# Talk:Hyperbolic geometry

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## 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. --C S 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! --C S (Talk) 03:25, 7 January 2006 (UTC)

## 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)

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)

Is this type of structure dealt with adequately here? As a non-mathematician I noticed it seldom discussed in articles about hyperbolic plane surfaces. epinet.anu.edu.au/mathematics/p_surface Mydogtrouble (talk) 19:28, 8 February 2010 (UTC)

## 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)

## 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.

### Attempted Fix

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

## 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. --C S 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)

## 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. --C S (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. --C S (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. --C S (Talk) 06:48, 15 January 2006 (UTC)

## 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. ---C S (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. --C S (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. --C S (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)

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)

## 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)

## 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)

## Hyperbolic Spaces > 2 dimensional

I found this page to be somewhat informative but then I think I might be in a little over my head too. I thought I would make this one comment though. To me (a person of limited geometry skills) the most disturbing aspect to this hyperbolic geometry is the expandability of it to at least a 3rd dimension. In Euclidean space it is fairly easy to imagine a 3rd dimensions being added to the 2nd. One could visualize it as the difference between a single sheet of paper and a stack of papers. If I try to imagine how to expand the hyperbolic 2 dimensional plane into a 3rd dimension I find that it doesn't work so well. In fact, it seems like the hyperbolic plane as a stack of curved papers still invades the Euclidean 3 dimensional space. Actually, come to think of it even the 2 dimensional hyperbolic space seems dependent on the Euclidean 3 dimensional space. If anyone would care to comment on how to visualize a 3 dimensional hyperbolic space I would be interested in reading it...

--Bitsync 02:14, 10 February 2007 (UTC)

I have seen a computer-generated image of a three dimensional hyperbolic space in which the space is filled with dodecahedrons which come together four along each edge and eight at each vertex. After seeing that, it is not so hard to imagine it. Or you could imagine generalizing the Poincaré disk model to three dimensions where it becomes a ball; and the "planes" are sections of spherical surfaces which intersect the boundary of the ball perpendicularly. JRSpriggs 09:03, 10 February 2007 (UTC)

Indeed! :) Tom Ruen 10:40, 10 February 2007 (UTC)

Thanks, Tom. Now, I am chagrined that I just sent an e-mail to Charles Gun at the Technical University of Berlin asking for his version of this 5,3,4 image which I saw in SCIENCE of 22 December 2006. JRSpriggs 12:29, 11 February 2007 (UTC)

It seems we are going back and forth on the inclusion of Systolic geometry in the "See Also" section. The link is somewhat relevant (and I think "spam" is a little harsh here), as there is mention on the target page of the use of systolic ideas in a hyperbolic setting. But the See Also section is getting pretty darn long, so it makes sense to open a discussion of how many links there should be, and which ones should be included. If it is too long, and some links should be pruned, then Systolic geometry would probably be one to go. After all, many topics in mathematics have as much connection to hyperbolic geometry as that does. Comments? -- Spireguy 02:18, 26 April 2007 (UTC)

## "hyperparallel"

This is the only place I've seen hyperparallel for lines sharing an ideal point. When I saw the word I took it to mean ultraparallel, i.e. diverging in both directions, and indeed that is how Mathworld defines hyperparallel. The most unambiguous word I've seen used for the sense intended is asymptotic. —Tamfang 00:59, 21 May 2007 (UTC)

Before I saw this article, I had always said "asymptotically parallel" for hyperparallel, and "having a common perpendicular" for ultraparallel when I wanted to distinguish them. JRSpriggs 10:02, 21 May 2007 (UTC)

What's the idea to repeat all the wikilinks in a separate "see also" section? It is redundant and makes the article ugly. --Pjacobi (talk) 10:58, 14 March 2008 (UTC)

