Talk:Elliptic geometry

Elliptic vs. Spherical
In some texts these are topologically distinct but with the same local curvature. Elliptic geometry is the one where the poles in spherical geometry are identified. These are the only two globally isotropic spaces of constant positive curvature but there are other compact topologies which are locally isotropic. This distinction is relevant for cosmological world models and FAIK the terminology is not very standard. Would anyone like to comment before I dive in and add a section on the issue? PaddyLeahy 15:40, 22 May 2007 (UTC)


 * I've certainly seen the distinction elsewhere, indeed there is a little about in the article. The whole article could certainly do with a bit of fleshing out. --Salix alba (talk) 17:40, 22 May 2007 (UTC)

" Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which asserts that there is exactly one line parallel to L passing through p. " This is not Euclid's parallel postulate, it's Playfair's axiom. The following is the parallel postulate:
 * " If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles. " —Preceding unsigned comment added by 82.148.66.254 (talk) 13:41, 26 February 2008 (UTC)
 * Quite right, although Euclid's parallel postulate is equivalent to one direction of Playfair's axiom, namely that there is at most one line parallel to L passing through p. Hyperbolic geometry violates this direction but not elliptic geometry which has no parallel lines to begin with, e.g. on the sphere the counterpart of a straight line is a geodesic or great circle, and all great circles intersect, in two places in fact, which by Euclid's definition of parallel makes them not parallel.  The reason Euclid omitted the other direction is that in proving Proposition 16 he thought he'd proved that direction, but his argument either was fallacious (the more likely reason in my view) or depended on a fast-talking interpretation of Postulate 2, that straight lines go on forever, which many have interpreted as ruling out great circles because they are bounded sets, i.e. of finite length, unlike the geodesics of the Euclidean plane and the hyperbolic plane which are unbounded sets, i.e. of infinite length.
 * Any objections to my fixing this? (Not in anywhere near that detail of course.)  --Vaughan Pratt (talk) 04:51, 12 December 2008 (UTC)

I think both of these problems are fixed in the present version of the article.--76.167.77.165 (talk) 19:27, 1 March 2009 (UTC)

I haven't contributed to wikipedia before, and I guess I'm not now since I don't have time to edit the article. Apologies in advance for any protocol I am violating. The claim made in the article that the first 28 propositions of Euclid are true in elliptic geometry is false. For example, propositions 16, 17, and 27 are not true for elliptic geometry. Either Euclid was making assumptions beyond the scope of his second postulate, or his second postulate isn't satisfied by elliptic geometry, take your pick. —Preceding unsigned comment added by 75.33.252.98 (talk) 08:42, 17 March 2009 (UTC)

re-revert
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)

Riemann Geometry?
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)


 * 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)

Models vs. types of geometry
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)

Alert
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)

Have introduced "Definition" section. Pruning to follow.Rgdboer (talk) 21:41, 29 September 2011 (UTC)

Alert: picture
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)

Alert: "Comparison with Euclidean Geometry Section"
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)

You might have to think of an infinite line as a line segment that goes around the sphere and meets up with itself, and then keeps on doing that, looping infinitely. If you think of a (infinite) line as just a great circle, I don't think the second postulate would hold, seeing as the circumference would be the longest distance, unless distances were changed in a similar way to what is done with two dimensional representations of hyperbolic geometry. My interpretation of what was said about the third postulate is that there's an upper bound on the radius of a circle, but the way I constructed lines, you can have a line segment of any length, so excluding "any real number" doesn't work. This is resolved when the third postulate is defined as "between any two points, there is one and only one circle that goes through one of the points and has the other as its center". If the two points are more than a quarter of the circumference away from each other, you make a circle with one point and the antipode of the other (which is the same point). I think even without defining a line as I did, defining "any real number" as "the length of any given line segment" doesn't work because even if the upper bound of a line segment were half the circumference (it could be me more if we didn't consider antipodes, but that wouldn't change anything here), we can't make a circle with a radius of that length. The biggest circle you can make has a radius equal to a quarter of the circumference of the sphere (a great circle). I could be mistaken about some of this, and criticism is welcome.

Elliptic Geometry vs. Projective Geometry
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)
 * Correct, thank you for noting this mistake. Changes have been made today.Rgdboer (talk) 22:47, 24 September 2013 (UTC)

How exactly are Non-Eucledian Geometries Independent from Euclidian Geometry
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)

24-cell
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)


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