Talk:Hyperbolic geometry

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Contents 1 MacFarlane 2 Hyperbolic plane vs. hyperbolic geometry 3 Escher circle limit III 4 i don't understand this 4.1 Attempted Fix 5 Intro 6 This section makes no sense 7 Hjelmslev transformation 8 Questions about hyperbolic 1-space 9 Relativity stuff 10 have you seen this curve?

[edit] MacFarlane

A fourth model is the Alexander MacFarlane model, which employs an 3-dimensional hyperboloid of revolution (of two sheets, but using one) embedded in 4-dimensional euclidean space. This model is sometimes ascribed to Karl Weierstrass. Macfarlane used hyperbolic quaternions to describe it in 1900.

I snipped the above from the article and replaced with something more accurate. I've never heard of this model being called this name and I can't even find a single reference on Google (except for wikipedia-related hits). Not to mention, MacFarlane is not the originator of this model, as far as I know. For example, John Stillwell credits Poincare; some credit Killing and say Poincare generalized it. I've never heard it being attributed to Weierstrass. I guess the part on hyperbolic quaternions may be ok, so I left it in. --Chan-Ho Suh 12:45, Dec 16, 2004 (UTC)

If Alexander MacFarlane's contribution to the Royal Society at Edinburgh was the first published description of the hyperboloid model, should he not be so credited ? Rgdboer 00:44, 2 Apr 2005 (UTC)

Sure, why not; but to avoid further controversy, it would be best to have a reference to the MacFarlane paper and/or to the secondary source that makes this claim. linas 02:44, 2 Apr 2005 (UTC)

Renolds reference is earliest suggestion so far Rgdboer 00:09, 7 January 2006 (UTC)

Nice job! --Chan-Ho (Talk) 03:25, 7 January 2006 (UTC)

[edit] Hyperbolic plane vs. hyperbolic geometry

I have often wanted to separate off the hyperbolic plane stuff to its own page. By hyperbolic plane I mean the unique simply connected two-dimensional surface with constant curvature −1. Hyperbolic geometry is a much more general term than just this. For one thing there are many hyperbolic Riemann surfaces that aren't simply connected (most in fact). Hyperbolic geometry should also refer to higher dimenisonal manifolds with similiar geometry (such as hyperbolic 3-space).

I started writing a draft of the hyperbolic plane article about six months ago (see User:Fropuff/Draft 2) but soon abandoned it and moved on to other things. I wasn't sure how to properly organize things. There are 4 main models of the hyperbolic plane:

1. The Minkowski model (which is usually taken as the definition)
2. The Klein model
3. The two Poincaré models (or conformal models)
   • The Poincaré half-plane model
   • The Poincaré disc model

My question was whether or not these should all be discussed in the same page or if we should have separate pages for each. If separate pages, should the Poincaré models be discussed on the same page or separate pages? I think at least the Minkowski model should be the same page as the hyperbolic plane page and taken as the definition. Anyone have any thoughts on the matter? -- Fropuff 17:04, 2005 Mar 10 (UTC)

I've written separate articles for all of the above, and then linked them to the article on hyperbolic space; however, there is still not a separate article on the hyperbolic plane. I think defining hyperbolic space as the hyperbolic model/Minkowski model is a bad idea, since there are various models. Moreover, the more fundamental model underlying both the hyperbolic and Klein models is the projective semialgebraic model, which defines a distance function on a projective semialgebraic variety; from this we get both the hyperbolic and Klein models by normalizing. I've put a discussion of this and the various models derivable from it in the hyperbolic space article. Gene Ward Smith 21:33, 19 April 2006 (UTC)

You asked for an opinion :) ... There should be an introductory article (such as this one) that gives a one or two paragraph into to each of the models, and mentions things like multiply connnected surfaces, etc. However, one can say a lot about the poincare models alone, and so the paragraphs should link off to the full-detail article on the poincare model.

Unfortunately, I am the one to blame for the current disconnected state of the Poincare model related articles. For a while I had one article with included both the Poincare metrics for both plane and disk, the symmetries for the plane and disk, the schwarz-alhfors-pick theorem for the plane and disk, and it just got so long that splitting it up seemed to be the right thing to do. So I split it up ... but now it just feels disconnected and scattered. I was going to continue cleaning up and integrating and shuffling the contents so that a nicer table of contents resulted, but never quite got around to it.

