# Talk:Non-Euclidean geometry

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Re. hyperbolic, parabolic, elliptic, these terms are still used when discussing differential equations 19 Feb 2012 Pendocehadron

spelling ... euclidean or euclidian make up your mind —Preceding unsigned comment added by 203.173.8.118 (talk) 06:40, 11 February 2008 (UTC)

"Einstein's theory of relativity describes space as generally flat (i.e., Euclidean), but elliptically curved (i.e., non-Euclidean) in regions near where matter is present." Isn't this statement incorrect? Flat spacetime is NOT Euclidean since the metric is not positive definite. 151.201.134.170 (talk) 05:10, 25 May 2008 (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.

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. --C S 12:18, Dec 16, 2004 (UTC)
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)
I didn't like it, either, and I'm glad to see someone else had already noted it. I rewrote the first paragraph to discuss the difference in terms of intersection (or lack of) instead of parallelism, and the second to mention the ultraparallel nomenclature. 171.64.71.123 00:38, 11 January 2007 (UTC)

## 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. --C S 08:49, Dec 19, 2004 (UTC)

Yes, good point. To clarify, you mean "(re-)discovered by the (Western) Europeans (during the Renaissance)"

What a fascinating article! I believe "re-discovered" is a good description. I always found the history of hyperbolic geometry Einstein fascinating. —Preceding unsigned comment added by 67.35.126.14 (talk) 19:14, 25 November 2007 (UTC) am not sure why someone got the bright idea to have the gaul to actaullt claim that "The theorems of Alhacen, Khayyam and al-Tusi", somehow constitute the begining of non-euclidian geometry. Here we go once again someone writting on islamic science and math taking one citation and just stretching beyond belief. The citation coming from Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, Vol. 2, p. 447-494 [470], Routledge, London and New York:. There is no reference in this citation that specifies that non-euclidian geometry was discovered before János Bolyai and Nikolai Ivanovich Lobachevsky. It merely claims that there were attempts at challenging the fifth postulate. If some one is planning on making bold claims, such as this one, its gonna require more then just one citation from one book that doesnt really prove what your claiming it does. —Preceding unsigned comment added by 24.36.181.171 (talk) 08:51, 14 December 2007 (UTC)

I agree with the above sentiment and would like to see collaborating references. However, back to the original question - we have in the commentary on Book I of the Elements by Proclus (A.D. 410- 485) the statement that the philosopher, scientist and historian Poseidonios (B.C. c.135 - c.51) suggested that the 5th Postulate could be proved if one replaced the definition of parallel lines of Euclid by one where the defining characteristic is "equidistance". This doesn't work as his "proof" is circular, but the point is that all of this clearly pre-dates the Renaissance. Heath's footnotes in his translation of the Elements goes into some detail on what Proclus wrote about this. Wcherowi (talk) 18:25, 21 August 2011 (UTC)

## 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"?

In Saccheri's work with quadrilaterals that we now call "Saccheri quadrilaterals" he investigated the measure of the summit angles. If he could prove that they were right angles, then he would have a proof of the 5th Postulate. His method for trying to do this was to eliminate all the other possibilities. After showing that the summit angles are equal, he only needed to get rid of the obtuse and acute angle cases. It did not take him very long to remove the obtuse angle case, but he never successfully eliminated the acute angle case. Today we realize that the obtuse angle case would give "elliptic geometry" if you also made some other changes to the axioms (the contradiction Saccheri arrived at was due to these other axioms, so it was a kind of accident that he got away with this.) To anwer your question, Saccheri had no inkling of what elliptic geometry was. The phrasing in the article is fine, but it is modern. The term "elliptic geometry" is due to Felix Klein in 1871. Wcherowi (talk) 18:47, 21 August 2011 (UTC)

## Diagram with hyperbolic

That diagram is only true for Poincare hyberolic disc lines, not other kind of hyperbolic geometries like the Klien model.

