Dancing Euclidean Proofs

 The first thing that surprises me in the paper is the reference “there are several dance companies who have used Euclid’s Elements as inspiration for abstract modern dance choreography”. I have never really danced or practice dance moves, and it is quite interesting that moves are designed with some mathematical properties in it. Perhaps the paths the body takes, or the arms or legs can have some nice properties that appeal to us without the audience knowing.

The second part I find interesting is “Note here that the logic is not exactly parallel to the definitional and axiomatic propositions in Euclid’s proof. Instead, we rely on our intuition that usually a person’s two arms are equal in length, and we rely on our imagination that the dancers’ arms are also equal”. I like how the author brings in the fact that intuition and imprecise imagination is often used to describe or understand difficult mathematical concepts. When we talk about higher level abstract topics often there is no clear way to present it. How do we even imagine 8-dimensions when we live in and detect 3-dimensional space? (Of course, we can try projections and such, but still that is not the whole picture).

The last intriguing note I like is “Any artistic representative endeavour, and any proof, involves decisions regarding clarity, practicality, and beauty, and our choreographic process demanded we make choices at forks in roads. We were constantly asking ourselves questions such as What is most beautiful? What makes the proof most clear? What is practical?”. I think this goes hand in hand with the second point, and I think this is precisely the reason why it is difficult for people to learn mathematics. It is unlike our language, or other things that we work with in everyday life, in the sense that it is mostly precise, accurate, and leave little room for errors. We are talking about objects, properties, and describe precisely the problems and solutions. It is not how we explore, for example we learn by trying, by exploring and by experience. We can try to build chairs, tools, and other things one step at a time, until we find what is precise and good. Mathematics is already at the end stage, trying to articulate and describe exactly, without error, what exactly it is going on, and this type of thinking is likely the difficult part of learning mathematics. (As an aside, this is probably why many math majors ended up in the computer science and programming industry. Computers need to take precise inputs and do things exactly; its behaviour is already defined. It just runs much faster than any human can do so. Simple tasks like sorting is actually quite interesting to re-learn from the computer perspective. Our modern day AI or natural language interpretation is really built on precise, robust algorithms and statistical methods, which are then back to the precise and accurate description category)