a section about angles with my younger son and it sure seemed as though there was a project for kids hiding somewhere in this post. I tried one project idea with my younger son this morning.
The first thing we did was look at Green’s opening paragraph – “It is easy to cut an equilateral triangle into four smaller equilateral triangles . . . ” Perfect, let’s talk about that! Right away we get to have a fun little conversation about triangles and counting.
Following my son’s idea of how to chop up an equilateral triangle into smaller triangles, I had him build the object he described out of our Zometool set and I built an example that used a (slightly) different idea. He sees a pattern in the number of triangles that goes 1, 4, 16, 64, . . . . When you include the triangle that I showed him you get a different pattern 1, 4, 9, 16 . . . . So, we get a couple of nice patterns to talk through.
We also talk briefly about the Monsky result at the end of this video.
After that brief introduction, we moved to the end of Green’s post and I had my son talk about some of the shapes he saw. It is always fun to hear ideas that kids have about math, and these tilings are so cool that I’m sure that kids will have all sorts of really fun things to say about them.
Finally, let’s talk about some angles. We used the shape that caught his attention and then tried to calculate what some of the angles in the tiling. The first angles that he noticed were the right angles, and then the octagon at the center of the tiling caught his attention – what are the angles in that octagon?
After finding the angles in the central octagon, we went looking for one last set of angles to calculate, and my son chose the angles in one of the hexagons. This calculation is a tiny bit more difficult because not all of the angles are the same. I love hearing his ideas about how to find these angles, and also his surprise that two of the angles are actually right angles!
So, a super fun geometry project based on Richard Green’s post. It isn’t that often that you can use ideas from current math research in conversations with kids, but the ideas in Green’s post were just too good to pass up. Thanks to Patrick Honner for sharing the post yesterday and thanks to Richard Green for pointing out and explaining this amazing geometry paper.