## 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)

## 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)

## 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? —Preceding unsigned comment added by 206.80.23.226 (talkcontribs)

I expect that most people who access this article are trying to find out basically what Non-Euclidean geometry is. I also expect that everyone who knows what affine geometry is already know what Non-Euclidean geometry is, so it is not helpful to include this term, and others of a similar level, in the introductory paragraph. Encyclopedia articles exist to help people who don't know the topic. Presuming advanced knowledge of the technical context in the first paragraph defeats this purpose. We, the uneducated masses, need an opening paragraph, or at least an opening sentence, that roughly describes the topic assuming the reader has a knowledge of geometry at or bellow the typical level. — Preceding unsigned comment added by 121.215.46.55 (talk) 03:49, 6 July 2016 (UTC)

An earlier version of the lead paragraphs was:

Non-Euclidean geometry is either of two specific geometries that are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much narrower than it appears to have in the general English language. There are a great many geometries which are not Euclidean geometry, but only these two[1] are referred to as the non-Euclidean geometries.

The essential difference between Euclidean and non-Euclidean geometry is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line and a point A, which is not on , there is exactly one line through A that does not intersect . In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting , while in elliptic geometry, any line through A intersects (see the entries on hyperbolic geometry, elliptic geometry, and absolute geometry for more information).

I unhappily accepted the current version as a compromise with another editor who wanted to make the subject "broader in scope". If you think that the original is better suited for the intended audience I can attempt to put it back (perhaps improved somewhat). --Bill Cherowitzo (talk) 16:48, 6 July 2016 (UTC)

-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 (usertalkcontribs) 14:32, 3 November 2006 (UTC)
An interesting way to look at it is by considering a mathematical system - a system is complete if it is reduced to its least necessary components,. Euclidean geometry could be defined by the first four postulates - hence, the fifth postulate could be derived from the other four. Non-euclidean geometries were developed by proposing the negative of the 5th postulate. For a line and a point not on the line, there is (1) zero lines going through that point that is parallel to the given line; and (2) there are more than one (or infinite number) of lines going through the point that is parallel to the given line,
Could this bit of history or analysis be helpful to the article?Jance 06:35, 20 December 2006 (UTC)
Yes, I think so. If it could be layperson-ized and put into the intro paragraph, it would help give a different theme to the article, instead of a strictly academia perspective. --Rhetth (talk) 01:16, 24 February 2008 (UTC)
In response to your saying that noneuclidean triangles don't exist ... as I understand it they very much do. For instance since the earth is a spherical surface a triangle drawn on that surface would basically be a spherical triangle and thus noneuclidean, with angles adding to more than 180 degrees. Triangles on a flat plane would have 180 degree angles, but on a curved surface a triangle is no longer defined in a flat plane. 70.114.210.149 (talk) 20:12, 20 July 2010 (UTC)
I think what it means is that in non-Euclidean geometry, you change the definition of "parallel" to in layman's terms mean "2 lines a mirror image of each other." Or to make it even more understandable, forget the whole parallel issue: non-Euclidean geometry is about shapes with curved lines. That's the best I can come up with. Even the Simple English wiki entry is cryptic. 24.155.230.38 (talk) 22:10, 15 February 2011 (UTC)

## Absolute Geometry

Absolute geometry does not take a position on Euclid's fifth postulate, the rule of Euclidean geometry that states that parrallel lines go on to infinity and never intersect. So it is wrong to refer to absolute geometry as a non-Euclidean geometry because it is compatible with both Euclidean and non-Euclidean geometry and the first 28 Euclidean theorems don't involve parralell lines so they could be considered absolute geometry so absolute geometry is really not neccessarilly a non-Euclidean geometry and should not be referred to as such in this article. NikolaiLobachevsky 6:03:14 1/4/2007 (UTC)

## long url in article body pointing to edit

This is in the article body, it seemed a little wierd (not the content itself, the big/edit url in the text):

Another practical model of hyperbolic space was developed by Dr. Diana Taimina in 1997 using crochet (please see http://en.wikipedia.org/w/index.php?title=Non-Euclidean_geometry&action=edit&section=5 for more information).

External reference [[1]] appears to be no good--Billymac00 (talk) 20:06, 24 April 2008 (UTC)

## Crochet model

"Another practical model of hyperbolic space was developed by Dr. Diana Taimina in 1997 using crochet."

