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 around it. I was surprised that the puzzle was only created a couple of centuries ago by Lucas in the 1800s, but nonetheless it fascinating to see what influenced the nature of the alternate names to this puzzle. I was fortunate to take a class on combinatorics and went over this puzzle in the recursion unit. It looks tricky to solve at first, but once what is behind the scenes is understood, it won't be as mindless/tricky to solve. I may use this as a fun project in the geometric series topic. First introduce it, talk about the steps behind on how to solve it and later deduce the closed form formula, which is a geometric sum. 

With respect to the course I really liked learning about the ancient Babylonian sexagesimal based 60 arithmetic. Playing around with fractions on the boards in the orchard garden were fun times. To understand different number bases is important, like binary in computing. There is also good reasons why 60 is a nice base number to work with. Getting in touch with the mathematical findings and work during the Islamic golden age was curious too. My least memorable moment is probably focusing too much on ancient Babylonian math, as most of the blog posts were on it. I think some suggestions is to do more hands on work and solving more problems for the future.

Assignment 3: Exploring a math history topic through making/the arts

Assignment 3 Tower of Hanoi

Nov 27 Medieval Islam Mathematics Reflection

The House of Wisdom is the first point that made me stop. While I was doing my presentation on historical integration, I came across Alhazen (Ibn al-Haytham) who lived over a thousand years ago. He had major contributions to optics and integration and was involved with the House of Wisdom. Initially I thought it was a group of collective individuals coming together to work on..discovering new things, though in fact it was primarily a library and a place for translation and research. The people in there would study ancient Greek mathematics/astronomy and further built on these topics. The one who established the House of Wisdom was Caliph alRashid. His son, Caliph al-Ma’mun, was the ruler who made the House of Wisdom so important. This caught my eye since the word "caliphate" means "an Islamic state, especially one ruled by a single religious and political leader". This word might've came from the two prominent aforementioned father/son. It also referred to the Islamic world during the medieval period, when Europe was in its dark ages while science and technology grew in the middle east. Furthermore it was interesting to how algebra and geometry flourished during the Islamic world which resulted in beautiful artwork captivating the eyes. A lot of the constructions with shapes inscribed within each other were mystifying too. Some circle geometry were shown as well with procedures I have never encountered in an elementary Euclidean geometry class. The good news is that many of these constructions can be replicated by hand with easy to follow instructions. This makes it friendly for school students to use so that they can create fascinating math works.

Nov 20 Maya Numerals

For the number "1729", Ramanujan noted that it was interesting because it was the smallest number expressible as the sum of two cubes in two different ways: Major's paper might discuss the role of intuition in mathematical discovery or the personal relationship that mathematicians have with their subject. Ramanujan's work often relied on deep intuition, which could serve as a prime example of this relationship. The collaboration between Hardy and Ramanujan, who came from very different educational backgrounds, could also be used to highlight how diverse perspectives can lead to significant mathematical advancements. If I would introduce this to secondary math class, mentioning these stories can be a captivating introduction to several math topics and would likely engage the students. The stories connects math to real world contexts, making it accessible and interesting. It shows that mathematics isn't just about solving abstract problems but can be playful and intriguing. For advanced students, exploring the proofs behind these math properties can deepen their understanding of mathematical rigor and methodology. In terms of numbers/letters having personalities, some might see the number 2 as a generator because it is even and small. It is also the smallest prime number. So many concepts seem to trace back to this number. The number 7 as more mysterious due to its associations in literature and folklore. The number 3 appears a lot as well and it's important since 3 is the smallest number of edges to form a closed polygon as well as having a triad stand to support a seat. In many languages the number 4 is dispised since it sounds like death. Dates such as Friday the 13th have had superstitious stories revolving around it which makes some people uneasy to write an exam at that time! 

Nov 8 Trivium and Quadrivium Article Response

The article introduced some interesting things. To begin, this article has a Latin title and it is separated into two parts, one of which has 3 topics and the other has 4. I initially saw this as a distinction between what is now known in modern day as the Humanities and STEM. This practice has been around since the medieval times. Either way, both of these parts required people to be literate and well versed cultured citizens. About 15%-30% of the population were like this back then. So there weren't that many opportunities for serfs and labor people to grow in their knowledge in these areas. However, looking at it today, most people can read and write, so the literacy rate has gone up. Many people are engaging in areas of both fields. Kids are learning a variety of everything in their young elementary years, and begin to specialize more and more as they get older. Another point that stood out was the different methods of division. There was the abacist method which was complex and cumbersome so it wasn't that favored, thus it got the name "iron division". It required a good understanding of the abacus and the ability to manipulate the beads to represent the division process physically. I personally had a hard time learning the abacus (mine was the Chinese suanpan) since there were some things to keep at the back of your head when doing the computations. On the other hand, the algorist method was simpler and more straightforward, thus it got the name "golden division". It used written calculations and was based on the Hindu-Arabic numeral system, which included the use of zero and place value, so it was more efficient to do the steps physically on paper. To get to this point wasn't that easy initially, as switching from the previously used Roman numerals to Hindu-Arabic numeral system took over two centuries. This newer method also allowed calculating the date for Easter much easier than before. I never knew why the date for Easter always changed annually, but it definitely is dependent on something, just like how the Spring Festival is dependent on the lunar cycles. Seeing now that Easter's date is determined by the Golden Number and the Dominical Letter and all the math involved behind it made me realize the complexity of the arithmetic people centuries ago had to perform. This, blended in with the fact that the clerics should know how to do the calculations emphasized the importance and significance this key date played in their faith.

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.

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.