I never got beyond calculus in school, and I’ve heard this term thrown around by smart math and science people bit have no clue what it means or why it’s special.
In: Mathematics
Warning: this is closer to ELI12, and a bit lengthy. Sorry; I’ve been [nerd sniped.](https://xkcd.com/356/)
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Right! We’ll need to take a peek at university math, in particular, an area called pure mathematics. While for majority of time we’re happy to assume things just work they do because… they do, pure math likes to investigate _how_ and _why_ they do, and could we stretch what ‘work’ means.
The usual way of defining something in maths is by stating a small set of unquestionable truths, called axioms – and then showing _everything_ else you’re used to knowing about this something follows. These aren’t always the most obvious statements, and usually they look suspiciously simple. In manufacturing terms, this set of axioms is the minimum viable product: if you have these, you inevitably end up with the right result – but they’re simplified and reduced to the point where any less and you won’t.
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As an example of axiom, our usual, day-to-day integer numbers are *well ordered*:
* if you give me a and b, either a<b, b<a, or a=b, with no alternatives.
The same applies to, e.g. real numbers as well. We don’t need to prove it; it’s an axiom we take for granted.
We use it to, for instance, prove that sqrt(2) is irrational: otherwise, we claim, there’s a way to write it as p/q, _and if there is, pick the way where p is smallest possible_. Then square it, 2q^2 = p^2, therefore p even, divide by 2, therefore q even, and we can simplify p/q by 2 – contradiction. A simple, neat proof, working by itself… well, not quite. The bit in italics actually calls on the well ordering of integers – if we couldn’t always decisively tell what’s smaller, we can’t rely on there being a smallest p.
So if we take a different set, for instance, 2D coordinates, where we know (1;2) > (0;1) and (100;1) > (-2;-5), but cannot actually tell the sign between (1;3) and (4;1) – neither’s greater, and they sure as shit ain’t equal – if we went and asked for proof that square root equivalent here is not a ratio, we’d need a different proof. It doesn’t mean it’s not still true – just that this path, which relies on an axiom we no longer have, won’t work.
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Now, onto your question. When Euclid first described your usual, on-a-piece-of-paper geometry rules, he didn’t get to a full list of axioms, at least not to rigour we’re used to. As later mathematicians tried to nail down what exactly is needed, they stumbled on this statement:
* Give me a line, and a dot not on that line. I can always draw one, and exactly one, line that’s parallel to one you gave through this dot.
It felt… odd. Wrong. Superfluous. And yet, attempts to get this statement out of other axioms consistently failed; so begrudgingly, mathematicians needed to take this as absolute truth for geometry to work.
But pure mathmos are the kind of creature which won’t leave a stone unturned, and easiest way to check if it’s truly necessary was seeing what we end up with without this axiom. And turns out, we can get a version of geometry that works with all other axioms, but not this one. In particular, we no longer get on-a-piece-of-paper situation; we get on-a-funny-shape situation. For a simplest example, let’s draw some lines on a globe.
We can still define what straight is – shortest distance between points. We can still measure angles. We can still define circles as all points with a set distance from a centre. A _lot_ of math works as you’d expect – thanks to other axioms we held on to. But if you give me equator, and ask for another line near it, it _will_ intersect. In fact, every full-length line is like a ring, possibly tilted – and will always intersect the equator. We can no longer draw any parallel lines.
And as a result, some proofs fall apart. Ever checked why we always claim triangle is 180 degrees? By drawing a line parallel to one side through the vertex not on that line, you can show all three angles add up to straight line. But we can’t do that anymore. And, as mentioned before, that doesn’t prove necessarily that triangles won’t have 180 degrees on a sphere, just that they might… and turns out, they don’t. In fact, give me 3 points – north pole, and two on equator quarter of circle apart – and I’ll draw you a triangle with three right angles for 270 total. South, East, North. All because we didn’t have a parallel line.
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So that’s known as non-Euclidean geometry. For a complete definition – it’s whatever system you end up with when you drop one or more axioms from our usual Euclidean. For useful examples, it’s what lines, circles, angles and all else does when your space isn’t ‘flat’ – it could be sphere, cone, saddle-shaped, whatever. It’s tricky to work with, because you need to double check your work – that you used since primary school – to ensure it still holds. But it’s inevitably useful – as an obvious example, all our maps are of a non-Euclidean space, and if you need, e.g., a country’s territory precisely, you need to know that a triangle you drew on the map is not exact same as triangle on the field.
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