What is Euclidean geometry, non euclidean geometry and any other types of geometry?
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Euclidean geometry is just “normal” every day geometry. Up is up, left is left, down is down, etc. You have your x, y, z-coordinates. If you move exactly up, only the z-value changes while x and y remain the same.
Classic geometry is also built around this. For example the angles of a triangle always add up to 180 degrees, and the angles of a square are all 90 degress.
But for example if you move on Earth, which is a sphere, over far distances, these rules are no longer obeyed. Start at the equator, move to the north pole, turn 90 degrees, move down to the equator, turn 90 degrees again, move to your starting point, and you have completed a triangle, but the angles add up to 90+90+90 = 270 degrees! We have broken the fundamental laws of *Euclidean* geometry!
https://qph.cf2.quoracdn.net/main-qimg-65dd7fa64e7cd74cea6e7263708e2a97.webp
This is an example of non-Euclidean geometry. What is perceived as a two dimensional plane is actually curved in space, so the Euclidean rules break down. But then you think, ok, but that is just an illusion. Earth is still a Euclidean object in space, it’s just very big and we don’t see that it is spherical when walking along it.
And that is fair, but the reality is that *space* itself can be non-Euclidean due to gravity and relativity. For example near a black hole with its enormous gravity, the fabric of spacetime, a 3D-fabric, is *curved*. This is why light, which is not affected by gravity, can take what we perceive as non-straight paths, when it is actually following “straight” paths through curved space.
[This is what they are trying to visualize with pictures like these.](https://imageio.forbes.com/blogs-images/startswithabang/files/2018/08/spacetimelu31.jpg?height=677&width=711&fit=bounds)
Non-Euclidean geometry is also often seen particularly in early 3D video games, often by accident because the engine was primitive and not perfect, leading to distortions of perspective. I think Quake (1996) was the first game to have a “perfect” Euclidean 3d-engine, which was a big deal at the time. Sometimes it’s used deliberately, in games like Portal, where you can open wormholes.
In the end, it’s all *geometry*, and non-Euclidean geometry is just more advanced mathematically, and we need to [involve tensors and heavy calculus to keep things consistent.](https://i.ytimg.com/vi/3ULt0IRkqPU/maxresdefault.jpg). This is why general relativity is so difficult for most students.
Euclidean geometry is geometry that occurs on a flat plane, like a piece of paper. It is why the angles of a triangle add up to 180 degrees. It is based on 4 really basic assumptions and a 5th which caused some drama. The 5th called parallel postulate which said
> If you have a line on your sheet of paper and some other point, also on the paper, then you can only draw one single line that is parallel to the first
Except he said it more complicated and probably in Greek or Latin. People thought that this was more complicated than the first 4 and tried to get rid of it by either proving it wrong or trying to prove it based on the first 4 assumptions. Nobody succeeded. It made many people unhappy because it always felt like a solution or refutation was just out of reach. Then some people tried to see what would happen if they changed it.
* What if there were no parallel lines you could draw through that point?
* What if there were 2?
* What if there were many?
And it turned out that you can build entirely consistent geometries from each of them. Eventually a guy named Riemann came along and tied all of the non Euclidean geometries together into a single larger theory.
And the funny thing is, is that this should have been more obvious from the start because you are already familiar with one non Euclidean geometry.
> What if there were no parallel lines you could draw through that point?
That is how a sphere works. If we both walked north on lines we thought were parallel, we would meet at the north pole. A triangle drawn on a sphere has angles that add to more than 180 degrees, we just don’t notice because its hard to draw a triangle big enough to see it. If we lived on a giant pringles chip instead of a sphere, we would get different rules and different triangles. Non Euclidean geometry is what explores all of those different spaces and how things change based on those differences
Have you been reading H.P. Lovecraft by any chance? That’s where I heard of non-Euclidean geometry. The other commenters are right in their definitions. But Lovecraft mostly used it as a shorthand for “the place you are in doesn’t follow the same rules as us.” Some of the drawings of M.C. Escher would also qualify, like [Waterfall](https://m.media-amazon.com/images/I/51vLQyrzsQL._AC_.jpg). Even though it is easy to draw, this is an impossible way for the world to behave. If you read about non-Euclidean geometry in fiction, that’s the kind of weirdness they’re trying to convey
To start off with Euclidean vs non-Euclidean Geometry, we have to take a small step back.
In Math, things are proven to be true by using other proofs as building blocks. But you eventually need something to start from, and those are called postulates. They are things that are so basic that they are simply given to be true.
When Euclid was writing his proof system for geometry, he boiled it down to five postulates that he could use to prove everything with.
1. A straight line segment can be drawn using any two points.
2. Any straight line segment can be extended indefinitely into a line.
3. Given any line segment, a circle can be drawn with one endpoint being the center, the other being on the edge, and the line segment being the radius.
4. All right angles are congruent.
5. If two lines which intersect a third line in such a way that the sum of the angles on one side is less than the sum of two right angles, then the two lines must eventually intersect.
Now that fifth one isn’t as given or obvious as the others. It’s defining how parallel lines work in the most succinct possible way. It’s not as simple as defining a line or a circle or a right angle. But it can’t be proven using the other 4, and it can’t be disproven either. But people said “Well what if it isn’t true?”.
It turns out, the fifth postulate only exists when you’re doing geometry on a flat plane (like a sheet of paper). If you’re doing geometry on a sphere (like the Earth), then it doesn’t hold true anymore.
Euclidean geometry is geometry that exists in forms where that fifth postulate is true (i.e., a flat plane), and non-Euclidean geometry is geometry that exists where the fifth postulate is not true (like on a sphere).
It is all basically about a single question: are all triangles the same?
In Euclidean geometry the answer is yes, and all other types of geometries (that aren’t based on shaped you can’t even make in the normal euclidean 3D space, like spheres or saddles) the answer is no.
An example for non-euclidean real life geometry is the geometry of the surface of a sphere, where triangles have more than 180°.
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