Talk:Elliptic geometry

message: User Shawnpoo reverted a bunch of work I did on the article. He left me a message on my talk page, but I couldn't tell what he thought the problem was. I re-reverted, and left a response on his talk page inviting him to discuss it here.--76.167.77.165 (talk) 19:05, 1 March 2009 (UTC)
thread_title: re-revert

message: The Wiki page on Non-Euclidean Geomotry states that "This kind of geometry, where the curvature changes from point to point, is called Riemannian geometry". I would take from this that Riemann Geometry is not Elliptic, although I may be wrong.
I'm going to remove that line, but feel free to put it back if it is in fact true.  —Preceding unsigned comment added by 86.149.136.187 (talk) 07:10, 1 April 2009 (UTC)
replies: text: I think it's fair to say:  Elliptic geometry is Riemannian but not all Riemannian geometries are elliptic. Zaslav (talk) 04:01, 29 July 2009 (UTC)
thread_title:  Riemann Geometry? 

message: I see a problem in the article.  The notion of a "model" of an axiom system is different from having inequivalent examples of the axioms.  The spherical model and the stereographic model are different "models" of the same abstract geometry.  The projective "model" is a different system of elliptic geometry.  I'm attempting to revise to clarify this very important point.  I hope others will correct any oversights, errors, or poor writing. Zaslav (talk) 04:05, 29 July 2009 (UTC)
thread_title:  Models vs. types of geometry 

message: This article is misleading. Consider the elliptic plane, the simplest example of elliptic geometry. There is no such thing as "spherical elliptic geometry", only spherical geometry. Two antipodal points on the sphere are identified to form the elliptic plane. The article real projective plane describes the actual "elliptic plane". See Coxeter, Introduction to Geometry (1969), pages 92 to 95. In conventional geometry, one presumes that two lines intersect in a single point or not at all. Great circles intersect in two points. The content of this article must be changed; perhaps I will.Rgdboer (talk) 20:50, 28 September 2011 (UTC)
thread_title:  Alert 

message: The picture is good, but the inset of Cape Cod should be labelled with approximation signs before the numbers, since on any finite area the triangle angles do NOT add up to exactly 180 degrees. Erasmuse (talk) 01:53, 9 March 2012 (UTC)
thread_title:  Alert: picture 

message: There's a problem in the "Comparison with Euclidean Geometry Section":
"For example, the first and fourth of Euclid's postulates, that there is a unique line between any two points and that all right angles are equal, hold in elliptic geometry. Postulate 3, that one can construct a circle with any given center and radius, fails if "any radius" is taken to mean "any real number," but holds if it is taken to mean "the length of any given line segment." Therefore any result in Euclidean geometry that follows from these three postulates will hold in elliptic geometry..."
This is garbled since it skips over Postulate 2, the infinite line postulate and one which needs more discussion in the context of elliptical geometry since whether it holds depends on one's interpretation of it. I'm not expert enough on this topic to make the changes, tho. Erasmuse (talk) 01:56, 9 March 2012 (UTC)
thread_title: Alert: "Comparison with Euclidean Geometry Section"

message: Hi.
I saw this:
"Mathematicians commonly refer to the elliptic plane as the real projective plane. Especially in spaces of higher dimension, elliptic geometry is called projective geometry."
But on the article for projective geometry, it says:
"It is not possible to talk about angles in projective geometry as it is in Euclidean geometry, because angle is an example of a concept not invariant under projective transformations, as is seen clearly in perspective drawing."
and
"Projective geometry is less restrictive than either Euclidean geometry or affine geometry. It is an intrinsically non-metrical geometry, whose facts are independent of any metric structure."
From Harold S.M. Coxeter's "Projective Geometry", in the intro it is said that projective geometry has "no circles, no distances, no angles, no intermediacy (or "betweenness"), and no parallelism".
Elliptic geometry, on the other hand, most certainly does deal with angles and distance (e.g. we can talk of the angle sum in a triangle and how it exceeds \pi). There are circles, and ordering (though it is like a cyclic order, rather than a linear order), as well. The only thing missing is parallelism. So elliptic geometry is not the same as projective geometry, even though the elliptic plane and the projective plane are topologically homeomorphic. mike4ty4 (talk) 01:24, 22 September 2013 (UTC)
replies: text: Correct, thank you for noting this mistake. Changes have been made today.Rgdboer (talk) 22:47, 24 September 2013 (UTC)
thread_title:  Elliptic Geometry vs. Projective Geometry 

message: Hello,
I have a colleage education in sciences, so I have some maths education, but am by no means a mathematician.
It must be the 10th time at different moments of my life I try to read-up on non-eucledian geometry to understand what it is about, and I always leave feeling that there is something profound I'm missing, after reading different Wikipedia articles on the subject.
1. I came to understand that defining a different metric than the euclidian metric, is equivalent to
2. Replacing Euclid's 5th postulate with something else.
The issue in understanding this that I have is that when explained like in 2., it sounds like a great revolution.
But when explained like in 1. it sounds like considering ℝ² and simply defining a distance or metric with a different formula than the "pythagoras theorem" metric.
And this leads my intuition towards this idea: you can always express non-euclidian geometry in terms of euclidian geometry, ie. elliptic geometry is actually just a sphere in Euclidian space where you do exotic computation and naming convention of what is the "plane" (it's the sphere), what is the "distance" and what is "parallel". When I hear the analogy that elliptic geometry is locally flat/euclidian, then I think to myself "oh we are just measuring things in Euclidian space and many terms arise because of the curvature, but 'locally' the distance computation will reduce to the euclidian distance because of o(x) types of consideration".
And so all these illustration of models of non-euclidian geometry in terms of euclidian objects I am familiar with both helps me understand, but also confuses me: it all seems like Euclidian geometry still is the canonical reference and these other geometries are handy/clever paradigm changes or referential changes within complex euclidian objects to be able to reexpress them with the simple "line and point" perspective.
Could someone debunk this for me ?
Thank you very much.
— Preceding unsigned comment added by ByteMe666 (talk • contribs) 08:54, 25 September 2020 (UTC)
thread_title:  How exactly are Non-Eucledian Geometries Independent from Euclidian Geometry 

message: According to D. M. Y. Sommerville, "the hyperspherical network corresponding to a 24-cell" is "a division of elliptic three-dimensional space into 24 octahedra". This statement occurs in section 14, Relations between the regular polytopes in four dimensions, of his Introduction to the Geometry of N Dimensions. link. Alternatively, the connection may be realized with versors representing the edges of the 24-cell. Rgdboer (talk) 01:00, 7 April 2024 (UTC)
replies: text: This is a very obscure connection which is not going to be at all obvious to readers, and doesn't in my opinion justify inclusion in a "see also" section (or mention in the elliptic geometry article). Perhaps this could be mentioned in the 24-cell article somewhere. –jacobolus (t) 02:40, 7 April 2024 (UTC)

text: Seems like a trivial observation applicable to all six regular 4-polytopes. —Tamfang (talk) 05:16, 7 April 2024 (UTC)
thread_title:  24-cell