Does not belong here I think. Maybe in a trivia section. I am removing it, please tell me if you do not agree. Fph (talk) 18:12, 13 January 2009 (UTC)

Well, I'd disagree; it's supposed to be a serious piece of work. try this, and this. Moonraker12 (talk) 13:07, 21 October 2009 (UTC)

## Scope

At present, the article presents non-Euclidean geometry in a vary narrow way. Inclusion today of the section on non-Euclidean spacetime geometry clashes with the narrowness. The problem is that metric space presumption has crept into the viewpoint. With the prefix "non" we must be ready to include almost any topological space or other structure which suggests itself as a geometry. Presently the opening sentence refers to a curvature; such presumptuous writing does not help us communicate to general readers. Much of the material here is found at hyperbolic geometry; we should avoid the common misconception that non-Euclidean geometry is just hyperbolic geometry.Rgdboer (talk) 00:39, 9 November 2009 (UTC)

Agreed that talking about the Riemann curvature tensor is presumptuous; Euclid (nor anybody in many centuries in between) wouldn't have a clue what this had to do with his geometry (or the negation thereof). I would say non-Euclidean geometry is geometry that is like Euclidean geometry, but without the parallel postulate. The lead should not open with too technical terminology. Please move this down to a more appropriate section. Marc van Leeuwen (talk) 06:00, 30 April 2011 (UTC)

I'll both agree and disagree with the above. The intro as it stands is both presumptuous and pretentious. It violates the core Wiki guidelines on what the intro section should do and I am amazed that this has not been changed. What is worse is that the opening statement is not common usage of the term, rather it is a specialized jargon used only by differential geometers. While it is linguistically correct to interpret the "non-" in non-Eulidean to mean not, it is not the generally accepted mathematical interpretation and this should be made clear to the readers of this page. A little historical background should clear up the intended meaning of the term. It was first used by Gauss to refer to what he considered his own invention, the only alternative to Euclidean geometry. By the time that Felix Klein introduced the terms hyperbolic, elliptic and parabolic it was already clear that these were subgeometries of projective geometry, which provided the unifying background. In this context, hyperbolic and elliptic geometry are the non-Euclidean geometries. The issue that is addressed by these geometries is the nature of parallelism, curvature of the space is not germane. Riemann's contribution is the minor example from his masterful address of elliptic geometry and no more. Wcherowi (talk) 19:57, 21 August 2011 (UTC)

## Careless wording

First paragraph: "Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line ℓ and a point A, which is not on ℓ, there is exactly one line through A that does not intersect ℓ. In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting ℓ [...]" The statements here about lines in Euclidian and hyperbolic geometry are absolutely equivalent to one another: in Euclid also, there are infinitely many lines that do not interesect the other specified line. This must be reworded but I don't feel quite competent to do so. Axel 19:26, 21 January 2010 (UTC) —Preceding unsigned comment added by AxelHarvey (talkcontribs)

Not through a given point. Given ℓ and A in a 2D Euclidean plane there is only one line through A that does not meet ℓ. In a space with hyperbolic geometry there are an infinite number of lines though A that don't meet ℓ. I.e. a Euclidean plane is one that satisfies Euclid's postulate. The first sentence you quote is a bit long but it's clear and correct, and it's summarised in a short list for clarity at the end of the introduction. Nothing obviously wrong to me. --JohnBlackburnewordsdeeds 20:11, 21 January 2010 (UTC)
Perhaps AxelHarvey is confused by Euclid's use of "line" to refer to what we now call a "line segment". Modern use of "line" refers to a straight line which extends infinitely in both directions. On a flat (Euclidean) plane, one and only one such line through a given point can be parallel (not intersecting) to a given such line (on which the given point does not lie). JRSpriggs (talk) 08:33, 22 January 2010 (UTC)
I hate to butt in at this point, being that I'm no expert and I'm just speculating on some things, but I think what Axel may have been arguing was using Euclidean geometry in a 3D space. In 3D Euclidean geometry, it would make sense that there are an infinate number of lines that pass through A that don't meet ℓ. So the counter that in a 2D Euclidean plane is also correct. But I don't like that JohnBlackburne says "in a space (3D) with hyperbolic geometry", to compare to the 2D Euclidean plane. I'm guessing that it refers to a plane in hyperbolic geometry (is that correct?), as it does not make sense to compare one geometry in 2D to another in 3D. Traveling matt 05:33, 16 March 2010 (UTC)
Yes, 'space' as in vector space or metric space, i.e. I was not implying it's 3D, just I think looking for a word more general than 'plane' which to me implies a flat and so Euclidian surface.

