Talk:Euclidean geometry

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Contents 1 plane versus solid 2 merging with non-Eucldean? 3 Is not enough! 4 statement of postulates 5 Parallel postulate 6 completeness 7 construction 8 removed part 9 grammar 10 Spelling 11 Inactive link 12 GA Re-Review and In-line citations 13 Non-Euclidean geometries? 14 As a description of physical reality 15 reversion of extensive anonymous revisions

[edit] plane versus solid

I disagree that Euclidean geometry refers primarily to plane geometry. Certainly I have never thought of it that way. Euclid also did not restrict him to plane geometry in his "Elements". So I hope you don't mind, but I changed that bit. mike40033 07:18, 12 Mar 2004 (UTC)

I would sugest at least

Euclidean geometry often means geometry in the plane

for me (I'm russian) Euclidean geometry means plane geometry if not stated otherwise, maybe for all of you it is different, but then it is not clear why the article on Euclidean geometry explains what plane geometry is...? Tosha 12:43, 29 Jul 2004 (UTC)

I've reworded everything so it's not necessary to make any statement about what it's generally taken to mean.--Bcrowell 19:00, 4 March 2006 (UTC)Hassan. Abdullahi.

[edit] merging with non-Eucldean?

Should this article be merged with Non-euclidean geometry? -- The Anome

No, I don't think so. This article ought to discuss Euclidean geometry. At the moment it doesn't actually say much about Euclidean geometry, and instead spends too much time discussing non-euclidean geometry, which is already discussed in Non-euclidean geometry. So it needs a lot of work, and some of it should be moved to Non-euclidean geometry, but it should remain a separate article. --Zundark, 2001 Dec 22

I agree with Zundark that it's not a good idea to merge the two articles. I also agree that the article is somewhat unbalanced by the large amount of discussion of non-Euclidean geometry. However, I disagree with Zundark about the best solution to the problem. I think the solution is simply to add more material about Euclidean geometry. The article would be incomplete if it didn't explain Euclidean geometry's relationship to other forms of geometry. I think it would also be a good idea to rework the article at some point so that all the fancy stuff (non-Euclidean geometry, Godel's theorem, higher-dimensional spaces) comes as late as possible in the article. However, I think it would be a waste of time to do that right now; we should do that after more material on Euclidean geometry itself is added. One obvious thing to add would be a sort of "greatest hits" list of important theorems in Euclidean geometry. It would also be cool to show at least one example of a nontrivial compass-and-straightedge construction.--Bcrowell 19:00, 4 March 2006 (UTC)

Oh, it does have a greatest-hits list at the bottom, but it doesn't tell us what these theorems say, or how important they are. It would be nice to have an example of a complete proof in Euclidean geometry, and I think a good example would be Book 1, Proposition 5, known as the "pons asinorum" or "bridge of fools."--Bcrowell 23:38, 4 March 2006 (UTC)Hassan. Abdullahi.

[edit] Is not enough!

In deed,the article is well constructed showing us the basics of Euclidian Geometry but i think is a little short for those who are intrested to learn more about this subject.I hope to find out more in the future!

[edit] statement of postulates

The five postulates are:

1. To draw a straight line from any point to any point.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any center and radius.
4. That all right angles equal one another.
5. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

only two perpendicular line can make a 90 degre angle.

Source: http://s13a.math.aca.mmu.ac.uk/Geometry/M23Geom/Euclid/Euclidbook1.html
looxix 21:51 Feb 23, 2003 (UTC)Hassan. Abdullahi

[edit] Parallel postulate

The fifth postulate is equivalent to parallel postulate, which can be phrased as follows

• Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.

I don't believe this is true, nor the correct way or naming things. The "parallel postulate" to be completely accurate should always refer to Euclid's axiom as he stated it. I believe what is being called the "parallel postulate" here is actually what is rightly called "Playfair's axiom", although to be perfectly honest, there is even some doubt in my mind as to whether this term is meant to mean "exactly one line" or "at most one line" (Playfair used the former phrase, Legendre the latter, so there is some confusion for me, or if it matters.) Revolver 06:30, 20 Mar 2004 (UTC)

The term "equivalent" here is being used, correctly, in a certain strict formal sense. It says that either form of the axiom can be used to prove the other form. The two statements are not obviously equivalent to someone who doesn't know the proof, but that's not what the article is asserting.--Bcrowell 19:00, 4 March 2006 (UTC)

[edit] completeness

Currently, the article includes:

 As Godel proved, all axiomatic systems -- excepting the very simplest -- 
 are either incomplete or contradict themselves, and this is no exception.

