Talk:Parallel (geometry): Difference between revisions
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A plane is by definition a flat (zero curvature) surface in Euclidean space.--[[User:Paul Wormer|Paul Wormer]] 17:04, 25 March 2010 (UTC) | A plane is by definition a flat (zero curvature) surface in Euclidean space.--[[User:Paul Wormer|Paul Wormer]] 17:04, 25 March 2010 (UTC) | ||
== Two remarks == | |||
"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. | |||
[[User:Boris Tsirelson|Boris Tsirelson]] 19:20, 27 March 2010 (UTC) | |||
:Please go ahead, fix it. --[[User:Paul Wormer|Paul Wormer]] 09:53, 28 March 2010 (UTC) | |||
::I did. [[User:Boris Tsirelson|Boris Tsirelson]] 12:14, 28 March 2010 (UTC) | |||
:::Thanks--[[User:Paul Wormer|Paul Wormer]] 15:26, 28 March 2010 (UTC) | |||
== Non-Euclidean parallels == | |||
Boris, the WP article you cited is only partially right. It is quite common to call all non-intersecting lines parallel (e.g., Hilbert). --[[User:Peter Schmitt|Peter Schmitt]] 21:22, 15 April 2010 (UTC) | |||
:Really? I did not know. Well, if so, change it accordingly. [[User:Boris Tsirelson|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. [[User:Howard C. Berkowitz|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"?). --[[User:Peter Schmitt|Peter Schmitt]] 10:22, 17 April 2010 (UTC) | |||
Latest revision as of 04:22, 17 April 2010
flat plane
A plane is by definition a flat (zero curvature) surface in Euclidean space.--Paul Wormer 17:04, 25 March 2010 (UTC)
Two remarks
"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)
Non-Euclidean parallels
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)