## Importance

The discovery of non-Euclidean geometries was a major turning point in the foundations of Western philosophy. It caused a re-examination of the concept of "truth" and the basic assumptions we had made about the world in which we live. This little piece of mathematics has had a profound impact on our culture and intellectual history, yet so many people are totally unaware of it. This section should be the centerpiece of the article, the reason for actually having this page instead of just redirects to the hyperbolic and elliptic geometry pages. The impact in physics certainly belongs here, but there is so much more that should be present. Wcherowi (talk) 22:01, 21 August 2011 (UTC)

## Scope2

Currently the article says in the lede:

This is one term which, for historical reasons, has a meaning in mathematics which is much narrower than it appears to have in the general English language. There are a great many geometries which are not Euclidean geometry, but only these two are referred to as the non-Euclidean geometries.

Some mathematics teachers may assert this claim, but generally it is not true. Galilean geometry and Minkowski geometry are non-Euclidean and widely addressed to support physical notions. Hyperbolic and elliptical geometry have a very high profile among the non-Euclidean, but they do not have exclusive dominion over the realm. Further, there are finite geometries and other geometrical structures that share some features with the Euclidean. The article is misleading.Rgdboer (talk) 22:26, 9 September 2011 (UTC)

Can you cite any references to support the position that "generally it is not true"? Every monograph that I have seen with titles referring to Non-Euclidean geometries use the term in exactly the way the lead now reads. From a broader perspective, to insist that non-Euclidean means not Euclidean reduces the term to something meaningless. The variety of geometries is so great that lumping everything which is not Euclidean into one pile does every one of them a disservice and perpetuates the myth that Euclidean geometry is somehow more "real" than other geometries. The historical significance of non-Euclidean geometry is precisely to disspell this myth. Wcherowi (talk) 17:22, 10 September 2011 (UTC)

Consider History of Non-Euclidean Geometry by B.A. Rosenfeld. His subtitle is Evolution of the Concept of a Geometric Space. Rosenfeld is sensitive to the washout that would follow from too broad a scope. So on the first page he opines that the "era of modern mathematics" beginning in the 1870s be recast as the "era of non-Euclidean mathematics". As it stands our article pivots on the concept of parallel (geometry) which is the sharp edge that exposed the other geometries. The topic treated in many books goes to foundations, for instance E.B. Golos Foundations of Euclidean and non-Euclidean Geometry (1968). It treats finite geometries in chapter 2 and consists mostly of axiomatic analysis. Rosenfeld felt compelled to include curvature of space (ch 8), groups of transformations (ch 9), and applications of algebras (ch 10), subjects that go beyond the scope. I agree that hyperbolic and elliptic geometry are two primary illustrations of non-Euclidean geometry, but there is considerable usage of the term non-Euclidean to describe spacetime geometry, both the Galilean geometry of absolute time and space and the geometry that describes special relativity. For instance, Isaak Yaglom, the friend to whom Rosenfeld dedicated his History, wrote A Simple Non-Euclidean Geometry and its Physical Basis in 1979 that deals with these two geometries that are used as spacetime models.Rgdboer (talk) 22:28, 10 September 2011 (UTC)

I don't think that you have made your case with these references. Both Rosenfeld and Yaglom are products of Moscow State University in the early 40's. The books you've cited are both translations from the Russian and it is not clear whether the use of "non-Euclidean" is due to the authors or to the translators. Even without that ambiguity the books are both a bit idiosyncratic, containing lots of good information put together in strange ways - hardly mainstream. Golos's book, which is not very influential (it doesn't even get a review in Math Reviews), is, as you say, concerned mostly with axiomatics and it is quite natural to talk about finite geometries in that context. He is using them to make points about axiomatics in a simplified system - this doesn't mean that they are to be considered non-Euclidean geometries. The term "non-Euclidean" has traditionally had a specific and narrow usage and is useful within that context. You seem to want to make it a synonym for "not Euclidean" and I ask you - what advantage is there for doing that? Wcherowi (talk) 06:19, 11 September 2011 (UTC)

