Euclid & Beauty

 Hello twins.

The following is a response to a biographical evaluation of Euclid's work from St. Andrew's University.

Euclid's Elements provided a synthesis of mathematical thought at that point in history. It compiled centuries of Babylonian, Egyptian and Greek mathematics into a single text. The book was crafted in a logical manner which provided the framework of mathematical thinking for centuries. He outlined the fundamentals which could serve as a friendly starting point for beginners interested in the field. He then builds upon those basics to explore more advanced topics relevant to more advanced practitioners It also introduced key ideas for points, lines, angles, and shapes which likely had applications to other disciplines. There was likely an aspect of convenience as one could refer to a single book instead of tracking down many independent works. There is an inherent satisfaction in an all-encompassing text for a subject and these summaries are highly valued in modern contexts.

As a math student, I have often heard instructors refer to an aspect of beauty or elegance in certain mathematical ideas. A skeptical outsider's perspective may be quick to dismiss this claim by questioning how a subject like math could ever contain an element of beauty. However, as I began to take more abstract math classes which dived deeper into the theory I discovered what they had described. There is beauty in the contrast between using a simple argument in an explanation for more advanced and complex ideas. There is also a deep satisfaction that can occur when math problems conclude cleanly and concisely. For our recent math history project, we had a puzzle that we presented to the class involving arc geometry. After solving the puzzle on my own, I messaged a group member saying that the puzzle was a beauty. I was referring to the extremely satisfying way in which the problem worked out when solving it. There was a variety of cancellations in the algebra which led to a very concise formula. Thus I suppose one must get their "hands dirty" by working with math to see the beauty that others describe.

Thanks for reading,

Josh

Dishes Puzzle

 Hello twins.

The following is my attempt at a 4th century Chinese puzzle involving dishes and meals.

Let r = # of rice plates, b = # broth plates, m = # meat plates.

Then r + b + m = 65. And since rice is shared in pairs, broth is shared in triplets, and meat is shared in quadruplets, we know that r = n/2, b = n/3, m = n/4 where n = # of people.

Plugging these in we get x/2 + x/3 + x/4 = 65. We can combine and simplify to get 13x/12 = 65.

Solving for x we find that x = 60 and so there were 60 guests in attendance.

Now I tried many ways of solving this without using algebra. I tried using modular arithmetic to deduce a pattern. I tried drawing venn diagrams and using properties of union/intersection to deduce a pattern. I also tried using floor functions in combination with some of these strategies but ultimately I was unsuccessful. Specifically, I struggled with the fact that I could easily go from # of people -> # of plates but I couldn't seem to figure out how to go in the other direction. There is likely a clever solution that lies just outside my reach at the moment but I am willing to accept my progress. It may be my heavy reliance on algebra that has clouded my view of alternative methods.

I definitely believe that the setup of a problem is an important part of enriching the student experience. It adds an extra layer of flavour beyond just typical math exercises you would find in a textbook. I found myself telling these problems to family members since the story behind the problem helps draw them in. It is human nature to enjoy stories and so I believe that incorporating more stories into math problems would be an effective practice for increasing student engagement.

Thanks for reading,

Josh

Math History Project

Hello twins.

Today I did a math history presentation on chord geometry in ancient times. Below is a short reflection of my thoughts on how the project went.

I have attached my slides below:

https://docs.google.com/presentation/d/1J7zrN0MaiTL_azNpEcC_MUBBF3leXjKa8AZ9lFbG5tw/edit#slide=id.g30835df58cd_3_107

The presentation went really well and I had fun teaching the material to my classmates. We were able to look deeper into the mathematics involved as well as the historical significance behind the problem. I personally enjoyed the research stages of this project and learning some of the ancient mythology behind the Ishtar Gate. The puzzle we posed to our classmates is one I would certainly utilize in future classes as it provides a very satisfying "aha" moment for students that I experienced myself. I was telling my partner that most of us math students are very drawn to "puzzles" as we immediately reach for our problem-solving toolkit. There is something innate in human nature for being drawn to solving puzzles and it is fascinating to consider how humans have been solving puzzles since ancient times. I've never been particularly interested in history but this class has unearthed an interest in learning more about the past. I wonder how engaged high school students would be in exploring history more in a math setting and whether they'd be open to it. Overall, I'm super grateful for how well this collaboration went and I enjoyed listening to each group's presentation.

Thanks for reading, 

Josh