I'm about to drop some ancient knowledge here, so pay attention.
Back in ancient Greece--I'm talking 300 B.C.--there was this cat named Euclid. This guy, when it came to geometry, was a straight baller. His main work was a book called Elements, which was basically the world's authoritative geometry text for well over a thousand years. Guy was legit.
So in Elements, before he sets about explaining just about everything in the goddamn universe, he needs to state a few postulates--5 of them. The fuck is a postulate? It's a statement that you assume to be true so you can get on with doing things that fucking matter. Anyway, so he makes 5 postulates at the beginning of the book to get it all started:
- Any two points can be connected with a line segment. Nothin' but a thang.
- Any line segment can be extended to create a line. A line goes on forever, for those of you who are really behind.
- A circle can be drawn with any line segment. Think of a spoke on a bike tire--one endpoint is the center, and the other traces the circumference.
- All right angles are congruent. If you don't know what a right angle is... look it up. I ain't got time to be explaining that basic shit right now.
So up until this point, this stuff is like Sunday morning. Easy. Now we come to that fucking 5th postulate...
- Say you have two lines that both intersect a third line. If the inner angles on one side of the third line add up to less than two right angles (180 degrees), then eventually those first two lines will cross each other on that side.
Alright, back the fuck up. What is this eventually bullshit? Mathematicians aren't big fans of assuming that something will eventually happen just because it feels right. It ain't rigorous, y'all. As a result, the Greeks Euclid hung around weren't big fans of big number 5. They were cool with the first four postulates because they are obvious as shit, but this one isn't what an educated person would call "self-evident." Euclid knew this, too.
There is probably a reason why this is the last postulate. Probably the same reason that he doesn't use the 5th postulate until the 29th fucking proposition in his damn book. It starts to look like Euclid knew damn well this postulate wasn't obvious, and he couldn't prove it was true, so he had to keep it as an "assumed truth" so he could move on with his shit. Even then, he waits as long as possible before he has to plead the 5th (I got jokes). What a sneaky bastard.
Not to say that the geometry he goes on to explain in Elements is useless. Hell naw. "Euclidean" geometry--the type of geometry based on the 5 postulates--is crazy good for all kinds of normal-ass problems here on Earth. Engineering? Architecture? Fucking tangrams?! For most people, Euclidean geometry is just perfect for what they need to do.
There is, however, another set of geometries called non-Euclidean geometries. After no one was able to prove that the 5th postulate was valid for fucking centuries, mathematicians started to experiment with doing geometry without it. Non-Euclidean geometries are based on the first 4 postulates, but they throw out the 5th and make some other crazy-ass assumption about parallel lines. Some of it sounds bat-shit crazy, but these geometries are just as valid as the "normal" Euclidean geometry. In fact, these geometries are better suited than Euclidean geometry for things like air navigation, Einstein's relativity, and--wait for it--celestial mechanics. What?!
There you have it--Euclid's 5th. Half building block of geometry, half bullshit. If you don't know, now you know.