Anyway, I have vague plans for continuing with the series, e.g. adding teichmuller spaces, and maybe in the process I'll clean up the 2d hyperbolic stuff a bit more. I know very very little about 3d & 4d hyperbolic things. (I gather no one does, outside of ed witten) linas 18:43, 20 Mar 2005 (UTC)

I'm not opposed to having separate pages for the two Poincaré models, although having a third page for the Poincaré metric seems overly redundant. Each model of the hyperbolic plane should discuss its own form of the metric.

As far as higher dimensions go, we need to have a separate page for hyperbolic space Hⁿ (which currently redirects here). Witten may know more the average guy about hyperbolic manifolds, but he is certainly not the expert. Entire books have been written about hyperbolic 3-manifolds. -- Fropuff 02:06, 2005 Mar 22 (UTC)

I was joking about Ed Witten. He's just mindblowing to listen to when he talks. Although 'tHooft is arguably even more fun. You'll notice I added a reference for a book by matsuzaki and tanaguchi for the 3-manifold case (its in the article Kleinian group). As the book deals primarily with 3-manifolds and secondarily with kleinian groups, feel free to copy the reference .. here ?linas 15:58, 24 Mar 2005 (UTC)

Oh,and just to be clear, I think the poincare half-plane and disk should be treated together in one article. Essentially all of the theorems and facts for one apply to the other; the're so closely mirrored it would be strange to separate them. I could try to merge the metric article back in with some other article. Not sure which. Low priority for me right now. I'll be reviewing Riemann surfaces shortly, and writing some articles about that, so maybe a clearer structure will emerge once I immersse my self in that a bit. linas 16:05, 24 Mar 2005 (UTC)

Oh, and one more comment: Klein model should probably redirect to Kleinian group and have one article discuss both. If/when that article bloats into something large, it could be split into two. But, for now, I think it would make sense to keep them together. linas 16:10, 24 Mar 2005 (UTC)

[edit] Escher circle limit III

I am removing the parenthetical statement about geodesics in Escher's circle limit III. It read:

The famous circle limit III and IV [2] drawings of M. C. Escher illustrate the unit disc version of the model quite well. In both one can clearly see the geodesics (in III they appear explicitly).

the reason for this is that the white lines in CLIII are not geodesics. The angles on the triangles are slightly less than 60 degrees and the angles on the squares (mmm ... equilateral equiangular quadrilaterals) are slightly more than 60 degrees. If they were geodesics they would meet the "bounding circle" at right angles. They clearly don't. Do a google search on "geodesics in circle limit III" and the link http://www.ajur.uni.edu/v3n4/Potter%20and%20Ribando%20pp%2021-28.pdf comes in near the top, and it explains it. Andrew Kepert 09:45, 23 Jun 2005 (UTC)

[edit] i don't understand this

Oh gosh, folks, this is a very informative article but it lacks any sort of introduction for the mathematically uninitiated as to what this stuff is, or why it's important.

[edit] Attempted Fix

There we go: I added an intro for you. Let me know what you think. --Rob 02:00, 4 October 2005 (UTC)

[edit] Intro

The intro has a lot of problems. Besides not being very well written and sounding very unencyclopedic, e.g. "things opened up big time", it perpetuates historical inaccuracies. A better idea may be just to merge stuff in from Non-Euclidean geometry and delete most of this intro. --Chan-Ho 09:30, 6 November 2005 (UTC)

I agree. -- Fropuff 04:06, 7 November 2005 (UTC)

It's also too long. Yanwen 21:13, 22 May 2006 (UTC)

[edit] This section makes no sense

"Another fun thing to do, when things are slow at the office, is to cut a couple of sheets of paper into a few dozen identically sized squares, and tape them together putting five squares at each corner. Then note how two "rows" of squares which are next to each other at on point will diverge until they are arbitrarily far apart."