It’s not supposed to be redundant, but rather to have links that are related, or of interest: increasing the accessibility. Even hypertext has see also section. Take for instance Elliptic geometry, which is linked to only in the see also section. If one didn’t know what elliptic geometry was related (and the article doesn’t say that it is, although there is a faint reference in the last section), they wouldn’t even know to search for it. Instead the link is there.
Some of them are redundant, according to Wikipedia: see also which says generally not to put in things that are linked to in the article, such as non-Euclidean geometry. In my opinion that doesn’t apply to things that are buried in the article where they might not be found though, such as Fuchsian model.
In any event, see also sections are a part of wikipedia that help users navigate. GromXXVII (talk) 11:16, 14 March 2008 (UTC)
I believe that "see also" sections serve a good purpose. They list in one convenient place many related articles. Some of these may not be linked elsewhere in this article. Others may have links which are difficult to find. Please do not remove the section again -- doing so is vandalism. JRSpriggs (talk) 11:23, 14 March 2008 (UTC)
It is completely ridiculous and bad style. But if it is enwiki policy to keep these abonimation, I cannot do anything but rolling my eyers. On dewiki the existence of a "see also" section is taken as sign of missing copy-editing. --Pjacobi (talk) 11:37, 14 March 2008 (UTC)
I view it as a sign of bad copyediting here too. It should only have links that are not included in the main text, and the existence of such links in a see-also section is a sign to me that the main text should say something more so that it could include those links more directly. —David Eppstein (talk) 15:45, 14 March 2008 (UTC)
I disagree. I think See also sections have an independent value. As I see it, there are three types of links, those that point to background information which may be considered prerequisite for the article, e.g.parallel postulate, those that point to tangential topics, e.g. people who attempted to prove the parallel postulate, and those that direct readers interested in the topic to articles that will broaden or deepen their understanding. The last, I believe, belong in a See also section so the reader does not have to pore back through the article to find each one. --agr (talk) 20:03, 14 March 2008 (UTC)
Maybe you should try to convince the editors of Wikipedia:See_also, who seem to believe otherwise. —David Eppstein (talk) 21:05, 14 March 2008 (UTC)
They style page doesn’t actually address the purpose of the section. Most of the time they include related topics that are either related and not mentioned, or related and mentioned frequently and thus warrant a link in the section in addition to the first usage in the article (as the subsequent uses are generally not linked to in the article, and it can be difficult to find the first usage if the article is long).
In any event, I think each link removed should be considered individually by somebody to see if it merits a link in the section (the arbitrary and broad removal of anything useful and true is rarely good). I don’t know enough about the topic to do this. GromXXVII (talk) 21:58, 14 March 2008 (UTC)

## "small scale" relative to what?

The following sentence is in the article:

"On small scales, therefore, an observer would have a hard time determining whether he is in the Euclidean or the hyperbolic plane."

At first reading, this didn't sit well with me. In Euclidean geometry, it doesn't matter at what scale you're working, and I don't think there's a way to specify the scale becaule the plane is infinite. (Does this make sense to anyone else?)

Then I figured that these "small scales" in hyperbolic geometry may be relative to the curvature of the hyperbolic space, or something like that. It seems that the more 'curved' the space is, the farther you have to 'zoom in' (and the smaller distances you have to consider) before the angle of parallelism approaches 90°. Given a hyperbolic space, does it have a parameter that describes its curvature? Might it be useful to add a few words on this topic to the article? Oliphaunt (talk) 12:55, 15 May 2008 (UTC)

In the hyperbolic plane, the angle defect of a triangle is the product of the curvature and the area of the triangle. In particular, it is directly proportional to the area. Hence, the smaller the area, the smaller the angular defect. Symbolically, by the Gauss-Bonnet theorem
$KA = \theta_1 + \theta_2 + \theta_3 - \pi$
where K is the curvature, A the area of the triangle, and the θi are the angles of the triangle. silly rabbit (talk) 13:31, 15 May 2008 (UTC)
OK, but do I then understand correctly that a hyperbolic plane is characterised by its curvature K? Shouldn't this concept then be more prominently discussed (or linked to) in the article? Oliphaunt (talk) 14:07, 15 May 2008 (UTC)
Yes, that's right. In fact a hyperbolic plane is the unique connected simply-connected complete Riemannian manifold of constant negative curvature K < 0. (Similar statements hold in higher dimensions, but with constant negative sectional curvature instead. See Space form.) This fact is rather important, and probably deserves to be mentioned somewhere. silly rabbit (talk) 14:27, 15 May 2008 (UTC)
As a result, each hyperbolic plane has a length scale which is $\frac{1}{\sqrt{-K}}$ where $K\!$ is the (constant) curvature of the plane. If the diameter of a region is small compared to that, then it will be approximately Euclidean. JRSpriggs (talk) 20:49, 15 May 2008 (UTC)
Thanks for the extra info. It would be nice if this could be added into the article somehow. Oliphaunt (talk) 21:20, 15 May 2008 (UTC)

## Exponential growth

The first is that the link relating to exponential growth points to the wrong page: it points to a page on a rather technical group theory topic rather than to the page on exponential growth.

