ELI5- How does non euclidean geometry work?
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So, on a flat surface, corresponding points on two parallel lines will always be the same distance from each other no matter how far up or down the line you go
This was the most awkwardly-worded of Euclid’s core rules of geometry. Non-Euclidean geometry is geometry where this rule doesn’t apply.
For example, suppose you have a spherical object which rotates along a single axis which passes through the sphere’s central point. Let’s call the points on the sphere where the axis of rotation intersects with it “poles” and the circle exactly perpendicular to the axis of rotation the “equator.” If you treat the sphere as a flat surface, lines on the sphere running from one pole to the other are all perpendicular to the equator (thus parallel to each other), but corresponding points are much closer together around the poles than at the equator. Math for a surface like this is one type of non-Euclidean geometry.
Most people spend a lot of time around objects like the sphere-rotating-on-axis that I described, so there can be many practical applications for that type of thing.
People have also come up with rules for how geometry works on hyperbolic, parabolic, or other non-flat surfaces.
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