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
Euclidean geometry is geometry on an infinite, flat, space.
Non-euclidean geometry is geometry on curved spaces, like the surfaces of spheres and donuts.
Non Euclidean = Not flat surface
Example: Earth, not flat, it’s a sphere, so drawing lines on it has different rules. It has spherical geometry
Eclidean geometry deals with planes, like drawing angles on a sheet of paper. Other geometries deal with drawing them on a sphere or a saddleback (hyperbolic).
When you draw 90° angles on a paper you’ll be pointing back at the angle you started after 4 times (a square). On a sphere, think of the earth, you can start at the north pole, go to the equator, turn 90°, ride the equator for a quarter of it, turn 90° again and go back to the north pole… Just turn 90° a third time and you’ll be pointing back to where you started.
One geometry sums to 360° the other just to 270°. But always with right angles and forming a closed figure.
Also on a sphere, two parallels meet.
I hope you get an idea how eclidean geometry differs from others (there are many more “rules” if you wanna dive a bit deeper).
Also, of course, a smaller square can be drawn on the surface of the earth, like your room. Also, two parallels at those sizes don’t meet xD
When you make assumptions about space you get a geometry (literally, “earth measure”). In math, it’s really a theory of earth-measure, not literally measuring. (That’s surveying.) If you make the same assumptions Euclid did, we call that geometry Euclidean. If you make any other assumptions, it’s “other than Euclidean”.
Geometry that is not in a flat plane, where Euclid’s work was.
For instance the surface of a sphere which is important to things like air travel.
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.
Watch this from Veritasium. Beautiful and simple explanation which will blow your mind.
What we were taught. 1) Get some good ol’ fashioned silly putty. 2) flatten it out. 3) draw a co-ordinate grid. 4) stretch out a side or a corner or something. 5) start doing maths.
As far as I have scrolled, many people gave you the same example of curved spaces that is non-euclidean: a sphere, where angles sum to more than 180 degrees and parallel lines always meet. Correct example, but only one of two main ones.
There is another example of curved space that behaves another way: if you live on the surface of a Pringle (that continues indefinitely, and is always curved like a Pringle). There, triangles have total angles of less than 180 degrees.
Also, since there is more area further away from the middle, the following holds: for a given straight line and one point elsewhere, you can draw multiple straught lines never intersecting the original one, that all pass through the point.
In the plane, this would be the parallel line through that point, which is unique. On a sphere, there can be no such line. But on a Pringle (mathematically speaking, in a space with negative curvature), you can draw multiple such lines, since with so much more space on the outside of the Pringle than on the inside, fitting more straight lines becomes easier.
It’s just geometry on a non flat surface. Stuff like a regular flat triangles internal angles add up to 180 bit, not if it’s on the surface of a sphere
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