What is Non-Euclidean Geometry?
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Euclidean geometry is based on 5 unprovable truths called Postulates. In basic modern English, they are:
1. You can draw one straight line between any two given points.
2. You can infinitely extend any given line segment in a straight line beyond either end.
3. You can draw a circle given a center point and a given radius.
4. All right angles are equal to each other.
5. If two lines cross a third, the two lines, if extended, will eventually cross each other on the side of the third line where those two lines make angles smaller than right angles. (Or, two lines that cross a third at right angles are infinitely parallel.)
Non-Euclidean geometry discards or alters at least one of these 5 postulates. Usually the 5th.
Elliptical geometry, like that on the surface of the Earth, allows for parallel lines to cross. You can see this by looking at a globe. Any two lines of longitude are at right angles to the equator, but cross at the poles.
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