Non-Euclidean geometry
Non-Euclidean geometry is what results from keeping Euclid's other assumptions and replacing his parallel postulate, which says that through a point not on a given line there passes exactly one line parallel to it. In hyperbolic geometry there are infinitely many such lines, and in elliptic geometry there are none. Both are internally consistent, so the parallel postulate cannot be a consequence of the other axioms, and the angles of a triangle need not add to 180 degrees.
Why it matters
It ended two thousand years of attempts to prove the parallel postulate and forced mathematics to treat axioms as assumptions rather than as descriptions of space.
Also written: hyperbolic geometry, elliptic geometry, parallel postulate
Where it comes up
Read about Non-Euclidean geometry on Wikipedia
Sources
- Non-Euclidean geometry Wikipedia
- Non-Euclidean geometry MacTutor History of MathematicsFollows the failed attempts to prove the postulate, which is the clearest way into the subject.
- Nineteenth Century Geometry Stanford Encyclopedia of Philosophy
