How can i understand Non-Euclidean Geometry?

Answer 1

http://en.wikipedia.org/wiki/Non-Euclidean:

The term non-Euclidean geometry (also spelled: non-euclidian geometry) describes hyperbolic, elliptic and absolute geometry, which are contrasted with Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the nature of parallel lines. In Euclidean geometry, if we start with a line l and a point A, which is not on l, then we can only draw one line through A that is parallel to l. In hyperbolic geometry, by contrast, there are infinitely many lines through A parallel to l, and in elliptic geometry, parallel lines do not exist. (See the entries on hyperbolic geometry and elliptic geometry for more information.)

Another way to describe the differences between these geometries is as follows: Consider two lines in a two-dimensional plane that are both perpendicular to a third line. In Euclidean and hyperbolic geometry the two lines are then parallel. In Euclidean geometry the lines remain at a constant distance, intersecting only in the infinite; while in hyperbolic geometry they "curve away" from each other, increasing their distance as one moves further from the point of intersection with the common perpendicular. In elliptic geometry the lines "curve toward" each other and eventually intersect. Therefore no parallel lines exist in elliptic geometry.

Answer 2

To understand Geometry you have to be experienced in doing proofs by contradiction, since most proofs above high school level geometry require this approach.

I think it was Susanne Epp that wrote a Discrete Math book that had good simple examples using integers for writing proofs, but there are other books out there.

Answer 3

watch back to the future