Talk:Non-Euclidean geometry
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| History needs dividing; needs more on the maths(!). Tompw 19:06, 5 October 2006 (UTC) | |||||
consider two lines in a plane that are both perpendicular to a third line. In Euclidean and hyperbolic geometry, the two lines are then parallel.
I am not at all happy with the above. As far as I know, in hyperbolic geometry two lines in a plane are either intersecting, or parallel, or ultraparallel. Two lines which are both perpendicular to a third line are ultraparallel, not parallel.
See e.g. http://s13a.math.aca.mmu.ac.uk/Geometry/M23Geom/NonEuclideanGeometry/NonEuclidean.html
I am concerned about the matter, also because the statement has been used (in translation) in the corresponding Swedish article.
Sebastjan
- This is a well known inconsistency in terminology. When specifically working with hyperbolic geometry, one does use the words parallel, ultraparallel, etc. However, when talking in general about geometry, such as in the above excerpt, parallel is taken to mean two lines that do not intersect. This shouldn't be a problem as long as this inconsistency is noted in the article on hyperbolic geometry; I'll check. --Chan-Ho Suh 12:18, Dec 16, 2004 (UTC)
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- I'm also unhappy with this, actually. I was always under the impression that in general, the concept of parallelism requires the provision that they remain equal distance apart as well as never intersecting. But I'll look it up again.--Cwiddofer 08:18, 24 June 2006 (UTC)
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[edit] When did 'debate' on parallel postulate begin?
From the history section:
- The debate that eventually led to the discovery of non-Euclidean geometries began almost as soon as Euclid's work Elements was written.
This slipped by me a couple times, but now I realized that this seems very suspicious. I can't think of any ancient Greek works debating or casting doubt, or whatever, on the parallel postulate. It seems to me that the debate, so to speak, began when Greek works, such as Euclid, were discovered by the Europeans. --Chan-Ho Suh 08:49, Dec 19, 2004 (UTC)
Yes, good point. To clarify, you mean "(re-)discovered by the (Western) Europeans (during the Renaissance)"
[edit] And When Was the First Work on Elliptic Geometry Done?
"In a work titled Euclides ab Omni Naevo Vindicatus (Euclid Freed from All Flaws), published in 1733, he quickly discarded elliptic geometry as a possibility"
Did he both discover elliptic geometry for the first time, and discard it -- or had someone previously investigated it?
If he discovered it, did he develop it thoroughly and refer to it by the term "elliptic geometry"?
[edit] Diagram with hyperbolic
That diagram is only true for Poincare hyberolic disc lines, not other kind of hyperbolic geometries like the Klien model.
[edit] References - correct title?
There is a reference to Beltrami's work Theoria fondamentale delgi spazil di curvatura constanta. This is incorrect Italian (but see my comment in the next paragraph). Correct Italian would be (with corrected words emboldened) Teoria fondamentale degli spazii di curvatura costanta.
That said, I have no idea what the actual title of the work is, however, and it is of course possible that Beltrami just spelled it wrong, or that it is in an Italian dialect, or that Italian really has changed as much as this since the 19th century. I think it certainly needs to be checked; if it is found to be correct as it stands, perhaps a [sic] should be added. — Paul G 11:03, 11 May 2006 (UTC)
[edit] Is the 'Fiction' Section apropriate?
I argue that the 'Fiction' section in this article is incongruous with the rest of the article. It has the look of an appendage, and would only be appropriate in an article that has a more expansive and varied content. Splendour 07:35, 28 June 2006 (UTC)
[edit] Message from the Masses
i just read this page in order to understand noneuclidean geometry, and i still don't. i don't think it did a good job of explaining to me. of course curved lines wouldn't be parallel (regarding elliptic) but how in the world could they BE parallel in hyperbolic? and isn't the definition of a line that it is straight? a curve is...well, a curve, not a line. perhaps if someone could recruit an expert? —The preceding unsigned comment was added by 206.80.23.226 (talk • contribs).
-I agree, an expert is needed here. I was hoping the article would discuss triangles. I was coming here to refresh my memory on that, and was surprised to see no info on triangles. Back when I had geometry in college, we were taught that one type of noneuclidean geometry contained triangles of less than 180, and the other contained more than 180. (To visualize this, since I guess it essentially doesn't exist, the "lines" ARE "curves.) There is a little about this in the triangles article under Non-planar triangles: http://en.wikipedia.org/wiki/Triangle
See article on manifold. Would spheres be non-euclidian? Seems I do recall postulates and formulas regarding them from high school geometry. Parallel lines meet at the poles but to what extent are they really parallel-at the equator?Tom Cod 08:31, 3 November 2006 (UTC)
- "On the spherical plane there is no such thing as a parallel line." from Parallel (geometry). ~a (user • talk • contribs) 14:32, 3 November 2006 (UTC)
