Dancing Euclidean Proofs

Image source: https://vimeo.com/330107264

One of the ideas that stood out to me was using "imagination" to assume both dancers arms were the same length. In proving one of Euclid's Propositions when I took a geometry course in my undergrad, I remember the professor putting emphasis on making sure our proofs followed logic over diagrams. At first, it seemed possible to draw a diagram that disproved the proposition, but once we stepped through the proof, we saw that it was impossible. In this way, I think these dancers could use their dance to show students that, although the "lengths" (arms) don't look equal, because they are defined as such, the proof still holds true. 

It was interesting to me that they pointed out that the math and choreography would fall into place after some rethinking. I hadn't really thought of the similarities between choreography and developing a proof before but after reading that, it seems quite clear. When initially attempting a proof, I often had a long-winded process of achieving a result but the more I worked with it, the cleaner and more pleasing it became. The same process happens with choreography - sometimes I would think, "I want to go from two horizontal lines into an "X" formation," and would probably change the way the dancers got to that position three to four times because I found a way that clicked together better and just made for a method that felt more natural overall. It was a realization that consciously or unconsciously, we're all developing intuition in a field by spending more time in it. To extend the similarities, I remember being a student in dance and being frustrated when the choreography would constantly change, but now I understand that it's less the teacher's inability to commit to a choreography but more them gaining familiarity with the song/students/situation and their increased understanding of the best execution of the dance that leads to those changes. Similarly, it seems like an exhausting and tedious task to rework a proof, but it often leads to a better understanding of the content and also a better "product" that others can more clearly understand and appreciate as well.

The idea of collectively being responsible and embodying the proof with someone else was a powerful one for me. I remember studying Euclid with my friends prior to tests and it was funny because even though we were all in the same course and all working on the same problem, each of us had a separate method for obtaining the result depending on how we began the process (for example, I tried to use circles wherever I could, while my other friend was partial towards angle relations). We often found it useful to explain our own processes to each other to make sure they made sense outside ourselves, and in case we noticed mistakes in each other's' reasoning. To be physically stepping through a proof with another person would not only be technically challenging, but also mentally challenging because you are having to follow a path that may not necessarily be your vision. However, this is quite valuable as well because there is a lesson in reliance and dedication to collaboration in deciding to pursue this method.