What is Euclidean geometry, non euclidean geometry and any other types of geometry?
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Euclidean geometry is geometry that occurs on a flat plane, like a piece of paper. It is why the angles of a triangle add up to 180 degrees. It is based on 4 really basic assumptions and a 5th which caused some drama. The 5th called parallel postulate which said
> If you have a line on your sheet of paper and some other point, also on the paper, then you can only draw one single line that is parallel to the first
Except he said it more complicated and probably in Greek or Latin. People thought that this was more complicated than the first 4 and tried to get rid of it by either proving it wrong or trying to prove it based on the first 4 assumptions. Nobody succeeded. It made many people unhappy because it always felt like a solution or refutation was just out of reach. Then some people tried to see what would happen if they changed it.
* What if there were no parallel lines you could draw through that point?
* What if there were 2?
* What if there were many?
And it turned out that you can build entirely consistent geometries from each of them. Eventually a guy named Riemann came along and tied all of the non Euclidean geometries together into a single larger theory.
And the funny thing is, is that this should have been more obvious from the start because you are already familiar with one non Euclidean geometry.
> What if there were no parallel lines you could draw through that point?
That is how a sphere works. If we both walked north on lines we thought were parallel, we would meet at the north pole. A triangle drawn on a sphere has angles that add to more than 180 degrees, we just don’t notice because its hard to draw a triangle big enough to see it. If we lived on a giant pringles chip instead of a sphere, we would get different rules and different triangles. Non Euclidean geometry is what explores all of those different spaces and how things change based on those differences
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