Talk:Parallel postulate
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Playfair's axiom has been redirected to here. The Anome 20:21 11 Jun 2003 (UTC)
The following sentence is not true. It's true for hyperbolic geometry, but not elliptic geometry.
- The parallel postulate is the only postulate of Euclidean geometry which fails for non-Euclidean geometry.
I'm removing the image, because it displays the Corresponding Angles Postulate, not the Parallel Postulate.--DroEsperanto 01:01, 24 June 2006 (UTC)
postulate I postulate II postulate III postulate IV postulate V
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[edit] Pedantic Note
It's a bit of a nitpick, but theorems aren't "proved". They're "proven". Saying that a theorem is "proved" rather than "proven" is like saying that a toaster is "broke" rather than "broken".--Flarity 06:14, 23 October 2006 (UTC)
[edit] Popular Culture
A good way to ruin a Wikipedia article is to show how the topic relates to what is euphemistically known as "popular culture." It is possible that contemporary "popular culture" is approaching an all-time nadir and is the most vile and decadent expression since the fall of the Roman Empire.Lestrade 15:39, 5 March 2007 (UTC)Lestrade
I looked on google & aol search and couldn't find 1 reference for that film, so i'm not to sure it even exists.No references for the director either Dinonerd 17:39, 5 June 2007 (UTC)Dinonerd
It's failed the google test so I am going to go ahead and remove it. As far as I can see it's just a bit of promotion for an otherwise unnotable minor project. Whoever wrote it up could come back with some references if they wish to put it back up, but it'll take a lot of convincing to show that this film/director which so far as I can tell, is basically unknown is 'popular' or notable. --I 05:34, 8 June 2007 (UTC)
[edit] Archimedes
I tagged the statement by Archimedes due to concerns about its accuracy. The full list of his treatises is given in the article Archimedes, and none is called On Parallel Lines. Some clarification is needed here or this information may be removed. --Ianmacm 18:13, 12 August 2007 (UTC)
[edit] Equivalence
There's a contradiction between this article and the one on Euclidean geometry. The latter says that Playfair's axiom is equivalent to Euclid's parallel postulate, and this article says that's not true. I did a lot of googling and reading to clarify this, but I'm not sure whether I've missed something and I'd like to discuss this here first before making changes.
The question is whether it follows from the other four postulates that there is at least one parallel. This articles denies this, whereas the German article on the parallel postulate explicitly affirms it.
As the article states, Euclid's Proposition 16, the Exterior Angle Theorem, plays a central role, since it is used in proving that there is at least one parallel. Cut the Knot has a clear analysis of Proposition 16 and the implicit assumption that Euclid made in proving it. (See also [1], p. 163.)
Clarification of the status of Proposition 16 is complicated by the ambiguity in the use of the term absolute geometry. The general idea is that absolute geometry is the geometry that results from Euclid's first four postulates without assuming the fifth (in fact this is how Mathworld defines absolute geometry); however, due to the recognition that there are some problems in Euclid's approach, various other axiom systems have been developed for absolute geometry -- see [2], which is the most extensive source on this I found.
Proposition 16 does not hold in elliptical geometry. Almost everyone seems to be in agreement that elliptical geometry is not an absolute geometry, but they rarely say which postulates they're assuming for absolute geometry. Indeed, many of the other axiom systems contain incidence axioms or plane separation axioms that can be used to prove Proposition 16 (see [3] ("Elliptic Geometry") and [4] (step 11)) and that are not satisfied in elliptical geometry. This does not, however, decide the question whether Proposition 16 follows from the first four of Euclid's original postulates.
Cut the Knot is the only source I found that explicitly says that elliptic geometry is an absolute geometry -- see [5]. (Interestingly, though they also say that Playfair's Axiom is equivalent to Euclid's parallel postulate: [6].) Also, this page takes the view that Proposition 16 is not a theorem of absolute geometry.
The Wikipedia articles on non-Euclidean geometry, elliptic geometry and hyperbolic geometry only talk about these geometries violating the parallel postulate; they make no statement about the other postulates, but it seems implicit that only the parallel postulate is violated.
The article on absolute geometry, which assumes only the first four postulates, mentions only hyperbolic geometry, not elliptic geometry, as an example, and states that Euclids first 28 propositions are valid in absolute geometry.
Another central question is whether Euclid's second postulate excludes elliptic geometry. (The other three of the first four apparently don't.) This depends on how you interpret "can be extended indefinitely in a straight line". One might need to understand the Greek original in order to tell whether "indefinitely" is meant here in a way that excludes great circles on a sphere that are finite but without ends. (This page takes the view that Proposition 16 is based on the second postulate.)
To summarize, there's a strong consensus on the Net that Playfair's axiom is equivalent to Euclid's parallel postulate (see e.g. [7], [8], [9], [10] / [11]) -- but it's not clear what this equivalence is relative to. It is certain that the equivalence holds given some of the modern axiom sets used in studying absolute geometry, but is unclear whether it holds given the first four of Euclid's original postulates.
Comments would be much appreciated.
