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.

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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At the very simplest –

In Euclidean Geometry two parallel lines will never intersect and the reason for this is because “Euclidean space is flat.”

Ok whatever that means.

In Non-Euclidean Geometry, two parallel lines ***can*** intersect and this is because “non-Euclidean space is curved”.

WTF?

Take a piece of paper and draw two parallel lines – they don’t cross, right? Because paper is flat.

Take a tennis ball and draw two vertical parallel lines at the middle, the lines will connect at the top and bottom of the ball, because the surface of a sphere is curved.

Another example – draw a triangle on paper and add up the degrees in all the corners, you’ll get 180 degrees. If you do the same on a tennis ball you can actually make a triangle with (3) 90-degree corners because it’s actually a wonky-triangle with curved sides.

Long story short – Euclidean geometry is the geometry of shapes drawn on a flat plane. Non-Euclidean geometry is the geometry of shapes drawn in curved space.