It seems to me that a sufficiently simple axiomatization of Euclidean geometry might actually be complete. I don't see any way to embed the natural numbers in Euclidean geometry, which is the usual way to verify that Gödel's theorem applies. -- Carl Witty

Carl Witty is correct, and the original remark in the article was incorrect. Godel's theorem doesn't apply here. The article now states it correctly: Euclidean geometry has been proved to be consistent and complete. I've provided a footnote with a reference to a book that discusses this in depth.--Bcrowell 19:00, 4 March 2006 (UTC)Hassan. Abdullahi

[edit] construction

Isn't there some work on showing how to make computations with ruler-and-compass constructions, providing you have a pre-existing "program"? The Anome 19:36 21 May 2003 (UTC)

It's:

• Simon Plouffe.The Computation of Certain Numbers Using a Ruler and Compass. Journal of Integer Sequences, Vol. 1 (1998), Article 98.1.3

The Anome 19:56 21 May 2003 (UTC)

[edit] removed part

I just removed new subsection, it was correct but irrelevent, might go somewhere else... Tosha 00:00, 30 Mar 2004 (UTC)

Thanks for stating your points of view regarding the correctness and relevence (sic) of the characterizations of physical spaces in terms of Euclidean geometry succinctly, yet explicitly and separately.

Thanks also for the generous scope of your suggestions where else within this encyclopedic representation of what's considered correct (at least: rather than "any where else but ...") this topic might be addressed instead.

Being left to narrow this considerable selection down, perhaps (at least) to

- the discussion of Euclidean planes and spaces by the author of A Modern View of Geometry, W. H. Freeman (1961),

- derivations and statements of certain expressions (such as that attributed to one Tartaglia) which some do seem to find noteworthy after all,

- considerations rather less frivolous than [[Talk:Why 10 dimensions]], or

- being an [[Wiktionary:Also-ran]] to what appears already established,

the choice appears nevertheless daunting ...

Regards, Frank W ~@) R 03:44, 30 Mar 2004 (UTC).

[edit] grammar

Some grammatical errors are in this article 209.155.121.101 13:46, 22 December 2005 (UTC)

I've tried to clear these up.--Bcrowell 19:01, 4 March 2006 (UTC)Hassan.Abdullahi

[edit] Spelling

Should it be "Euclidean geometry" or "euclidean geometry"?

[edit] Inactive link

The link titled "In English" under "The Elements" (http://aleph0.clarku.edu/~djoyce/java/elements/toc.html) is inactive. Someone restore it or just remove it.

Pizzadeliveryboy 08:40, 11 August 2006 (UTC)

[edit] GA Re-Review and In-line citations

Members of the Wikipedia:WikiProject Good articles are in the process of doing a re-review of current Good Article listings to ensure compliance with the standards of the Good Article Criteria. (Discussion of the changes and re-review can be found here). A significant change to the GA criteria is the mandatory use of some sort of in-line citation (In accordance to WP:CITE) to be used in order for an article to pass the verification and reference criteria. Currently this article does not include in-line citations. It is recommended that the article's editors take a look at the inclusion of in-line citations as well as how the article stacks up against the rest of the Good Article criteria. GA reviewers will give you at least a week's time from the date of this notice to work on the in-line citations before doing a full re-review and deciding if the article still merits being considered a Good Article or would need to be de-listed. If you have any questions, please don't hesitate to contact us on the Good Article project talk page or you may contact me personally. On behalf of the Good Articles Project, I want to thank you for all the time and effort that you have put into working on this article and improving the overall quality of the Wikipedia project. Agne 05:55, 26 September 2006 (UTC)

[edit] Non-Euclidean geometries?

