ELI5- How does non euclidean geometry work?
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For any non-Euclidean surface (typically non-flat surfaces), we start by first trying to describe what a straight line is. The way we do this is by basically just asking, “if an ant were to walk along this surface without leaning left or right, what path would it make?” So for example, if you put an ant on a ball, a straight line for the ant will be to walk around the equator of the ball, or any other “equator” (more formally called a “[geodesic](https://en.wikipedia.org/wiki/Geodesic),” basically any giant circle around the ball, but *not* any smaller circles, since those require leaning left or right to make).
From there, we consider all the stuff that’s true about Euclidean surfaces and see what’s different. For example, one of Euclid’s postulates is that all lines have exactly one line parallel to them that passes through any specific point. On a sphere though, there just aren’t any parallel lines at all, since they always come back around the sphere to intersect each other.
There are other non-Euclidean surfaces outside of just spheres, such as a wavy sheet basically looks like [a pringle up close](https://study.com/cimages/multimages/16/hyperbolic_paraboloid.png) (this is called [hyperbolic geometry](https://en.wikipedia.org/wiki/Hyperbolic_geometry)), but this is basically our process for examining any new surface. The point of this is to allow us to extend what we know about geometry to talk about things that aren’t flat because we do not live in a completely flat world.
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