Perspectives in Mathematics

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"Hay naku, two periods ng Math today?"

 “Oo nga eh, may bagong lesson daw.”

“New lesson ulit? Sayang lang oras natin.” 

“I knowww. I mean, kailan ba natin ‘to gagamitin sa totoong buhay?”

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The dialogue above is a sentiment that is all too familiar to those who have endured the tedious waves of mathematics. The feeling of boredom and frustration that plagues students during a dull Math class leaves them lost in a sea of numbers and equations. It’s grueling to find the motivation to push through and appreciate the M in MTAP, in STEM, or even in the IMO, especially when its relevance in one’s life is not  immediately apparent. Contrary to popular belief, mathematics is not just a cycle of monotonous concepts and formulas to be remembered for in an exam. Yet oftentimes, the true essence of mathematics is obscured by the false narratives and misconceptions that surround it.

Now is the perfect time to remove the perception of mathematics as mere calculations and memorizations, and to see it for what it truly is.

Unveiling its hidden aesthetics

For those who have experienced math as a repetitive series of formulas and equations, the idea that it can hold inherent beauty may seem difficult to grasp. The emphasis on rote memorization and problem-solving in most traditional education systems can make the subject feel very uninspiring. Many students, myself included, have to admit that we’ve mindlessly memorized for the sake of passing a test, never truly grasping the lesson’s purpose. As such, we are missing out on its elegant nature.

An example of an aesthetically pleasing math concept would be eiπ + 1 = 0, also known as Euler’s Identity.

Usually taught in Pre-calculus classes, Euler’s Identity, a consequence of a more generalized Euler’s formula, is widely considered the epitome of mathematical elegance. It interweaves some of the most fundamental units of mathematics into one satisfying equation. From circles (π) to calculus (e) to complex (i) and counting (0 and 1) numbers, Euler’s Identity demonstrates the interconnectedness and interdependence of different fields of mathematics into much deeper, and intricate notions. As  “The Math Guy” Keith Delvin perfectly puts it:

“Five different numbers, with different origins, built on very different 

mental conceptions, invented to address very different issues. And yet, 

all come together in one glorious, intricate equation, each playing 

with perfect pitch to blend and bind together to form a single whole 

that is far greater than any of the parts. A perfect mathematical composition.”

Unfortunately, many students do not see the beauty of Euler’s Identity due to how mathematics is traditionally taught in schools. As teachers tend to present this equation out of the context of broader Mathematical concepts or with proofs that are too complex and bashy, students fail to view this “God Equation” in a better light.

Another example of Mathematical beauty in higher studies can be found in the Lorenz Attractor.

Another example of Mathematical beauty in higher studies can be foundFirst discovered by American mathematician Edward Norton Lorenz, this butterfly-like shape is created due to the interplay of three different variables. The mathematically inclined might be able to notice them from the following system of differential equations

dx/dt = a(y - x)

dy/dt = x(b - z) - y

dz/dt = xz - cz 

for some constant a, b, and c. Anyhow, the aesthetic nature of this pattern lies within its recurrent three-dimensional visuals, also known as “fractals”. Fractals are these geometric shapes that are created by repeating a simple pattern over and over again, but at different sizes.

Imagine a coastline — when you look at it from far away, it appears smooth and simple, but as you zoom in, you begin to see more and more intricate details, such as small bays and coves. This is the same concept of fractals, specifically the Lorenz Attractor; one can zoom in and zoom out for an infinite amount of times, yet the pattern will continuously twist and flow like a never-ending dance.

Now what?

I’m certain some of you may be scratching your head, wondering, "Why did I just learn all that? Sure, it's pretty and all with those moving shapes and dots, but I’m never gonna use it, right?” 

You'd be absolutely correct. 

I’m sure no one would use Euler’s Identity or the Lorenz Attractor to calculate taxes, maintain a conversation, or solve relationship issues. But that’s not the point. Oftentimes, it’s necessary to think of them as works of fiction— something that is appreciated for its beauty and not its practicality. After all, one does not read a novel wondering:

                           When will I ever need to throw extravagant parties in the hopes of winning back a lost love?

Similar to how stories appeal to our emotions and longing for resolution, mathematics should aim to evoke a feeling of awe, wonder, and the pure pleasure of discovery. As French polymath Henri Poincare once said:

“The mathematician does not study pure mathematics because it is useful;

 he studies it because he delights in it, and he delights in it because it is 

beautiful.” 

Even so, the purpose of each and every concept of mathematics follows suit quite quickly, no matter how hard it tries to hide itself. 

Purpose in the real world

Euler’s formula, a generalization of Euler’s Identity, has a multitude of notable real-life applications.

For instance, Euler’s formula is used in electrical engineering, specifically in the design of alternating current (AC) circuits. The formula allows engineers to represent complex waveforms, such as sine and cosine waves, in a compact and easy-to-manipulate form. This makes it easier for engineers to analyze and design circuits that can be used in various applications, from power transmission to audio systems. 

Another example of Euler’s formula being put to good use is in the field of physics, specifically quantum mechanics. The movement of subatomic particles can be modeled through the formula. This makes it possible for physicists to analyze and understand the behavior of these particles, which is essential for creating new technologies such as quantum computers and sensors.

Notice that Euler’s Identity, or I suppose Euler’s formula too, is simultaneously aesthetically pleasing and extremely useful in the real world. From the chaos of randomness, we not only find beauty, but also meaning. 

The Lorenz Attractor also shares this same striking sense of this dualism. The system of differential equations is not only a visual feast for the eyes, but it is also a powerful model that illuminates the most chaotic systems of this world. Yes, you heard that right. Those sets of moving dots are actually a depiction of population dynamics, atmospheric turbulence, financial markets, and many more intricacies. 

A subject of timeless beauty that simultaneously unlocks the mysteries of the world. A vast concept that is not just a work of art, but also a powerful tool that unravels the intricate challenges of the world today. Truly, mathematics should not be something to be survived, but one that is thrived.