If you change the fifth postulate while keeping the other four, you get the hyperbolic, elliptical, and absolute geometries. But what sort of geometry do you get if you change the fourth postulate while keeping the others? --Carnildo 07:21, 12 December 2006 (UTC)

[edit] As a description of physical reality

I think this section should somehow be reworked. It is mostly about GR and seems to repeatedly overstate things. The article states: "no possible physical test (that) can do any better than a beam of light", but this failed long before GR. Newtonian gravity would cause light to bend. The theories differ in how much the light bends. Light has always been known to bend through glass. Even saying the light was "bent by the Sun's gravity" is not consistent. It really highlights that there IS a way to consider something straighter than light - that being the unbent path one might have thought the light could have taken. So it seems absurd that we might have to "reject the entire notion of physical tests of the axioms of geometry" because light bends. Also the picture caption starts with the word "proof". This word has very special meaning for this page and should not be used so casually. Using the phrase "shows that the true geometry" seems to be using common words that tend to be stronger even than "proves". I think that the article should somehow say that GR POSITS a curved space. Ned Phipps 23:01, 29 December 2006 (UTC)

Light has always been known to bend through glass. The article is talking about light in a vacuum. It really highlights that there IS a way to consider something straighter than light - that being the unbent path one might have thought the light could have taken. Your idea doesn't work, because the concept "could have taken" is undefined. What the article says about GR is completely standard; check any textbook on GR.--75.83.140.254 01:39, 20 January 2007 (UTC)

Actually, the concept "could have taken" is /very/ well defined and is the meaning of having measured the amount of bending (or GR's apparent bending) and having seen the that amount is double the amount that Newton's gravity predicted (light bending within Euclidean space /in a vacuum/). The text reading "is bent by the Sun's gravity" raises, all by itself, the concept that something is considered straighter. I am quite aware of standard GR and that GR says the parabolic path that a baseball takes is apparent only. Finding words to describe /that/ bending as fundamental would be much more descriptive of GR. It is the extreme wording that I object to. "reject the entire notion of physical tests of the axioms of geometry". Physical tests can never rise to the level of /proof/ that is so special to Euclidean Geometry. They should never be completely accepted or rejected.Ned Phipps 02:09, 25 January 2007 (UTC)

"reject the entire notion of physical tests of the axioms of geometry". Physical tests can never rise to the level of /proof/ that is so special to Euclidean Geometry. The text you're quoting seems to agree with your statement. The text reading "is bent by the Sun's gravity" raises, all by itself, the concept that something is considered straighter. If a geodesic passes from point A to point B, through the sun's gravitational field, then there is no other geodesic from A to B that is more straight than that one. The bending being referred to occurs for an observer who has chosen a particular set of coordinates (which are asymptotically flat); a different observer could choose a different set of coordinates in which the ray of light was not considered to be bent, but in those coordinates spacetime would not be asymptotically flat. Obviously this is not an article on general relativity, so it's not practical to go into great detail on this sort of thing. Any short discussion that doesn't use precise, technical terminology and notation is going to be subject to incorrect interpretations.--75.83.140.254 01:35, 29 January 2007 (UTC)

I think that the terminology should be reversed. Something like:

With GR the concept of a path of an object bending is discarded. Instead, objects go straight (geodesics) and the space, itself, is considered curved.

Using the path of light as the example of what bends is very misleading since that is essentially straight whereas the path of a baseball is considered a bend in one and not the other.

My other concern has to do with the fact that experimental verification of this theory is only in the very first order deviations from "flatness" in space and time. Hardly a "proof". There's quite a difference between a physicists theory that has supporting evidence a mathematical proof as introduced by Euclid and which make Euclidean Geometry so special.64.161.207.162 00:10, 1 February 2007 (UTC)

Einstein himself deals elegantly with the jusification for discarding Euclidean geometry as a description of space-time in his 1938 book The evolution of Physics pp 222-234. It is not that one or the other is "True". It is possible to model the universe either way (or in any number of other ways). The strength of General Relativity is in the simplicity and elegance of its underlying postulates, whilst still providing the most accurate match yet obtained for observations made in the real universe. DaveApter 13:16, 20 March 2007 (UTC)

[edit] reversion of extensive anonymous revisions

I am reverting the edits by anonymous user 75.83.140.254 .

I would request that anyone contemplating such extensive edits be logged in, and prepared to discuss the merits of the changes with other editors to reach consensus. DaveApter 14:34, 25 January 2007 (UTC)

I'm available via my talk page, and I'm ready to discuss anything you want to discuss. How about discussing edits on their merits, rather than carelessly undoing other people's work? Reverting.--75.83.140.254 01:16, 29 January 2007 (UTC)

Considering that the major portion of your edit (apart from a few minor wording changes) was the wholesale removal of a substantial section of the article which other editors considered relevant and interesting, I respectfully suggest that you take your own advice. DaveApter 13:05, 20 March 2007 (UTC)