My other point was silly. It didn't even dawn on me that it was referring to the hyperbolic circumference divided by the hyperbolic radius. Sorry 'bout that. --Hurkyl (talk) 20:53, 10 February 2009 (UTC)

On the hyperbolic plane, the circumference of a circle is
$C (r) = 2 \pi \sinh (r) \,$
assuming that the curvature is -1. Integrating to get the area gives
$A (R) = \int_{0}^{R} C (r) d r = 2 \pi (\cosh (R) - 1) \approx \pi \exp (R) \,.$
Clear? JRSpriggs (talk) 09:40, 11 February 2009 (UTC)

## Gyrovector spaces - please remove

The section on gyrovector spaces is very poorly written, and there doesn't seem to be any evidence that this theory yields new insight into hyperbolic geometry. As far as I can tell from the linked article, someone has recently stumbled upon a less convenient formulation of symmetric spaces (a very well-established theory), and proposed to explain sundry mysteries of the universe (e.g., the nature of dark matter) by some ill-defined means. It looks like some combination of crackpot and original research which, as I understand, is frowned upon here. —Preceding unsigned comment added by 18.87.1.150 (talk) 21:30, 15 February 2010 (UTC)

Perhaps the section should be reduced to a sentence. The main article, Gyrovector spaces, is now well-developed with several significant sources. One of them, Scott Walter (2002), indicates the vital subject of relativity and modern mechanics in hyperbolic geometry. Walter credits the creator of gyrotheory with cultivating this vital thread in science. But his mention of an "elegant non-associative algebraic formalism" reminds me of hyperbolic quaternions; non-associativity is seldom elegant. Furthermore, Walter says the gyrotheory is most useful for comparing "elegance of respective proofs of selected theorems".Rgdboer (talk) 02:34, 16 February 2010 (UTC)
I've rewritten the section. I think it should stay. Charvest (talk) 07:36, 17 February 2010 (UTC)
What, there's a formalism for measuring elegance? —Tamfang (talk) 19:06, 29 September 2010 (UTC)

The difference between hyperbolic vectors and euclidean vectors is that addition of vectors in hyperbolic geometry is nonassociative and noncommutative.

Could this be reworded in a way that doesn't contradict the standard axiomatic definition of a vector (Euclidean or otherwise)? Dependent Variable (talk) 04:28, 28 September 2010 (UTC)
No. A vector is something with magnitude and direction. Something with magnitude and direction doesn't have to be compatible with the axiomatic definition of a vector space. Relativistic velocity addition is not associative or commutative (except when the velocities are in the same direction). Therefore relativistic velocities demonstrate that the axiomatic notion of vector space does not encompass all types of vectors. 84.13.72.71 (talk) 08:11, 28 September 2010 (UTC)
How about replacing the word "vector" with "gyrovector" here (given that a vector space is a special case of a gyrovector space, rather than the other way around, according to Gyrovector space)? A nonassociative or noncommutative vector is just a contradiction in terms.
"Something with magnitude and direction" is an appeal to intuition used in elementary texts, rather than an actual definition. The axioms give a precise definition which accounts for how the word is actually used in mathematical literature. A car has a direction and a magnitude, but that doesn't make it a vector. A polynomial, a color (in the context of a color space), a linear transformation, an infinite sequence of real or complex numbers, a discrete probability distribution can all be characterised as vectors, with appropriate definitions of addition and scalar multiplication. A vector space doesn't have to have a norm (defining magnitude) or an inner product (defining angles and hence a notion of directions).
If physicists call relativistic velocities "vectors", it could be for historical reasons (because the concept descends from that of velocity in Newtonian mechanics), and because they're represented as triples of real numbers (such objects being widely treated in other contexts as vectors). For some authors, they may be defined as vectors in terms of some overarching structure, such as spacetime algebra (a kind of geometric/Clifford algebra). It is often pointed out that relativistic velocity is not a vector associated with Minkowski space, only the spatial components of one in a particular, arbitrary basis. Dependent Variable (talk) 11:42, 7 October 2010 (UTC)
I've ammended the article to say gyrovectors rather than vectors are noncommutative. Of course in newtonian mechanics both position and velocity can be represented by the same type of vector and geometry, whereas in relativity position is unbounded but velocity is bounded so they have different geometries and notions of vector (n-tuples of R).
And yes it is the right way round. Every vector space is a gyrovector space. The axioms for gyrovectors are less strict as they don't require full associativity. So gyrovectors encompass a larger class of objects. Hyperbolic space can be seen as deformed euclidean space. Euclidean space is just a gyrovector space with a zero amount of deformation. 78.148.177.200 (talk) 13:54, 7 October 2010 (UTC)
It would be better to represent Lorentz boosts in Minkowski space by multiplication of matrices rather than addition of gyro-vectors. JRSpriggs (talk) 13:19, 28 September 2010 (UTC)
Yes. Gyrovector addition does not represent boosts. "Gyrovector addition" just means 3D-velocity composition. A boost is a transformation, between two different frames of reference, of the 4D-spacetime coordinates of events. Boost composition is associative. 3D-velocity composition is not associative. 84.13.64.158 (talk) 19:41, 1 October 2010 (UTC)
What is velocity composition if it is not following one boost by another? JRSpriggs (talk) 20:40, 1 October 2010 (UTC)
Velocities are closed under composition - a velocity followed by a velocity results in a velocity.
Boosts are not closed. A boost in one direction followed by a boost in a different direction is not just a boost in a new direction, but a boost plus a rotation. 84.13.208.35 (talk) 10:31, 2 October 2010 (UTC)

