A plane is by definition a flat (zero curvature) surface in Euclidean space.--Paul Wormer 17:04, 25 March 2010 (UTC)
"do not cross at any point, not even at infinity" — in elementary texts there is no such notion as intersection at infinity; in non-elementary texts (say, projective geometry) such notion exists, and it appears that parallel lines do intersect at infinity.
"parallel lines satisfy a transitivity relation" — no, it is not, unless we agree that each line is parallel to itself.
Boris Tsirelson 19:20, 27 March 2010 (UTC)
- Please go ahead, fix it. --Paul Wormer 09:53, 28 March 2010 (UTC)
- I did. Boris Tsirelson 12:14, 28 March 2010 (UTC)
- Thanks--Paul Wormer 15:26, 28 March 2010 (UTC)
Boris, the WP article you cited is only partially right. It is quite common to call all non-intersecting lines parallel (e.g., Hilbert). --Peter Schmitt 21:22, 15 April 2010 (UTC)
- Really? I did not know. Well, if so, change it accordingly. Boris Tsirelson 05:20, 16 April 2010 (UTC)
Axiom or postulate?
Is the proper Euclidean term the "Parallel Axiom" or "Parallel Postulate"? I learned it as the latter, which I think is traditional although axiom would be more correct modern mathematical terminology. Howard C. Berkowitz 01:00, 17 April 2010 (UTC)
- You are right: "Parallel Postulate" is frequently used. But there is a subtle difference. In his Elements, Euclid has a "postulate" which is -- in the context of his geometry -- equivalent to (but different from) the "uniqueness of a parallel" (which can be used in a much more general context). I think that (finally) there can be (and should be) both a page on Parallel Postulate and Parallel Axiom (or "Parallel axiom"?). --Peter Schmitt 10:22, 17 April 2010 (UTC)