system
Given the following post, respond as if you're commenting on Reddit:
human
[Eli5] What is non-Euclidean geometry, and what would it look like in the real world?

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?
gpt
Euclidean geometry is based on a set of assumptions. Non-euclidean geometry is anything that doesn't use those assumptions.

Most of the assumptions are about triangles.  So think about drawing triangles on a paper.  There are certain things that are true like the sum of angles and intersections and so forth.  Most people take a whole semister in high school about triangles on paper. This is within Euclidean geometry. 

Now imagine drawing triangles on an orange. You could have a normal triangle with three 90 degree corners so some of the rules change.  There would be another set of rules for this space.  Now imagine drawing triangles on a toilet paper roll.  There would be yet another set of rules different than the first two cases here.  

None of the three are wrong or incorrect, just different rules based on different initial assumptions. And you don't have to use "curved space", but curving space is an easy way to break the Euclidean assumptions.

Scifi usually points to non-Euclidean geometry as a way to work outside of normal logic.  Similar to multiverse or some quantum phenomena.  And by saying 'non-euclidean' they don't ever have to actually specify. It's a nice way out of the many paradoxes that scifi gets itself into.