There's Felix Klein (1928) Vorlesungen uber Nicht-Euklidische Geometrie which has a section on applicaton of projective mensuration to relativity theory (SS. 316 to 319). He makes clear that he also thinks there are just two non-Euclidean geometries (Kapitel VII, S. 211). Here we are talking mathematics. But Wikipedia is not divided into departments like a university. Physical science has a systematic science of time, space, and events. As seen in the reference "Synthetic Spacetime" there is a deductive approach that mimics the Euclidean sequence. From the mathematical viewpoint this system isn't even a topology since points are not separated by the spacetime interval. Every event is identified with its light cone. The preferred term may be pseudo-Euclidean space, but writers often say non-Euclidean instead. The exclusivity that you claim (spaces of non-zero constant curvature) may hold in the halls of mathematics but physical scientists have had their say too and even Felix Klein acknowledged that.Rgdboer (talk) 23:58, 12 September 2011 (UTC)

The new section (today) on Planar algebras as a source of non-Euclidean geometry is mathematics. It seemed necessary to show the basis for use of the term on another premise than curvature and parallelism. I have added Yaglom as a reference. Some appropriation of the term non-Euclidean by physical scientists may be defended through these alternative complex planes.Rgdboer (talk) 03:21, 15 September 2011 (UTC)

Yes this is mathematics, but again, a deeper reading does not support your position. As I've mentioned before, Yaglom is not considered main-stream, but he certainly has some interesting ideas. The geometries that he constructs from two of his three algebras he does call "non-Euclidean", but there are only 8 of them and he does not use the term for anything beyond these 8. In the supplement to Geometric Transformations III, he states, "Projective geometry is not merely not Euclidean geometry; it is 'very much non-Euclidean'." This is a clear indication that he does not consider "not Euclidean" and "non-Euclidean" to mean the same thing. You can not defend the misuse of this term on the basis of Yaglom's work. Wcherowi (talk) 05:21, 15 September 2011 (UTC)
I have made seperate edits to put the article more in line with the lead. I removed the concepts section since after eliminating the not Euclidean stuff there was nothing left that wasn't already said in the lead. I moved the spacetime section to a lower position in the article. I removed the paragraph on Grassman as it is irrelevant to the development of non-Euclidean geometry. I would suggest, as I have seen someone else do before, that the section on fiction be removed as being inappropriate to this article. I think that a seperate article on mathematics in fiction writing might be a good article, concentrating on those works where mathematics is essential to the plot rather than just using the terminology to invoke complexity and mystery. I might even consider contributing to such an article (Piers Anthony's early work - Orn, Ox, Omnivore, Rudy Rucker's stuff, and a Clifton Fadiman collection, "Mathematical Magpies" I and II come to mind) but I wouldn't want to write it. That would be a natural home for this section. Wcherowi (talk) 18:17, 20 September 2011 (UTC)

The lede was changed today to reflect a slightly broader scope of the term non-Euclidean geometry in the twentieth century after it was defined in the nineteenth.Rgdboer (talk) 02:09, 18 November 2011 (UTC)

I'm not going to say that I think the old lead was perfect, but this version is not an improvement for several reasons. First of all you've pumped up the lead with excessive amounts of jargon that only an expert would appreciate, losing the function of a lead to introduce the topic to a general audience. Secondly, you are pushing a minority point of view in violation of WP:NPOV. Also, you've removed the context in which the diagram of the lead can be interpreted. I'd revert this edit, but I'm not interested in getting into an edit war... so I will just ask you to reconsider, for the nonce. Bill Cherowitzo (talk) 04:14, 18 November 2011 (UTC)

Considering your contribution of 23 October, "a nod toward other uses", it seemed that your absolute view re elliptic or hyperbolic, had changed. That is why the lede was changed. "excessive amounts of jargon that only an expert would appreciate" is not true: links are provided so that a reader can appreciate non-Euclidean geometry. "minority point of view" is also incorrect: the affine geometry of spacetime is supported by references in the Kinematic geometries section. The old lede was interpreting the diagram, which you mention, but this diagram is on the subject of parallelism. The new lede refers to twentieth-century understanding, the old did not. Let us continue the discussion; the change was made with consideration.Rgdboer (talk) 20:26, 18 November 2011 (UTC)

