A-LIGNing Perspectives: Lines and Planes

What exactly is a line?

Let’s start with the basics. In mathematics, a line is a simple, yet powerful concept. It’s the idea of an infinitely thin, perfectly straight path that stretches out endlessly in both directions. No curves, no breaks, just a pure, continuous path that goes on forever. Now, when we imagine a line in our heads, it might seem like something straightforward—just a stroke, maybe, or a wire extending without end. But in the realm of mathematics, this simple idea has much more depth. You have thought of lines before, and perhaps often. And I don’t mean the mathematical, literal definition. Maybe not consciously, but whenever you’ve felt like you were heading in “one direction” in life, or stuck moving forward on a narrow path with no way to turn around, you were imagining a line.

In mathematical terms, we define a line by choosing a single starting point—just one spot in space where the line could begin—and then extending it using something called a vector. A vector is like an arrow pointing in a particular direction. It tells us which way the line moves and how it extends infinitely beyond the starting point. The beautiful thing about this definition is that, by combining a starting point and a vector, we can take that line you’re imagining and express it in a precise way that mathematicians can work with. This allows us to turn an abstract idea into something that can be analyzed and understood.

But lines don’t exist in isolation. What happens when we start placing multiple lines together? What if they’re not all moving in the same direction? Well, if two lines point in different directions, there’s a chance they might intersect—cross paths at a single point.

In life, we experience these intersections all the time. You meet people, share moments, and then move on, continuing along your own path. With some interactions, your lives may cross for just a moment, a brief exchange, before diverging again. But here’s something interesting to note: in most cases, two lines won’t intersect more than once. That might seem strange because in life, it feels like we cross paths with certain people again and again. There are people who influence us constantly, who seem to run alongside us in our journeys. But in the strict mathematical sense, if two lines are truly distinct and not perfectly aligned with each other, they’ll meet once, and that’s it. Where’s the catch?

This is where the idea of planes comes into play. A plane is like a vast, flat surface that stretches out in all directions. You can think of it as adding an extra dimension to your line. Instead of moving in just one direction, you can now move in two directions—imagine standing on a flat surface and being able to walk both forward and sideways. Just like with a line, we can create a plane by taking our starting point and extending it not just with one vector, but with two vectors pointing in different directions. Together, those two vectors create an entire plane—an expanded world of possibilities, where you’re no longer confined to a single path.

Now, when two planes intersect, they don’t just meet at a single point like two lines do. Instead, they intersect along an entire line of points. Imagine two sheets of paper connected at their edge—where they meet, they form a long, continuous line. That’s the idea of planes intersecting. And just like how Isaac’s path crossed with Jaylen’s in a single moment, sometimes the planes of our lives intersect with others in a deeper, more lasting way. These intersections may create a shared space where our influences on each other extend beyond just one brief encounter.

In this way, mathematics gives us a way to understand not just abstract concepts, but real-life experiences. The relationships we form, the paths we take, and the ways we change over time can all be described with the same clarity and structure as lines and planes. Mathematicians aren’t just working with numbers and equations—they’re finding ways to simplify and explain the world we live in. As William Thurston once said, "Mathematics is an art of human understanding." At its core, math helps us see patterns and connections that might otherwise go unnoticed, giving us a framework to make sense of our reality.

This brings us to the next question: What about the physical world? The math we’ve been talking about so far is abstract, but in physics, we take that mathematical framework and apply it to the patterns we observe in nature. Physics helps us explain why things happen the way they do, revealing the rules that the universe follows without fail. As we move forward, you’ll begin to see how these mathematical ideas translate into real, physical laws—the principles that govern everything from the motion of planets to the way we move through space and time.

But for now, hold on to the idea that the paths we walk—whether they’re straight, intersecting, or expanding into new dimensions—are more than just abstract concepts. They’re reflections of how we live, how we connect, and how we understand the world around us.