Friday, October 20, 2023

Nov 6 Dancing Euclidean Proofs Reflection

This whole concept of using dancing to display Euclidean geometry proofs in multisensory ways does help us understand it in new ways. It is visual and unique. I’ve never seen math proofs done using dance before so watching it for the first time was interesting. There is harmony to it and seeing how the human body, as intricate and complex as it is, is able to do what a compass and straightedge can do - and much more! Watching the dances on screen created a zen atmosphere so it was quite relaxing. This integration could help the general audience appreciate the beauty of mathematics. I have seen folk dance being used in a computing science course when I took it at the UofA. The lessons were on sorting algorithms and there were some supplementary videos that showed us a row of dancers moving around like how objects would move around depending on which sorting algorithm was used. Here is the link to the playlist! Another thing that made me fathom was that a dot on a page is just as inaccurate of a way to physically represent a point as a dancer’s body. The dot on the page takes up some two dimensional space in reality, so it isn’t a totally accurate physical image of a point, just like how a sheet of paper isn’t really 2D, but is 3D even though the thickness of it is super thin. The dot and the body are both representations of the true abstract concept. Dancing allows one to connect with that abstract concept through embodying it with phyical movements. I think it is nice to have students up and moving around since students around high school sit for a long time in front of a computer screen to do homework, research, watch shows, or play games. Having these body motions allows students to remember certain concepts better and associate its value to oneself. Also there is no technology used during this process of dancing with Euclidean proofs, so this practice would have also been implemented historically too. However I do see that not every student is into dancing since they may not feel comfortable with it. Maybe for traditional reasons they can’t have a partner to create these proofs visually with the body (from the video there were two people “dancing out the proofs”). Other students might just not care or don’t enjoy it. I probably won’t do dancing fully, but maybe some physical actions moving arms and hands around (maybe a bit of the legs). I think unless they really don’t want to, they don’t do it, but I encourage those who can (who may just be somewhat indifferent or unamused by it) to move around and try it out.

Tuesday, October 17, 2023

Oct 18 Reflection on Euclid's Works

I don’t remember when was the first time I heard of Euclid’s moniker, but there was a math contest from the University of Waterloo named after him. People who were applying there [for math/CS/engineering] in grade 12 were recommended to take that exam. Younger feverish students also took it for fun (or for fame/rewards). The early bits were doable, but the difficulty got harder as the questions progressed. There were some very interesting geometry questions later on that I’ve never seen before. The fact that this contest (which was a supplementary material for evaluating application forms for UWaterloo) was named after him shows how lasting his name is embedded in our academia institutions. I later studied Euclidean geometry in university, a topic I haven’t revisited since grade 9. Euclid’s book (The Elements) has indeed left a great mark on mathematics and the way we think about abstract concepts. His methods are quite systematic. He presented a subject in a logical manner based on definitions, axioms, and theorems. This systematic approach became a model for subsequent mathematical works and the scientific method in general. Starting with small lego blocks, larger things could be built from those blocks which in turn build into even larger things. Geometric proofs are also quite visual, so those who are more visual thinkers would have a better time understanding them than say a proof from real analysis. The geometric proofs are also presented in a clear manner. This clarity has made it a favored choice for educators over time. Overall it lays out the foundations of plane geometry nicely and succinctly. Although beauty is quite subjective, it often appears with elements of simplicity, symmetry, clarity, depth, and harmony. When applied to abstract concepts like mathematics, beauty often emerges from the elegance of a certain topic, the depth of insight it offers, or the surprising connections it makes (how well things go together). Because of this, many find beauty in the Euclidean axioms. Some of the proofs and constructions that use those axioms are thought to be beautiful, like some sort of minimalist perfection where every step feels necessary with no redundancy. Beauty isn’t all about a physical image that is captivating to the eyes, but it’s also about mathematical notions working together and fitting together nicely like jigsaw pieces.

Friday, October 13, 2023

Oct 16 Was Pythagoras Chinese? Reflection

    I think it’s important to acknowledge non-Euro math sources because it allows people to better understand the universality of this field of study. That groups who were not in contact with each other all discovered the same concepts using different methods. The complexity and ingenuity of the ways the ancient peoples/cultures used act as a bedrock to what we are learning today. People now can’t begin to come up with ideas on how certain problems were solved with no technology. A lot of theorems and findings are already done from the past, we are so used to just using them in our daily lives. From ancient China, the significance of these foundational texts, especially the "Jiu Zhang" and "Zhoubi suanjing," persisted for millennia. Around the time these works were completed, the Chinese civil service grew under the Eastern Han dynasty. So China’s prosperity didn’t come from using knowledge gleaned from Euro math sources. If I mention and give credit to how the ancient Babylonians, Indians, Egyptians, Chinese, and Myans, it displays integrity to those groups. Seeing from the lens they derived some rules from opens our minds and makes us scratch our heads to think in new directions. It also gives a breath of fresh air to know that new ideas aren't confined to one area of the world.

    The "Zhoubi suanjing” focused on astronomy and mathematics related to land surveying and construction. It introduces the gou-gu theorem, which resembles the Pythagorean theorem. The notion that of a right-triangle the sum of the squares of its legs equals the square of its hypotenuse isn’t solely a Pythagoras (or ancient Greek) finding. The ancient Babylonians centuries before Pythagoras also listed Pythagorean triples, which is not a trivial task. Other mathematical terms like Pascal’s triangle fall into the same category. Pascal wasn’t the first one to discover it. It is also known as the Staircase of Mount Meru which pays tribute to Indian mathematician Pingala. Clear evidence/sources from ancient China mentions it as Yang Hui’s triangle. It’s the same triangle we know today but is depicted using rod numerals similar to zongs and hengs. The reason why we use Pythagorean theorem and Pascal’s triangle is because it’s well embedded into the literature and there is less ambiguity when talking about them. People will know automatically what they mean, whereas if I say the Staircase to Mount Meru, others might think I’m referring to some attraction site. So it’s helpful to mention these things.

Wednesday, October 11, 2023

Oct 11 Assignment 1 Reflections

Researching about how the ancient Babylonians solved this difficult problem on generating reduced Pythagorean triples was no easy task. Lots of rich information was lost from the primary source as the left part of the tablet was broken off. In the end people had to resort to proposals on how the ancient Babylonians actually did it based off of their other works and discoveries. The proposal I used was the reciprocal pairs method, however it used a lot of modern day algebra to deduce what happened, and didn't utilize much of the notations and numeral systems the ancient Babylonians used. It was nice to revist the cuneiform script in sexagesimals (base 60) again, however I didn't explicitly use them. I still think it was neat to know that they knew such big Pythagorean triples with the limited amount of tools and knowledge they had. I worked well with my group mates. Each person had their own task and it was a nice transition between us presenting. I felt we could've spent more time on interactive activities, but due to time constraints we briefly brushed through them. Giving the participants/audience more time to think and go through what was happening would've been helpful too. I am glad to present on a topic that I am familiar with, something I had a fun time learning about when I took number theory one year ago. I saw the intricate details of proving the Pythagorean triples theorem. Boiling it down to a simple fun history math presentation was a mild way to express the interesting aspect of this topic.

Oct 11 Two nice visual proofs of the Pythagorean Theorem

I like Eddie Woo's teachings and character and below are two of his videos on visual proofs of the Pythagorean Theorem:

Visual proof of the Pythagorean theorem

Another Visual proof

Edward Woo is an Australian secondary school teacher and writer best known for his online mathematics lessons published on YouTube.

Assignment 3 and Course Reflection

The project on Tower of Hanoi was interesting since I was able to learn why the puzzle was named that way and the stories that revolved arou...