[Eli5] What is non-Euclidean geometry, and what would it look like in the real world?

Question

This term seems to come up in science fiction every so often, and the layman description is that it’s just 2d geometry on a curved surface?

In: Mathematics

Answer 1

Euclidean geometry: geometry on a flat thing.
Non-euclidean geometry: geometry on a curved thing. (At least the non-euclidean geometry that the most mathematicians find interesting)

We just so happen to live on a globe, not on a flat thing.
Notice how on flight maps the course between places isn’t a straight line, it’s a curve?
That’s because it’s a straight line on a curved surface, and someone just did some non-euclidean geometry to figure out where to go.