My position has not changed, I was trying to accommodate yours. Talking about perpendicularity and hyperbolic orthogonality is clearly excessive jargon. Just take a look at the issues that have been raised on this talk page ... they indicate basic misunderstandings of the fundamental concepts, perpendicularity is just shooting way over the heads of the typical audience for this article. Your bringing up affine geometry is of interest. Coxeter does not support what you are implying he supports. Pages 18-19 (in my edition, the MAA reprint) do not mention using the term non-Euclidean geometry in reference to affine or Minkowski geometry and furthermore, on page 4 we have the explicit statement, "We now learn of many different geometries, but for historic reasons we reserve the name non-Euclidean for two special kinds: hyperbolic geometry, in which all the "self-evident" postulates I-IV are satisfied though Postulate V is denied, and elliptic geometry, in which the traditional interpretation of Postulate II is modified so as to allow the total length of a line to be infinite." Parallelism is what non-Euclidean geometry is all about. I am not taking issue with the importance of affine, Minkowski, or any other type of geometry. I am in 100% agreement with the last two sentences of your lead – it is just that they are totally irrelevant to the topic. I am sorry that you do not like the generally accepted mathematical meaning of this term, but that does not give you the right to change it in accordance with your own lights.Bill Cherowitzo (talk) 21:31, 18 November 2011 (UTC)

Perhaps the page numbers have changed in the new edition of Coxeter's textbook, my edition is from 1942. It is Chapter 2, Real projective geometry: foundations, section 2.1 Definitions and axioms. Four pages in he gives a tree diagram with elliptic, affine, and hyperbolic on one level with Euclidean and Minkowskian as decendents of affine geometry. These lights you ascribe to me are lights of the twentieth century.Rgdboer (talk) 02:03, 19 November 2011 (UTC)

Precisely the section I was referring to. Again, there is no mention of referring to these other geometries as being non-Euclidean. You are engaging in WP:SYNTH. They are of course not Euclidean geometries, as are most of the cutting edge geometric theories of the current day. I do not understand your implication that a certain simple definition is "old hat". I do 21st century cutting edge geometric research, but I use definitions and ideas that may be centuries old. That's the way mathematics is. Good ideas do not age, new ideas are not always advances. Concepts are judged on how useful they are, not how old they are. Unlike the physical sciences, mathematics builds on itself – there is very little of the "out with the old, in with the new" mentality to be found. If faster than light travel is shown to be real, the foundations of modern physics will be shaken. A comparable discovery in mathematics could never imply that there was a fundamental flaw in the calculus. Mathematics is just built in a different way. Bill Cherowitzo (talk) 05:38, 19 November 2011 (UTC)

Its not "out with the old", but rather to "see the old with broader perspective". Coxeter displays the affine Minkowski geometry in the family of nearly Euclidean geometries. Reissue of his book in 1998 by MAA tells me this source is not dismissible. Further, authors Wilson & Lewis and Yaglom use the label "non-Euclidean geometry" to identify their spacetime work that appeared in the twentieth century, thus showing that the term had been expanded beyond the usage by Gauss, Klein, and Beltrami. Your attachment to parallelism as the key concept for non-Euclidean geometry is our problem, as the sources show that not to be so. For instance, in Synthetic Spacetime you see that axiom 3 is Playfair’s form of the parallel postulate. As Coxeter says, parallelism only gets you to affine space; to resolve Euclidean geometry one must select perpendicular orthogonality additionally.Rgdboer (talk) 20:59, 19 November 2011 (UTC)

Coxeter was one of the 20th century's greatest geometers ... I certainly hope that you wouldn't dismiss him. I am perplexed by your comment. I quoted verbatim what he had to say on the issue in the very book that you are claiming as support for your position ... and it could not have been made any clearer – he disagrees with you! Bill Cherowitzo (talk) 23:33, 19 November 2011 (UTC)

