What was the question?
The premise of our project was to study what can be constructed with ruler and compass and then with origami. Why are there differences and how can we achieve the different constructions?
The format of this mentorship program was that of a reading course with some experiments carried out by us.
What motivated me?
I have always been interested in ruler and compass constructions since I learn of them in the Olympiad. At some point I learned that with origami one could extend the constructions (in particular the trisection of the angle). Also, while reading about Galois Theory, I learned that the answer to the question: which polygons can be constructed with ruler and compass? has the same answer for the circle as for the lemniscate. That was incredible to me. I suggested the students to learn about this together and they agreed.
What did the students do?
We studied algebra and the concept of fields and field extensions to make sure we understood what was happening in the field extensions. They studied about complex numbers, algebraic constructions and their meaning in constructability, as well as perform several exercises on related topics.
For example, they proved a famous result, in a series of exercises, that all the division points, for any n, can be constructed with ruler and compass for the cardioid.
Special moments I remember š
There is a way with origami, with certain extra rule, to divide an angle into five equal parts. The important change is the number of folds that can occur simultaneously. In this part of the work, we followed the exposition of Robert Lang given here. We were trying to perform the steps for a straight hour and a half because the paper we needed to use was enormous. It was quite thrilling.
Outcomes of the work
The students presented successfully in what they had learned in the final presentation day of the mentorship program.
Malors 