What is a Sangaku?: A Learning and Reasoning Memoir in Attempting to Find the Answer

Opening up the course page for our “Day 20” work, I found myself a bit at a loss with what I saw. On the page were some colorful, yet complicated, diagrams of geometric shapes. At first, reading about this Sangaku I was required to make, I thought, “It’s a circle inside of a triangle. What’s so hard about that?” Then I saw that the shapes were supposed to stay inside one another, growing and changing as another shape grew and changed. This is when I thought, “How am I supposed to make that?”

With our work time in class, I was still at a loss. Professor Golden had made a fairly valiant attempt at explaining what to do, but since I have a small issue called “impatience,” I sort of paid attention while trying to make headway on the Sangaku. This ended up being much more difficult than I expected. For one thing, the circle didn’t grow and change as my triangle grew and changed, and I wasn’t sure why.

Asking Professor Golden, he directed me to the conics section, where I was to create a parabola and use that to find the centers of circles. By the end of class, I had a pretty good diagram.

Then I made a mistake. I didn’t save it, I forgot how to do it, and then I went on my merry way. The other mistake there was that I had had a test the next Thursday, so I devoted all my efforts to understanding abstract algebra and pushed Sangaku reasoning to the back of my mind. Then, once I went to thinking about it again at 9 pm on a Thursday night, I had completely forgotten how to create the shape I had been working on.

After a lot of thinking, Chrome crashes, and headaches, I finally came up with a shape that satisfied the definition of Sangaku. It grows and changes if I move the elements of the shape, and it looks pretty cool besides.

How did I do this? Well, let’s walk through it.

I started by creating a square. This is a lot harder than one might think since it involves quite a bit of moving around to get all the lines to be the same length.

I then plotted both angle bisectors and perpendicular bisectors. The ultimate goal of doing this was to see where center of the circle that is designed to “kiss all sides” of the square would go.

I then plotted a center point in the middle of my angle and perpendicular bisectors. I used this point as the “focus” of my parabolas and plotted four parabolas, each with a directrix that was each side of the square. The points where the parabolas intersected the square, also plotted below, served as the points where the circle “kissed” the sides of my square.

I used this circle’s “kiss” points to plot four more parabolas, where each point was the focus and the directrix was the corresponding bisector line that was straight across from it. I also plotted the intersection points where the parabolas used to plot the original circle and the new parabolas met, which would now serve as the centers of my two new circles. I then plotted the circles, using the intersection points as my centers and the “kiss” points as the point where the circle stopped.

Finally, I plotted two more parabolas. I used the end of one parabola, with endpoint labeled “N”, and another with the endpoint “P”, as my foci. My directrixes were the corresponding lines of my square. I then plotted my two final circles using where the parabola’s vertex crossed the perpendicular bisector, which I labeled accordingly.

I then used the endpoints of my square to make semicircular arcs. This ends up forming the whole Sangaku.

So, this demonstrates how I started with absolutely zero knowledge of how to approach the problem, but worked through it using my reasoning and my knowledge about parabolas. I am still rather confused as to how hyperbolas and ellipses would help to plot similar shapes, but I am happy that I at least began to form reasoning behind why a Sangaku would look this way.

I hope that I could perhaps use a similar exercise to teach my class about why conics are applicable to more than just forming equations. Additionally, although it requires a lot of critical thinking, which in and of itself is valuable to a math classroom, this is a pretty fun activity and I think students would enjoy doing it. I hope that they find themselves thinking strongly about their reasoning like I did, as well. 

I would love to hear any feedback on how someone else did the Sangaku, and I would like some feedback on how to understand conics a bit more. Additionally, I would love to hear how others would teach this. 

Note: I used GeoGebra to plot these circles. I imagine kids could use paper or whiteboards as well, but I think GeoGebra shows the visual uniquness of Sangaku a bit more.