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.
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