The Mysterious Potential Of The Plane

In recent years I've embarked on a sort of journey of rediscovery with respect to mathematics. As a kid I was somewhat good at math and I spent my college years doing lots of it—to the point where I grew to really dislike it. But even in those times, mathematics—while becoming increasingly difficult—always seemed like it was trying to tell me something. These days, it fascinates me.

A common observation among those who study classical engineering fields is that many (if not most) of the problem domains are modelled with very similar techniques. The motion of a cantilevered beam, a road bridge, a plane wing, a mass on a spring, a musical note, and even a planet's orbit can be described by very similar mathematics. Indeed so many physical phenomena can be described by these equations that it can make you wonder if you've stumbled into some great secret of the universe.

But that wonderful truth is only a gateway to something much deeper, more fundamental—a true capital-S Secret of the Universe. Mathematics tends to deceive with simplicity, and there's few things in this world that hide more secrets than this simplest of concepts.

Consider the Plane

As children we're taught about the 2-D Plane.¹ At first, students are confused in a manner similar to how they felt when being told was a number now, and oftentimes that confusion grows into consternation. At worst, plotting and manipulating curves on this plane feels pointless, arbitrary, and unnatural. For those who come to know the plane as a tool, it can be quite an enlightening process. As an undergrad engineering student, you become familiar with the plane and its rules; you learn to shape curves and solve perspective puzzles with it. Modern video animations make this so much more intuitive because they're able to show a figure on the plane change over time: you can watch it squish and stretch as you adjust the coefficients of a quadratic.

What's more Geometry holds so much deeper meaning in our world than I think most people imagine. Vast numbers of people can use mathematics to do every day tasks: measure the square-footage of a room, scale up a recipe for a family dinner, or plan their finances, but Geometry does so much more. Simple rules about finding the angles in a triangle can help measure the curvature of the universe, and simple sines and cosines (which form the backbone of many vector operations) enable the ability to encode the meaning of abstract linguistic concepts in modern Large Language Models. Our universe is made of math in truly fascinating ways.

But this isn't my point. So far, this has all been backstory.

Exploring the Plane

What's truly wondrous about the 2-D Plane isn't what you can plot, but what you can discover. There are infinitely many lines, curves, and shapes one can draw on the 2-D Plane and while most of them are useless, quite a few are incredible. Those are the ones we plot to find the motion of a ball in the air, or the speed of a planet's orbit over time. But there are some curves that hold incredible secrets. We know of a few and there are likely many, many more. These magical curves transform our understanding of mathematics itself.

Consider the logarithm:

This is a function that can be used to do all sorts of practical, useful operations, but it can also convert multiplication into addition.

Plotted, the logarithm function makes a curve on the 2-D Plane. This means that there exists a curve on the plane which converts multiplication to addition! What other magical curves exist with incredible properties? Perhaps there's a curve that plots itself? Oh wait! There is!

At it's core, a function is a thing that takes a number(s) and returns another. This sequence of values can be plotted to form a curve on the plane. There's a curve mapping subtraction to multiplication, one for the GDP of the U.S. over time, and (given some encoding trickery) a curve describing the lifespan in seconds of each and every being that will ever exist! They're all in there somewhere; we just have to find them.

This is one of the true wonders of mathematics², and more specifically of geometry. Who would think that simple shapes and curves could tell such great truths? Yet so much hides within the expanse. In this way, the work of finding such curves is one of exploration in a treacherous wilderness. With simple tools, one carves a path of discovery amid the endless vastness, returning to share the knowledge they have found.

The Secrets Within

Held within the depths of this infinite expanse there are perhaps infinitely many fundamental secrets. We only find them through careful study, or occasionally by accident, and we have no idea how many we might be missing—even amidst the sectors of that expanse we've so thoroughly explored.

It's so wondrous to me that such a simple concept could hold such great truths and that shapes and curves upon the plane could so accurately model the natural world. Perhaps it should be the Plane or the Triangle that stands beside Fire as our most powerful invention.