(unindent) If you are in spaceship O, and there is a spaceship A with velocity v relative to you, and a spaceship B with velocity u relative to A, then you can ask: "What is the velocity of B relative to you?". That's velocity composition.

Suppose the two velocities are not in a line but in different directions, then:

The coordinate transformation between O and A is a boost,
the coordinate transformation between A and B is a boost.
"What is the transformation between O and B ?". That's boost composition.
The answer is that the coordinate transformation between O and B is not a boost - if the axes at A are aligned with O and the axes at B are aligned with A, then the axes of B will be rotated relative to O. (Thomas precession). 84.13.66.230 (talk) 12:00, 2 October 2010 (UTC)

Thank you for the link to Thomas precession. I agree that the composition of two boosts is frequently not a pure boost, but includes rotation. I will have to think some more about what this means for velocities. JRSpriggs (talk) 12:27, 3 October 2010 (UTC)
The details of the relationship between boost composition and velocity composition have now been added to Velocity-addition formula#Velocity composition paradox. 89.241.232.46 (talk) 18:23, 1 November 2010 (UTC)
Yes but it's wrong because boost (short for Lorentz boost) is a matrix. The velocity that it acts on is the Minkowski 4-velocity so you should be talking about composing 4-velocities not boost matrices. Actually all the troubles originate from of the attempt to combine Cartesian vectors supplemented by the artificial gyro rotation. Gyrovectors are not needed because velocity is actually a vector in hyperbolic space as in the theory of Varicak and for such vectors there is a natural rotational effect when they are combined caused by parallel transport in curved space. Already in 1913 Borel remarked on this rotation which was later confirmed with the Thomas precession (see the article of Scott-Walker cited in the Varicak article).JFB80 (talk) 21:31, 5 December 2010 (UTC)
This is getting really off-topic for this page which is the talk page for hyperbolic geometry. However, yes it's true that a boost is usually represented as a 4x4 matrix. The boost matrix B(v) means the boost B that uses the components of v, v1, v2, v3 in the entries of the matrix, or rather the components of v/c in the representation that is used in the section Lorentz transformation#Matrix form. (To digress slightly: The formulae in the matrix entries depends on the way spacetime coordinates are represented. That article and most standard texts represent spacetime as (ct,x,y,z) and putting c=1 this becomes (t,x,y,z), but you could just as easily use (t,x,y,z) to start with (in which c is not used here not even c=1) and then you'd have to change the matrix entries to work with (t,x,y,z) instead of (ct,x,y,z), but that is a digression.) The point is that the matrix entries depend on the components of the 3-velocity v, and that's what the notation B(v) means. You could argue that the entries depend on the components of the 4-velocity because 3 of the entries of the 4-velocity are the same as the entries of the 3-velocity, but the usefulness of parameterizing the boost by 3-velocity is that the resultant boost you get from the composition of two boosts uses the components of the 3-velocity composition u$\oplus$v in the 4x4 matrix B(u$\oplus$v). But the resultant boost also needs to be multiplied by a rotation matrix because boost compostion (i.e. the multiplication of two 4x4 matrices) results not in a pure boost but a boost and a rotation, i.e. a 4x4 matrix that corresponds to the rotation gyr[u,v] to get B(u$\oplus$v) = B(u$\oplus$v)gyr[u,v] = gyr[u,v]B(v$\oplus$u). Admittedly the notation gyr[u,v] was previously used as the rotation applied to a 3-velocity, whereas here the same notation is used to mean the rotation of spacetime coordinates, and this has caused confusion, but it is at root talking about the same rotation: namely the rotation that arises from the composition of two boosts. The notation gyr in context refers to the rotation, whether the rotation of 3-velocities or the rotation of 4-coordinates, it refers to the rotation. I've added this explanation to the velocity-addition article to clear up the confusion. 2.97.18.221 (talk) 08:37, 7 December 2010 (UTC)
