About

An expert is a [person] who has made all the mistakes which can be made, in a narrow field.
- Niels Bohr

View the old About page from 2021 here.

About Me

Hi! I am Darren, a freshman from the Phillips Academy in Andover (class of 2024). I spend most of my free time entertaining myself with fascinating ideas in mathematics. Apart from math, I am also interested in linguistics, music theory, computer science, and physics. I have always spoken English and Chinese, and I have tried to pick up various other languages before. When I’m outside, I like to run, cycle, or take relaxing walks.

Over the past two years, I have also enjoyed speedcubing¹ a lot. I only started my “speedcubing career” in my early teens, which was comparatively quite late in this sport, but the Rubik’s Cube and the sense of logical challenge have always enthralled me as a kid (that is before I found out about “algorithms”, see this post). As of right now, I can solve every WCA puzzle with the exception of the Square-1. My best events are the 3x3x3, 4x4x4, and 5x5x5 with modest PBs (Personal Best times) of 15.7”, 1’58”, and 3’52” respectively. In other words, I can solve any of these puzzles within the time it takes me to run a mile (5’25”). If I have more time to speedcube in the future, I would try to get better at the Skewb and 3x3x3 Blindfolded. I am also very intrigued by the representations of twisty puzzles as permutation groups, for example, realizing that the beginner 3-blind² method shares many similarities with cycle decompositions of permutations within symmetry groups. And that is something I also wish to learn more about in the future.

I’ve tried to take on various languages in the past with little success, but I have always been intrigued by the relationships between various languages. Such relationships reveal a lot of information about the formation and evolution of languages, as well as the migration and settlement of peoples, the power dynamic between nations, and the exchange of cultures. Since 2016, I have been trying to teach myself Russian. With the opportunity to come to Phillips Academy, I was able to continue studying the Russian language after placement into an appropriate, intermediate difficulty at school. I am really grateful for the amazing Russian Program of which I had the fortune of being a part at Phillips Academy. However, the years that I spent studying Russian prior to studying at Phillips Academy gave me the freedom to explore and learn in any fashion I wanted, giving me the opportunity to experiment with autodidactic language learning methodologies and simultaneously learning a lot both about myself (the way I learn) and languages.

At school, I also ran for the cross country team. Cross country has been a passion of mine since 2019, and I always try to challenge myself and push myself further. For example, in November 2019, I kept up a streak of running over 10 kilometers per day (25+ laps around the track) for 30 days. Although my performances were not very good–I’m stuck at the 20 minute barrier for the 5K and have a personal best mile of 5’28”–wherever I went, my cross country team felt like my family. So it was a difficult decision to adjourn my pursuit for cross country and turn my attention to track and field. During the track and field season of the spring of 2021, I took a break from the distance events and tried my luck with the jumps. I tried three events: long jump, high jump, and triple jump. And I got lucky with triple jump. Having practiced for a total of one hour across two days, I was able to jump 36’11” and win the A/E (Andover vs. Exeter) junior track meet. Now, I still try to keep up with my running mileage while training for the triple jump and other track events. Although I am no longer actively part of the cross country team, they will always be like family to me.

Although I am not very musically gifted, I have always been intrigued by music and music theory. Out of interest, I started playing the piano (although quite inconsistently) at age four. Although I had to “give up” on many occasions due to various reasons, I have always found myself drifting back to it. When I was in middle school, I wanted to join my school’s band, something a pianist cannot do, because playing a coordinated piece with many people sounded like a cool concept to me. Hence, I picked an intrument that no one in the band played – the Bassoon. The Bassoon was certainly a weird instrument to me. Not only one of the only double-reed instruments in the band, it also had one of the lowest note registers, quite a broad range, and an unique, identifiable timbre. But playing the Bassoon in my band was certainly an unique and enjoyable experience. I also chose an instrument with a lower range because I got to play the secondary (supporting) role to the melody. Before picking up the Bassoon, I participated in the school’s casual middle school choir for three months. I was a Baritone, which meant I almost never got the main melody. But having to learn and sing a part that complements the melody instead gave me a new perspective. Before, when I listen to music, I seldom pay attention to the supporting roles that complement the main melody and how much it impacted the music. But, this ear-opening experience eventually led me to choose amongst the lower instruments and settling with the Bassoon. I am also interested in the arithmetic behind tuning theory. Mathematics and music are both my passions; however, I did not know just how much they were connected (through the physics of soundwaves and harmonics). Learning that harmonizations occur when there are “good,” whole ratios between frequencies, that octave intervals sound like the same note because they differ by a factor of two, and that equal temperment uses irrational numbers to approximate pythagorean ratios between intervals (a cent is roughly a geometrical ratio of 1.00058, and this has some interesting connections with measure theory) shocked me at first. Perhaps the most intriguing fact about the history of music theory was why pythagorean tuning “doesn’t work.” Sixteenth-century Italian musician and mathmatician Giambattista Benedetti discovered what became known as Benedetti’s puzzles, a set of puzzles that “broke” pythagorean tuning, which he published in a paper entitled “The Cosmic Joke.” In his puzzles he desmonstrated that based on pythagorean tuning ratios he could derive the absurd conclusion \[\frac{81}{80} = 1\] and allow a repeating melody the shift the tune ad infinitum. \[\lim_{n\to\infty} \left(\frac{81}{80}\right)^n = \infty\]

There are many more gadgets and cool concepts that interest me, and I spend my free time researching and investigating them. In particular, I am interested in the process of mastering a skill, no matter how relevant it is. And I devote this blog to writing about those fascinations.

1. The “sport” of attempting to solve puzzles, especially Rubik’s Cube related twisty puzzles, and especially those officially recognized by the World Cubing Association (WCA), as fast as possible. There are other WCA categories that do not focus on minimizing time, such as the Fewest Moves Challenge (FMC), that are also considered to be speedcubing. ↩

2. An abbreviation for 3x3x3 Blindfolded commonly used within the speedcubing community. ↩