How is non-euclidian geometry like?

Answer 1

Non euclidean geometries differ from Euclidean geometries in the 5th postulate. Euclid's 5th postulate is that given a line and a point not on that line there is exactly 1 line through the point that does not intersect the line.

For hundreds of years mathematicians distrusted this postulate. They considered it very different from the others. which seemed evident, and there were many attempts to prove the 5th postulate, thereby showing it was not needed.

All these attempts failed, and it wasn't until the 19th century that Lobachevsky explained why. He showed that, by choosing a different parallel postulate, you could come up with a perfectly consistent geometry, even though it behaved differently than Euclidean. His parallel postulate allowed 2 parallel lines (and, as could be proven from there, infinitely many).

Riemann later used the other extreme, that there were 0 parallels through the point. This makes it very easy to see that it is consostent, since Riemann's geometry is simply the geometry on the surface of a sphere, with a great circle being a line.

Answer 2

Strictly speaking, the surface of the Earth is an example of a no-Euclidian geometry. For example, triangles drawn on the surface of a sphere always have angles that sum to greater then 180: in a Euclidean geometry the sum is always 180.

Answer 3

Your first introduction to formal proofs. Pay particular attention here if you plan to enroll in advanced calculus and differential equations courses. Discrete mathematics is also proof intensive.