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[edit] plane versus solidI 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
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
[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! Wikipedia can be edited by anyone. Give it a go.--THobern 21:54, 18 June 2007 (UTC) [edit] statement of postulatesThe five postulates are:
only two perpendicular line can make a 90 degre angle. Source: http://s13a.math.aca.mmu.ac.uk/Geometry/M23Geom/Euclid/Euclidbook1.html [edit] Parallel postulateThe fifth postulate is equivalent to parallel postulate, which can be phrased as follows
[edit] Non intuitiveness of the fifth postulateFrom the article: To the ancients, the parallel postulate seemed less obvious than the others; verifying it physically would require us to inspect two lines to check that they never intersected, even at some very distant point, and this inspection could potentially take an infinite amount of time. This sounds dubious to me. First Euclid's fifth postulate was not formulated in terms of parallel lines. Second I don't think that there is any evidence that the ancients thought that way. And finally why doesn't this apply to the second postulate? Doesn't it take infinite time to examine that a straight line can be indefinitely extended? Nikos.ap 13:15, 12 June 2007 (UTC) [edit] completenessCurrently, 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
[edit] constructionIsn'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:
The Anome 19:56 21 May 2003 (UTC) [edit] removed partI just removed new subsection, it was correct but irrelevent, might go somewhere else... Tosha 00:00, 30 Mar 2004 (UTC)
[edit] grammarSome grammatical errors are in this article 209.155.121.101 13:46, 22 December 2005 (UTC)
[edit] SpellingShould it be "Euclidean geometry" or "euclidean geometry"? [edit] Inactive linkThe 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 citationsMembers 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) I'm not aware of anyone ever attempting that. The original rational for trying alternatives to the fifth one was a feeling that the other four were intuitively compelling, but that was not to the same degree (of course this was in the context of the assumption of the time, that the system was both an axiomatic schema and an accurate description of physical space). DaveApter (talk) 15:32, 16 October 2008 (UTC) [edit] As a description of physical realityI 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)
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)
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) Hmm, the statements about the standard model and GR are all in the class of 'sort-of-true'. When I have some time I will try to rewrite the entire paragraph. The problem is that, for instance, the standard model is not *based* on Euclidean geometry. There is a mathematical trick which could be interpreted as that (which goes under the name 'Wick rotation'), but it can be shown that this is needed to make the theory well defined (more precisely, to define the right causal propagator). I would say that there are far more fundamental problems than choosing the geometry. And even then, the geometry should be a result of the theory, not an input (see background independence). —Preceding unsigned comment added by 130.225.212.4 (talk) 08:33, 27 May 2008 (UTC) [edit] reversion of extensive anonymous revisionsI 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)
[edit] figure of Euclidthis is of course not a true depiction of the historical Euclid, whose identity is contested as it is. I think this should be mentioned in the figure caption. --128.139.226.37 12:55, 23 April 2007 (UTC) [edit] Parabolic and elliptical?The recent edits by Ray Chason [[1]] seem to me to have replaced two statements that were factually accurate with two that were not. I just thought I'd check before reverting them? Any comments? DaveApter 16:40, 22 May 2007 (UTC) I have gone ahead and corrected them. DaveApter 12:52, 24 May 2007 (UTC)
[edit] Euclid's axioms v.s. postulatesI thought that Euclid's axioms were separate from his 5 postulates. And consisted of his basic definitions for lines, points, and circles. Lonjers 06:30, 8 June 2007 (UTC) [edit] Gravitational theoryI have attempted to rewrite this section with several goals in mind.
One fine technical point: It is traditional when discussing the ellipse/parabola/hyperbola trichotomy to speak in terms of eccentricity; the concept of intersections with infinity, while anachronistically modern (compared to Kepler), seems to more directly clarify the connection to orbits. Perhaps the revision is a little better than its predecessors, if not ideal. I would hope that future edits retain some mention of the curvature of space. --KSmrqT 18:49, 16 June 2007 (UTC)
Thank you, KSmrq. I would agree that many of the more arcane points you make may be appropriate to an article on astronomy or celestial mechanics, but not to this one that is about geometry. However, I do feel that there are some aspects of this topic that are relevant to this article, and I will expand on this below. Furthermore, the restriction of the discussion to comets seems to detract from the relevance to the subject of geometry. Let us start by setting out the points on which we agree: 1) The conic-section shapes of orbits and trajectories are only approximations to what is observed in the real world (even on Newtonian assumptions). Reasons for this include (a) The fact that the bodies are not point masses, (b) the irregularity in shape and density of real bodies, (c) the gravitational interference from other bodies, and (d) where relevant, the effect of atmospheric viscous drag. Agreed. On the other hand, Kepler had plotted the orbits of the known planets as being ellipses, to a high degree of accuracy. 2) A body moving tangentially to a larger body at greater than escape velocity will follow a hyperbolic trajectory (subject to the above limits to the approximation), and disappear into deep space never to return. Agreed. The paths of comets had been observed to be compatible with being either hyperbolic, parabolic or elongated elliptical - but with insufficient precision in the 17th century to distinguish which. 3) I agree that I was incorrect in my statement above that an orbiting body moving at sub-escape speeds would move in a parabolic trajectory. I also now see that my original characterisation of the differences in velocity in relation to the shape was confused. 