Why do they have to be geometries of their own kind? For example, instead of talking about lines in spherical geometry, can’t we just imagine circles on a literal sphere in Euclidean space?
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> Don’t Euclidean shapes like sphere and hyperbola suffice?
Many non-Euclidean geometries can be embedded in higher-dimensional Euclidean spaces, but sometimes you need to add a lot more dimensions, and often there are aspects of the geometry that are more convenient to describe using the lower-dimensional non-Euclidean description. And nowadays there are lots of complicated geometries people study that simply cannot be described using a higher-dimensional Euclidean space.
Because a circle on a sphere in Euclidean space is a circle and is treated as such. It has the properties of a circle as defined by Euclid.
But a circle on a sphere in spherical space is a straight line and has the properties of a straight line as defined by Euclid.
Look at a map of the earth. Draw a straight line between say London and Los Angeles. Is that the shortest path? It’s not, because the surface of the Earth is not a Euclidean surface and a map of the Earth (which is technically Euclidean) is a distortion of the actual Earth. At local distances the earth is flat enough for Euclidean geometry to be good enough but it doesn’t take that far for curvature to start to matter.
We actually don’t need it for most purposes. On Earth, it’s mostly to help map longitude and latitude. Non-euclidian geometries really come into their own when mapping out Gravity since Gravity warps space. The very idea of straight lines gets a bit wonky.
Sure, you can, in the same way that you can think of 9x as x+x+x+x+x+x+x+x+x. But you wouldn’t want to, because it either denies, or makes very inconvenient, a lot of tools you’d want to use to attack problems.
Ultimately, everything in math is a huge pyramid scheme built on top of only a few basic logical symbols. The point of math is to abstract away the lower levels of symbol with simpler, higher levels of symbol that reduce the number of steps and the amount of effort you need to see and talk about the behavior of mathematical objects.
This approach turns out to be especially inconvenient when you’re talking about curved geometry that isn’t somehow “sitting in a higher dimensional space”. For example, space-time in our Universe is curved, in the sense that its geometry is non-Euclidean, but it’s not “inside some bigger space” as far as we know. We can mathematically describe it as one if we really want to (it’s called an [embedding](https://en.wikipedia.org/wiki/Embedding#Differential_topology)), but then we’re inventing some higher-dimensional fiction for no reason (and it turns out that, in general, you [need twice as many dimensions for this to work](https://en.wikipedia.org/wiki/Whitney_embedding_theorem) – a sphere can do with 3, but other curved 2d surfaces like the Klein bottle need 4; 4d spacetime requires, in general, eight!).
If we have a coordinate system in say 2D Euclidean space, the distance ds between two points at location (x,y) and (x+dx,y+dy) is ds^2 = dx^2 + dy^2. That changes in non-Euclidean spaces. On the surface of the Earth, you have ds^2 = dlat^2 + cos(lat)^2 dlon^2, where dlat and dlon are the (small) differences in latitutde/longitude, and cos(lat) is the cosine of the latitude. There’s no way to come up with a coordinate system in 2D that converts those distances to the Euclidean ones, so just by measuring distances and lat/lon, you can tell you are on a curved surface. This extends to higher dimensions as well, we could have (but don’t) live on a universe where the 3D space looks like the surface of a 4D sphere. Very often, it’s easier to do math taking advantage of that fact! Because my spherical trig sucks, I still measure distance between points on the Earth’s surface by going back into 3D.
However, in real life, the situation rapidly becomes more complex. If you work in say cosmology, or with black holes, you can still write down the distances, but they no longer correspond neatly to a simple higher dimensional space where it’s easy to do math. Picture the surface of the Earth as it really is – with mountains, valleys etc. If you want to measure distances, you need to take that into account. You *could* do it by including an altitude map of the Earth, but very often what happens is that the math/physics tell you how to calculate the distance, so it would be an awful lot of work to go from that back to the 3D surface when you could always just work in lat/lon, where elevation just gets taken care of from the physics.
If this sounds complicated, it is! That’s why people don’t take courses in general relativity until they’re getting graduate degrees in physics.
The Earth is a sphere in Euclidean space.
The surface of the earth however, is not.
If you travel along a sheet of paper, and make 4, 90 degree turns, the path you make will be a rectangle since you would be moving in Euclidean space. If you did the same thing on the surface of Earth, your path will not be a rectangle, hence a very simple example of why we need non-euclidean geometry.
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