It's not that encyclopedic, for starters, and is it just me, or is it impossible to understand? Perhaps it should be made more formal, for one thing. Take out "when things are slow at the office," perhaps? I'm not sure if the "fun" part is okay or not; it's not like this paragraph is meant to be solid, technical facts, but an encyclopedia isn't meant to be so informal. And more importantly, I don't understand what this proposes you can do. "Putting five squares at each corner" doesn't make sense. If anyone understands these instructions, could they please clarify them? Gyakuten 00:23, 12 January 2006 (UTC)

"At each corner" means that you try fitting five squares to go around a corner insteaad of the usual four you would have if you fit them together to give a flat sheet. The angle excess gives negative curvature at that corner. --Chan-Ho (Talk) 01:46, 12 January 2006 (UTC)

Sorry, I was in a bit of a rush last time. Anyway, that paragraph might as well be deleted. The content is basically that one can make combinatorial models of hyperbolic geometry and realize them as nice paper models. To write about that in a nice way would take quite a bit of work and probably be deserving of its own article really, e.g. "physical models of the hyperbolic plane", which could include crocheting the hyperbolic plane. --Chan-Ho (Talk) 07:37, 12 January 2006 (UTC)

Anyway, I'm going to remove that paragraph for now, since it is not encyclopedic in form and not written very clearly. --Chan-Ho (Talk) 06:48, 15 January 2006 (UTC)

[edit] Hjelmslev transformation

# The fifth model is the Hjelmslev transformation. This model is able to represent an entire hyperbolic plane within a finite circle. This model, however, must exist on the same plane which it maps, and therefore non-Euclidean rules still apply to it.

This appears to have been written by someone unfamiliar with the other models of hyperbolic geometry, as both the projective disk and conformal disk models have the properties stated. In addition, I've never heard of a fifth model called the Hjelmslev transformation (it would be strange to call a model a transformation, in any case), although there is another model called the upper hemisphere model that I've been meaning to add. Also, when I look at Hjelmslev transformation, it just describes the conformal disk model (also called Klein model), so it would seem to be redundant. My literature search makes me suspect that the Hjelmslev transformation refers to a map between two of the known models. In addition, the only references of Hjelmslev I could find were in reference to Hjelmslev planes and axioms which appear to be a more general setting than that of hyperbolic geometry. ---Chan-Ho (Talk) 08:41, 17 January 2006 (UTC)

Uh... yeah, sorry. I added that entry on Hjelmslev's transformation, and I wrote what is at Hjelmslev transformation so far. I cannot say that I know much about the Klein model, except that I got the impression that the disk was, in fact, Euclidiean. The Hjelmslev transformation may be a step between the the actual plane and the tidy Klein model, but it is also an independent model of it's own. A geometer needn't only do geometry in an Euclidean plane to be certain of his results. I remind you that Lobachevsky did his whole work on hyperbolic geometry without the aid of models. If one were accustomed to working in this geometry, the Hjelmslev transformation could be seen as a tremendous time-saver and proof-simplifier. Proofs about parallel and ultra-parallel lines become very clear as a result of this transformation. Since I do not understand the rules which govern the Klein model, and no article has been created explaining it, I think it is hasty to remove this passage about this transformation. All I know is that this is a legitimate model of hyperbolic geometry. If it belongs under some other heading, I think there ought to be an article created for the sake of this. And the explanation of exactly how these other models work. Oh, I don't know. --- SJCstudent 07:23, 18 January 2006 (UTC)

I'm on something of a wiki-break, so I really shouldn't be editing this :-) but I thought you should get a prompt response. I don't understand your statement about the Klein model being "Euclidean". It isn't, as all the "models" of hyperbolic geometry are in fact respresenting hyperbolic geometry. If you mean that geodesics in the Klein model look like straight chords in the disc, your Hjelmslev model has that same feature from your pictures and description. In fact, your pictures of parallel, ultra-parallel, etc. look like the standard pictures of parallel, ultra-parallel, etc. in the Klein model! Not to mention that the properties you state at the end of Hjelmslev transformation:

1. The image of a circle sharing the center of the transformation will be a circle about this same center.

2. As a result, the images of all the right angles with one side passing through the center will be right angles.

3. Any angle with the center of the transformation as its vertex will be preserved.

4. The image of any straight line will be a finite straight line segment.

5. Likewise, the point order is maintained throughout a transformation, i.e. if B is between A and C, the image of B will be between the image of A and the image of C.

6. The image of a rectilinear angle is a rectilinear angle.

also hold for the Klein model. So I'm afraid you haven't said anything to clarify why this model is different than the Klein model. In addition, I can't find any mention of the Hjelmslev model in any of my books on hyperbolic geometry. The only mention I find on the Internet is of something seemingly more general. Also, my search of published mathematical papers by Hjelmslev doesn't turn up a model of hyperbolic geometry. His papers on infinitesimal and projective geometry, as I said, appears to be of a more general foundational nature.