Your understanding of Coxeter is too brief. We have been discussing the contents of his book Non-Euclidean Geometry where I have shown you a passage pointing to more expansive of use of the term. On page one of his The Real Projective Plane (1955) he says "projective geometry can be developed into the various kinds of ‘non-Euclidean’ geometry that are relevant to more modern ideas of relativistic cosmology."Rgdboer (talk) 20:28, 23 November 2011 (UTC)

You should consider the meaning of the quotes in that statement. See Quotation mark - Signaling unusual usage Bill Cherowitzo (talk) 04:39, 29 November 2011 (UTC)

Today an external link was provided to the encyclopedia maintained by Springer and the European Math Soc. It contains a section on "Group theoretic context". Coxeter's use of quotes signals that by 1955 there was discussion of the scope of the topic.Rgdboer (talk) 22:07, 2 February 2012 (UTC)

What are you talking about? That section provides, in group-theoretic language, Klein's description of hyperbolic, elliptic and parabolic geometry. How are you pulling out of that any reference to Coxeter or 1955? The section does not support any wider scope than what I have been talking about. The only comment that I have about that article is that it was clearly written by a differential geometer with a bias towards the Russian point of view. Bill Cherowitzo (talk) 04:25, 3 February 2012 (UTC)

Today Morris Kline's Mathematical Thought from Ancient to Modern Times was added to the References. On page 919 Kline gives a tree diagram similar to Coxeter's discussed above. He includes "Other subdivisions of affine geometry" in the tree and adjacent to Euclidean geometry. On page 880 of the chapter cited he says "a form of non-Euclidean geometry which we have yet to examine has been used in the theory of relativity." Kline views the place of the study in a later chapter, and says on page 922 "The advent of the theory of relativity forced a drastic change in the attitude toward non-Euclidean geometry". The scope of this article has been stated to narrowly.Rgdboer (talk) 00:35, 15 August 2012 (UTC)

Well, its taken you seven months to find Kline ... I guess this says something about how widespread this point of view is. We have been at this for about a year now and I do weary of it. I certainly have taken a hard-line stance on the traditional meaning of the term, but this was mainly to counter the extreme position that you advocated. There is a compromise position. One can state the traditional meaning of the term and identify it as such. One then says something like, "there are some who would extend the meaning of non-Euclidean to encompass those geometries which attempt to capture more of the true nature of physical space than Euclidean geometry does, such as ...{list your favorite geometries} ... This opens up the article to discussion of these ideas. However, I will not condone any statement that says non-Euclidean can mean anything that is not Euclidean. Bill Cherowitzo (talk) 03:09, 15 August 2012 (UTC)

## Kinematic geometries

Regarding spacetime, today's edit renames the section previously on spacetime. With 98 watchers of this article, perhaps others will comment (here). Wcherowi, the references clearly show that authors see this topic as non-Euclidean geometry. Since the hyperboloid model is so important to relativity, it provides the bridge.Rgdboer (talk) 21:12, 23 September 2011 (UTC)

If we projectivise Minkowski space, we have the projective model of Mobius (conformal)geometry. This is the same construction as is used in (conformal)geometric algebra making use of a null basis that provides a point representation equivalent to inverse stereographic projection. Thereby neatly connecting the hyperboloid model to everything else.Selfstudier (talk) 20:03, 14 November 2011 (UTC)

## Vague

Vague terms like "most notorious", "complicated" and "complexity" appear in the article. — Preceding unsigned comment added by 86.180.152.221 (talk) 13:51, 28 November 2013 (UTC)

Possibly a postulate came in for attention because it is only used once and only to prove a theorem the converse of which is proved without it. — Preceding unsigned comment added by 86.180.152.221 (talk) 14:12, 28 November 2013 (UTC)

## Recent reverts

An IP has recently been removing a sentence about the effect of the paradigm shift on theology, and I have been replacing it. I would like to remind the IP that we do not pass judgement on the validity of the statements in WP, rather we report on what is said in the literature. The fact that this IP disagrees with the referenced statement is of no concern. The way to handle this is to cite opposing viewpoints in the literature. Bill Cherowitzo (talk) 04:20, 6 September 2014 (UTC)

1. ^ this is well established, in the references see Blumenthal, Coxeter, Greenberg, Meschkowski, Richards, Smart and Trudeau. Also see the terminology subsection of the History section.