Geometric Constructions with Origami

Yesterday I saw a great tweet from Evelyn Lamb:

I’d never see constructions using origami before, so the idea that you could trisect an angle using paper folding was brand new to me. One of Zsuzsanna Dancso’s comments in the Numberphile video made me believe that you could also solve the “doubling a cube” problem using paper folding. Sure enough that construction is here:

How to double a cube with origami from Cutoutfoldup.com

The combination of that construction and the construction in the Numberphile video made for a great Family Math topic for today.

I started by talking through some basic ideas of compass and straight edge constructions from Euclidean geometry. My older son and I have touched on this topic in our geometry work, but my younger son has never seen it. Because this topic was new to my younger son I didn’t want to go that much into detail. The main idea for today was to introduce the three famous impossible constructions: (1) doubling a cube, (2) trisecting an angle, and (3) squaring a circle and then show how to solve (1) and (2) in the last two videos.

Our first origami construction was solving the “doubling a cube” problem by constructing $\sqrt[3]{2}.$ The directions I linked above are really easy to follow, so each of us made our own version of this one:

The last thing we did was attempt the “trisect an angle” origami construction that Zsuzsanna Dancso demonstrates in the Numberphile video. This construction is a little bit more difficult than the doubling a cube one, so we worked on this one together rather than making three separate constructions. I also used a ruler to draw in some of the lines just to speed things up, but it is easy to see that we could make the same lines by folding.

So, not as much mathematical detail in this one, but some fun history and some fun constructions. I wasn’t aware of the idea of origami constructions before seeing the Numberphile video, so this project had a little bit of extra fun for me because the kids and I got to learn something new together!

Surprises you get watching kids do math

My older son is finishing up a chapter about right triangles in his Geometry book.  The last section in the chapter discusses how to construct perpendicular lines.  This topic has come up a few times previously, so I dedicated a bit more time to the previously section – Heron’s formula – before starting in with the new section this morning.  I also left one of the examples in the book for him to work through as part of his homework.

One of the other homework questions was to construct a segment with length $\sqrt{3}$ given a segment of length 1.  I thought this might be a little bit of a challenge and was pretty interested to see how he would approach it.

When I got home from work this afternoon I got a nice surprise:

Me: Were you able to construct the 30-60-90 triangle?

Him:  That wasn’t one of the homework questions.

Me:  Oh, I thought there was one about constructing a length equal to $\sqrt{3}.$

Him:  I made a 1 – $\sqrt{2}$$\sqrt{3}$ triangle for that one.

Funny how focused your mind gets when you “know” how to do something.  Wouldn’t have thought of this approach in a million years.  Nice little construction!