[WP1] Math, Gardens, and the Power of Visualization in the Pursuit of Knowledge
What makes something ‘aesthetic’? Something with order? Something that just has natural beauty? Maybe something that follows a certain pattern? Something colorful, contrasting, and unique? To me, something that contains all of these is aesthetically pleasing. Off the top of my head, I can think of two examples, both of which I have a personal connection to. Mathematics, which was created to simply observe patterns we find in nature, and the Huntington Gardens in Pasadena, a beautiful, vast group of gardens, each based on a different part of the world. Although seemingly completely unrelated, I see them both in the same way, there is a deep correlation between them. I may sound crazy, but because of the time I’ve spent working on Math research at the gardens, I can visualize my work within the gardens themselves.
The first similarity is very clear to see- both the gardens and math are extremely vast, and have huge depths to them. Looking at the first picture depicting one of many ponds in the gardens, I see each one of those ponds as a subject in math, for example my field of research, automorphic forms could be one of these ponds, furthermore, think of each type of fish and plant found in that pond to be a mathematical object, or a tool to use in finding more, and every grain of sand on the bed is an unsolved problem, but from these tools and unsolved problems, new fish and plants emerge, creating an ecosystem, just like when a problem is solved, a new tool to learn even more is produced. For the vastness of math, it is almost impossible to fathom. My entire field of research is under the mathematical subject classification 11Fxx and spans 1.5 of 224 pages [1]. Back to the garden, encapsulating multiple ponds, we are able to see beyond the pond, into the entire field of math, with some mighty structures standing tall in a large pool of water, and bridges connecting them, surrounded by a diverse set of large trees. These structures are the fundamentals of math itself, such as Number Theory, and Analysis. The bridges connecting them are seen in math too, and are connections between these fields, to conquer and cross the large body of water, something I like to call the sea of possibilities, constantly getting larger with water flowing in from a nearby waterfall. It was with one of these bridges that one of the greatest problems was solved- Fermat’s Last Theorem [2] by Andrew Wiles, using the connection between Elliptic Curves and Modular Forms. As with the gardens currently, one of the largest projects in modern mathematical research is the Langlands Program, which aims at finding a bridge between Number Theory and Geometry.
Taking a deeper look at Fermat’s Last Theorem, which states that there are no integers for n>2 such that a^n + b^n = c^n. The case for n=2 is the Pythagorean Theorem, which everyone has used before, and is just a special case of this theorem, to show its importance. Fermat’s Last Theorem went unsolved for about 350 years, until it was finally shown to be true by Andrew Wiles. The key to proving it was in a 1–1 correspondence between two mathematical abstractions, namely ‘Elliptic Curves’ and ‘Modular Forms’. The visual beauty of it was captured in a video by Aleph 0 [3]. It took that long to build the foundations of math that were NEEDED to solve the theorem, and nobody expected these objects to even be related. They were two completely different branches of math entirely- which you can see in the picture below, and the idea that each elliptic curve could correspond to a modular form in such a way was a brilliant one, which led to this crucial problem being solved. The proof in itself was extremely long, rigorous and ugly, and took Wiles around 7 years to write, being 129 pages of just math. However, the beauty of being able to show, using extremely complex tools to find an underlying beautiful formula that could be simply stated, as done by Fermat makes it one of the most elegant proofs in history.
Personally, I gain satisfaction from solving extremely challenging problems, and, in contrast to different fields such as Physics or Engineering, which I have also worked on, whose answers are usually extremely ugly, with so many factors having to be taken into account, but with math, the solution is always elegant, and easy to state (although the working is dirty) leading to a beautiful solution- (JUST LIKE GARDEN CONSTRUCTIONS!). Take my published paper [4] for example, which asks the question “when is the product of E_2 and another eigenform again an eigenform?”, and the answer is extremely simple- if the result is one dimensional, otherwise never. It spans 12 pages of complicated math, but again, the answer itself is simply stated, and tends to make sense, even if it seems unintuitive, compared to more complicated responses in almost every other field, which I believe enhances its beauty. The main difference between the norm for ‘aesthetically pleasing’ and math in my opinion is that math is a form of ‘abstract beauty’, and is very hard to understand unless the necessary steps have been taken to learn the subject intricately.
To emphasize simply the application of just beauty, and how just being able to clearly, truly, SEE math as something beautiful, we recall the story of David Hilbert, a famous mathematician and a story that I could relate to, surprisingly a lot. He was a mathematician who tried to keep math as ‘pure’ as possible, devoid of any application, done simply for the sake of learning something new and enchanting. Well, or so he thought. That math that he did, we now use in Gene editing, turns out it could explain a lot of modern biology today. This is how math is, we do it for the sake of beauty, but at some point, no matter how hard we try to gatekeep it, with complex symbols and crazy formulas, it will always have applications, and the entire world is built on the foundations of math itself. To further that, another weird bridge is the current ‘crown jewel of math’- the Riemann Hypothesis, which states that all the nontrivial zeroes of the zeta function lie on the line with real part ½ in the complex plane, and that somehow is related to the distribution of prime numbers, which is somehow related to cryptography, which would lead to unhackable locks and so on, which just completely blows my mind. I believe that the first step to uncovering the truths of the world is by uncovering what math truly is, rather than avoiding it as much as possible, and to see it in the same beauty we see gardens, to not get lost in the complexities of the subject, but to look at the big picture, and to see its creations, the very structure we live in, and to embrace it.
The main goal of this essay was to provide a larger audience an explanation of how math could be seen to make it less unappealing, to show that there is more to math than just solving problems, by providing a more concrete way to perceive it, and show that a large amount of why we are at the place we are today is largely due to a lot of people who didn’t give up when things got hard kept going, in pursuit of finding, even creating more beauty, just for their own satisfaction. In my opinion, math should be seen as just the study of abstract beauty, not the way it is taught in school- as a tool to solve problems. I hope that this, in some way, inspires more people to do math, to seek out unintuitive abstractions that may in turn satisfy them, or at least, to not view math as an annoying tool to use in calculations because it really is not. I would strongly recommend watching some videos by 3Blue1Brown [5], which is a channel that is completely based on visualizing all the complex mathematical formulae into more concrete objects, and inspired me to do number theory myself- specifically the video on the Riemann Zeta Function and Analytic Continuation. To conclude, a quote from a great mathematician: “The arithmetical symbols are written diagrams and the geometrical figures are graphic formulas!” -David Hilbert.
Sources Used:
[1] Mathematics Subject Classification 2020. Msc2020.org, 2020, msc2020.org/.
[2] 26 Fermat’s Last Theorem. 2017, LectureNotes26.pdf (mit.edu).
[3] “Elliptic Curves and Modular Forms | the Proof of Fermat’s Last Theorem.” Www.youtube.com, www.youtube.com/watch?v=grzFM5XciAY.
[4] XUE, Hui, and Naveen PRABHATH. “ACTION of E2 on HECKE EIGENFORMS.” Kyushu Journal of Mathematics, vol. 77, no. 2, 1 Jan. 2023, pp. 401–412, https://doi.org/10.2206/kyushujm.77.401.
[5] 3Blue1Brown. “3Blue1Brown.” 3Blue1Brown, 2019, www.3blue1brown.com/.