To allay your fears, let me mention that yes, I am familiar that geometers work in spaces more general than a Euclidean plane and I am aware Lobachevsky did not use a model. Despite all this, I have not heard of the Hjelmslev tranformation, nor do I know of some model of hyperbolic geometry by Hjelmslev that looks so much like the Klein model. I request that you cite a source you are using to create your Hjelmslev transformation article. As it stands, it looks like a ripe candidate to be merged/moved into Klein model with correct historical attributions. --Chan-Ho (Talk) 22:37, 23 January 2006 (UTC)

Ok, here is the deal. 1. The difference I am trying to highlight is this: the Klein Model places an infinite hyperbolic plane within a finite euclidean circle, the Hjelmslev transformation places an infinite hyperbolic plane within a finite Lobachevskian circle. The lines in the example circles look straight because they are straight, I have preserved the image of straightness in the finite diagrams. Either way... 2. I am looking for some textual basis for this transformation outside of my non-euclidean college textbook. I have written the author of the manual and spoken to other professors who are in charge of the department. I will either provide you with proper evidence soon, or alter my college's curriculum. Either way, I will have something soon. Thank you for your patience. SJCstudent 17:15, 25 January 2006 (UTC)

If I understand the article correctly, it is equivalent to what you would get by taking the Klein model and shrinking it by some scaling factor k; that is, each vector ||u||<1 in the Klein model becomes ku, where 0<k<1. In terms of the hyperbolic space, we have a model of hyperbolic space in a finite ball in hyperbolic space. As k-->0, this approaches the Klein model since the curvature of the region covered by the ball goes to zero. Gene Ward Smith 22:26, 20 April 2006 (UTC)

FWIW, I have seen older books (books concentrating on ruler-compass constructions, as opposed to "modern" algebraic constructions) that have a model of the hyperbolic plane with the geodesics being straight (angles are of course not preserved). I just don't remember the name of that "model". linas 01:05, 26 January 2006 (UTC)

That model is called the Klein-Beltrami model or the projective disk model. SJCstudent claims to be talking about something different. But what he is talking about I'm not quite sure yet. -- Fropuff 01:41, 26 January 2006 (UTC)

Ok, I did some extensive research. But, before I reveal what I have found, I think there are a few misunderstandings I should attempt to clarify first. One: for the last time... the Klein model projects an hyperbolic plane into a EUCLIDIAN circle, the Hjelmslev transformation projects an hyperbolic plane into an HYPERBOLIC circle. I do not know how I can make the distinction more clear than this. Two: I am not denying that these two models are very similar in alot of ways, this does not however make one more primary. For example... I could have easily said that the Klein model is just some ripoff of the Hjelmslev transformation. Either way....

The 16th volume of the mathematical series "International Series of Monographs in Pure and Applied Mathematics" is entitled "Non-Euclidean Geometry" and is written by Stefan Kulczyscki. It was trasnlated from Polish by Dr. Stanslaw Knapowski of the University of Pozan. Copywright 1961 by Panstwowe Wydawnictwo Naukowe Warszawa. It was originally prinited in Poland. Its Library of Congress Card Number is 60-14187.

In this book, sections 9 and 10 detail the creation and use of the Hjelmslev transformation. Please respond asap. I feel that this outside textual source justifies the reinsertion of the transformation into the article. SJCstudent 18:39, 3 February 2006 (UTC)

H.S.M. Coxeter writes in his review of that book (in the American Mathematical Monthly Vol. 69 No. 9 p. 937 available through JSTOR:)

The mathematical development begins with Hjelmslev's theorem (see e.g. Coxeter, Introduction to geometry, Wiley, NY, pp. 47, 269), which enables the author (following Hjelmslev himself) to prove that a particular transformation of hyperbolic space ("mapping j") is a collineation. O being a fixed point, each ...[Coxeter explains the Hjelmslev transformation]...The whole space is thus transformed into the interior of a sphere whose chords represent whole lines. We thus have the Beltrami-Klein projective model imbedded in the hyperbolic space itself!