4) I agree that strictly speaking a terrestrial trajectory (in vacuo) is an ellipse rather than a parabola, due to the variation of gravitational force with altitude and to the change in direction to the center of the earth with horizontal movement. But for practical purposes the parabola is an accurate approximation (except for cases such as intercontinental ballistic missiles). Galileo had demonstrated by purely Euclidean arguments that a body moving without resistance in a uniform gravitational field would describe a parabola. Newton used Euclidean arguments to demonstrate to contemporaries the explanatory power of his laws to predict Kepler's observations. It seems to me that the significance all this to an article on geometry relates to the deep and mysterious inter-relationship between the abstract world of mathematics and the working of the material universe, to the fact that purely geometrical reasoning can provide proofs that we would derive analytically nowadays, and to the role that the availability of these tools had in the development of Newton's laws of motion and gravitation and the unification of celestial with terrestrial mechanics. DaveApter 10:11, 14 August 2007 (UTC) Hear, hear. I don't see why "the full truth" is a pain to get across. Neglecting vicissitudes (friction, viscosity, truncation of the parabola by collision with a window or the ground, etc.) is a standard assumption, as is the distinction between local behavior (throwing a baseball) and global behavior (orbiting satellites). It suffices to say that baseballs follow parabolic orbits because gravity in the small is essentially flat while satellites follow conical orbits because gravity in the large is essentially spherical. Both small and large are just approximations; when they are very good approximations geometry as a model of reality can be very helpful. --Vaughan Pratt 21:58, 11 September 2007 (UTC) [edit] What is this article about?I came to this article because in terms of the good article criteria it is one of the weakest math GAs: indeed an attempt was made to "boldly" delist it recently, but this caused a bit of a stir, so the delistment was temporarily overturned. I have been hesitant to return to the issue, because this is a great subject (well I would say that), and the article is full of lots of interesting stuff, thanks to the dedicated work of the editors involved. Nevertheless, I can see why it found itself in the delistment firing line, as it is woefully undersourced, and seems, on the face of it, to contain quite a bit of original analysis. Even though the GA criteria are more relaxed about citations now, this article states not just mathematical facts, but historical and analytical material that really needs citation. If I were to tag it, it would look like several fact bombs had exploded: not a pretty sight :-) But forget about GA: if that process has proven demonstrably incapable of assessing mathematics articles, then who gives a fig? I should be happy to delist the article, make it a Bplus on the mathematics scale, and forget about it. Except, on reading the article more carefully, I find there are other more important problems with it. There is a lot of nice stuff here, indeed, but rather too little of it is actually about Euclidean geometry. There are extensive digressions on the relations between Euclidean geometry and logic, and between Euclidean geometry and physics. What is an article on Euclidean geometry doing talking about mereology, Godel's theorem, general relativity and grand unified theories? Why does it not even mention inner product spaces or the Euclidean group? Felix Klein must be rolling in his grave. Triangles are hardly mentioned except to make logical points. Conic sections are introduced in two sentences (the second of which analyses them from the point of view of projective, not Euclidean, geometry) and the rest of the paragraph is all about physics, not geometry. All very well, but does it really belong here? Geometry guy 16:34, 30 September 2007 (UTC) [edit] GA ReviewFor what it is worth, here is what a GA review would look like based on the current article, in my opinion.
The issues are: the lead contains material that is not elaborated in the article; the lack of sources and citations; and the lack of breadth, the off-topic digressions, and the bias towards axiomatic and applied points of view, which I have already discussed. I won't delist for a while. If I get time, I will try and fix some of the breadth issues, but semester is just starting, so I thought I would raise the above issues here, in the hope that someone else will have more time than me... Geometry guy 16:34, 30 September 2007 (UTC) I've now replaced the GA with Bplus. Geometry guy 18:48, 6 October 2007 (UTC) [edit] Fourth postulate: equal, not congruentI have amended the fourth postulate to read this way, changing congruent to equal:
In standard usage congruence never applies to angles or sides, only to figures. This is all very straightforward and clear, since the standard term equal works perfectly well for angles and sides that are simply measured numerically. See Congruence (geometry); see also major British and American dictionaries: SOED, and M-W Collegiate (Congruent "2: superposable so as to be coincident throughout"; there is no "throughout" for sides or angles, since they are not compound as geometrical figures are). The original Greek uses the word ἴσος (or strictly, its feminine plural form ἴσας; see Elements, p. 6, where the Greek is given and the translation is with equal). This word plainly means "equal". It needs to mean that, since Euclid immediately goes on to say things like this: "11. An obtuse angle is greater than a right-angle." Relations like greater than make sense in the context of relations like equality, but not of compounded qualities like congruence. – Noetica♬♩ Talk 21:47, 10 December 2007 (UTC) [edit] intrepenuerǒawtztae kayop
[edit] EquivalenceThere's a contradiction between this article and the one on the parallel postulate. Please contribute to the discussion at Talk:Parallel postulate#Equivalence. Joriki (talk) 15:14, 9 January 2008 (UTC) [edit] Parabolic GeometryI am wondering what was wrong with the Parabolic Geometry page. It seemed like a legit disambiguation to me. Bob the Wikipedian, the Tree of Life WikiDragon (talk) 20:10, 15 April 2008 (UTC)
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