What are the undefined objects in Euclid’s geometry?

Points, Lines, and… Nothing? The Weird Foundation of Geometry

Ever think about what really makes geometry tick? I mean, we all remember triangles and circles from school, but what about the stuff those shapes are made of? Turns out, the very foundation of geometry rests on things that… well, we can’t actually define them. Sounds crazy, right? But stick with me.

Euclidean geometry – that’s the geometry we mostly learn in school, the one with flat surfaces and straight lines – is built on three key undefined terms: point, line, and plane. Yep, the most basic elements are the ones we can’t pin down with a neat definition. Why? Because if you tried to define everything, you’d end up chasing your tail in a circle of definitions. It’d be like trying to explain “up” without using “down,” “left,” or “right.” You just know what it is. As someone cleverly put it online, “Something has to be undefined. Otherwise you end up with a circle of definitions…That is as good as defining nothing.” Exactly!

So, we can’t define them, but we can describe them. Think of it this way:

• Point: Imagine a tiny, infinitely small location. It’s just a spot. A place. It has no size, no width, no height. Just… a point. We usually mark them with a dot and label them with a capital letter, like “A” or “B.” But the dot is just a visual aid; the point itself is dimensionless.

• Line: Now, picture that point stretching out straight forever in both directions. That’s a line. It’s got length, sure, but no width or thickness. Think of it as an infinitely long, perfectly straight piece of spaghetti, but without the girth. We usually draw arrows on the ends to show it goes on forever. You can name it with a lowercase letter or by picking two points on it.

• Plane: Okay, now imagine a perfectly flat surface, like a giant, infinitely thin sheet of paper that stretches on forever in every direction. That’s a plane. It has length and width, but zero thickness. Think of a perfectly smooth tabletop that never ends. You can name it with a capital letter or by using three points that aren’t on the same line.

These undefined terms get their meaning from how they relate to each other, and that’s where axioms come in. Axioms are like the basic rules of the game, statements we accept as true without needing to prove them. For example, “You can draw a straight line between any two points.” That’s an axiom that tells us how points and lines play together.

Euclid, way back when, laid out a bunch of these axioms in his book Elements. These rules, along with the undefined terms, give us a framework to build on, to prove things, and to figure out geometric truths.

Now, fast forward a couple of centuries. Mathematicians started to realize Euclid’s system wasn’t perfect. David Hilbert came along and made it more rigorous with a better set of axioms. He really hammered home the importance of those undefined terms and how they relate to each other.

And then things got really interesting. People started playing around with the axioms themselves, creating non-Euclidean geometries. Turns out, you can change the rules and still have a perfectly valid system, just a different one. This showed that points, lines, and planes aren’t tied to some absolute reality; they’re abstract ideas defined by their relationships within a specific set of rules.

So, what’s the takeaway? Those undefined terms – point, line, and plane – aren’t a weakness in geometry. They’re its strength! They’re the bedrock on which everything else is built. By accepting them as they are, we can avoid endless loops and create a solid, consistent understanding of space. It’s a testament to the power of abstract thought and the surprising beauty of math. Who knew “nothing” could be so important?