From the start, I thought it was something like this, a transformation, rather than a new different model of hyperbolic geometry. It should be noted that maps between (and into) different models are not uncommon, and I am familiar with several of these. For example, the Klein model is often times embedded into the projective plane (historically, this was one way it was discovered by Klein), and can consequently show up in the visual sphere of an observer in a higher dimensional hyperbolic space (as viewed through the upper half space model). The Lorentz model also offers different ways to view the Klein and Poincare disc models inside the Lorentz space.

My conclusion is that despite your reference (Coxeter mentions it is a good book by the way), it does not justify listing Hjelmslev transformation as a separate model of hyperbolic geometry. It certainly deserves some mention but perhaps under "see also" or some other explanatory section. --Chan-Ho (Talk) 22:17, 3 February 2006 (UTC)

This seems satisfying. I apologize for not fully recognizing the distinction between "model" and "transformation" earlier. A "See also" section is suiting... Sorry for all the hassle. SJCstudent 01:31, 4 February 2006 (UTC)

[edit] Questions about hyperbolic 1-space

For all the talk about the hyperbolic plane and even hyperbolic space (where people might generally be thinking of hyperbolic space of more than two dimensions, although I know hyperbolic 1-space is an example of hyperbolic space), hyperbolic 1-space, the "maximally symmetric, simply connected [of course any connected 1-dimensional manifold, even the circle, is simply connected]," 1-dimensional "Riemannian manifold with constant sectional curvature −1," is not often talked about, and I am curious about it. What is it called? The hyperbolic line? I can tell that hyperbolas are not examples of hyperbolic-1 space as they are not connected and clearly do not have constant curvature. How many dimensions of Euclidean space does it take to isometrically embed hyperbolic 1-space? Or what I'm really looking for is: how many dimensions of Euclidean space does it take for hyperbolic 1-space to be embedded in and be as much itself, if you know what I mean, as the circle is in the Euclidean plane and the [i]n[/i]-sphere is in Euclidean [i]n[/i]+1-space? That may be eqivilent in all cases to a manifold being able to be isometrically embedded in a certain space, but I'm not sure. It seems like that number of dimensions must be greater than two, because a curve having constant nonzero curvature and being confined to the Euclidean plane would seem to have to be a circle. But would three Euclidean dimensions be enough for hyperbolic 1-space to "naturally" fit? The number of dimensions it takes to isometrically embed hyperbolic 1-space could shed some insight into the number of dimensions it takes to isometrically embed hyperbolic 2-space, which I believe has been narrowed down to 4 or 5 now but I'm not sure if it's been proven that it doesn't take 6 dimensions. Any answers to these questions would be greatly appreciated. Kevin Lamoreau 05:20, 4 June 2006 (UTC)

I'm overwhelmed by the full question, but I don't think it makes sense to talk of negative curvature of a 1-dimensional surface. Guassian curvature is intrinsic property as the product of the principle curvatures, and curvature defined as the reciprical of the radius of tangent circles. The sign is only meaningful if there are 2 or more dimensions to the surface. So you can ask for a 1-dimensional surface of constant curvatures either +1 OR -1, but both are simple unit circles embedded in a Euclidean 2-space or higher. I guess that doesn't help much on your question. Tom Ruen 17:57, 4 June 2006 (UTC)

See my reply at Talk:Hyperbolic space. -- Fropuff 18:17, 4 June 2006 (UTC)

Thanks for both of your replies, Tomruen and Fropuff. I've looked at the curvature article and I get what both of you are saying. If it wasn't for the absolute value function, the curvature of the circle would be positive if the circle were defined parametrically c(t) = (x(t),y(t)) with the circle going counter-clockwise as t increases, and negative the circle were difined parametrically with the circle going clockwise as t increases. I also, after some pondering (I can have a thick head sometimes), get Fropuff's point in Talk:Hyperbolic space about all one-dimensional Reimannian manifolds being locally isometric, not just locally homeomorphic. So both of your replies were helpful. Thanks again, Kevin Lamoreau 20:02, 5 June 2006 (UTC)

[edit] Relativity stuff

I removed some of the stuff about relativity, which was not encyclopedically written and was too specific--I don't think the numbers made anything clearer. I corrected the remaining observation and combined it with the existing note about observers. -- Spireguy 02:32, 10 October 2006 (UTC)

[edit] have you seen this curve?

I'm surprised that there is an article for hypercycle (aka equidistant) but none for horocycle. —Tamfang 07:17, 21 October 2006 (UTC)