What is Euclidean geometry, non euclidean geometry and any other types of geometry?
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To start off with Euclidean vs non-Euclidean Geometry, we have to take a small step back.
In Math, things are proven to be true by using other proofs as building blocks. But you eventually need something to start from, and those are called postulates. They are things that are so basic that they are simply given to be true.
When Euclid was writing his proof system for geometry, he boiled it down to five postulates that he could use to prove everything with.
1. A straight line segment can be drawn using any two points.
2. Any straight line segment can be extended indefinitely into a line.
3. Given any line segment, a circle can be drawn with one endpoint being the center, the other being on the edge, and the line segment being the radius.
4. All right angles are congruent.
5. If two lines which intersect a third line in such a way that the sum of the angles on one side is less than the sum of two right angles, then the two lines must eventually intersect.
Now that fifth one isn’t as given or obvious as the others. It’s defining how parallel lines work in the most succinct possible way. It’s not as simple as defining a line or a circle or a right angle. But it can’t be proven using the other 4, and it can’t be disproven either. But people said “Well what if it isn’t true?”.
It turns out, the fifth postulate only exists when you’re doing geometry on a flat plane (like a sheet of paper). If you’re doing geometry on a sphere (like the Earth), then it doesn’t hold true anymore.
Euclidean geometry is geometry that exists in forms where that fifth postulate is true (i.e., a flat plane), and non-Euclidean geometry is geometry that exists where the fifth postulate is not true (like on a sphere).
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