I dont think you understood what I said which is very relevant to hyperbolic geometry particularly the subject of parallel transport which I believe offers the correct explanation of what are called gyrovectors. See particularly the diagram parallel transport.png.JFB80 (talk) 17:17, 7 December 2010 (UTC)
In your previous comment you say "Gyrovectors are not needed because velocity is actually a vector in hyperbolic space", but gyrovectors are vectors in hyperbolic space. You also say "as in the theory of Varicak", but Varicak in his 1924 book abandoned attempts to formulate a vector algebra in hyperbolic geometry because he couldn't get vector algebra to work, which Scott Walter points out in the "concluding remarks" of the article you mention.
As you point out, parallel transport of a vector around a closed loop back to the starting point in curved space leaves the final vector pointing in a different direction than the starting vector. For relativistic velocities, this is Thomas precession - the "natural rotation" you talk of. The rotation makes, as you put it, "the attempt to combine Cartesian vectors", nonassociative and noncommutative (I assume this is what you are referring to by "troubles"), but the whole point of the gyrovector approach is to have a Cartesian vector algebra for hyperbolic geometry. 89.241.225.113 (talk) 20:20, 7 December 2010 (UTC)
I've changed the articles to use upper case Gyr for the 4-spacetime rotation and lowercase gyr for 3-space rotation. I had mistakenly used gyr for both. 2.97.24.14 (talk) 10:30, 9 December 2010 (UTC)

(1) You said: 'gyrovectors are vectors in hyperbolic space'. That is just an assertion, it needs demonstration and proof (which should have appeared in the article on gyro-vectors.) I have not seen a proof and dont believe it is possible to give one. . (2) What S Walter (1999) said was: 'adapting ordinary vector algebra for use in hyperbolic space was just not feasible, as Varicak himself had to admit'. which is not quite what you said and is also not what Varićak (1924) said which was 'In Lobachevsky space, in contrast to Euclidean space, there is a large variability in the form of geometrical constructions. For this reason, one encounters difficulties in transferring theorems from the usual vector algebra to Lobachevsky space' (Quoted verbatim from Kracklauer's 2006 translation). Varićak did formulate a vector algebra after discussing these difficulties. (3) Thomas precession occurs during motion in a path with continuous tangent. Parallel transport of vectors round paths with discontinuous tangents e.g. triangles, gives the sort of behaviour ascribed to gyrovectors. On a spherical analogy, compare the diagram I quoted (parallel transport.png) with a corresponding diagram in Sommerfeld's 1909 Phys. Z paper (Wikisource) (4) You said 'but the whole point of the gyrovector approach is to have a Cartesian vector algebra for hyperbolic geometry. The quotations (2) are saying that is not possible, which is also my opinion. If you maintain it is possible you should prove it.JFB80 (talk) 22:07, 9 December 2010 (UTC)

You're right. The Gyrovector space article should explain the link between a gyrovector space and a hyperbolic space and not just assert that they are the same. It should show that angles, lines, and distances in a gyrovector space are the same as in the model of hyperbolic geometry that the gyrovector space is supposed to represent; that a gyrovector which is a pair of points and the geodesic between those points are the same as hyperbolic geodesic linking two points. So that gyrolines, gyroangles and gyrometric match the hyperbolic lines, angle and metric. Ungar's books do show that the line element, curvature, geodesics etc are the same. There is also some info in this paper. I'll see what else I can find amongst the gyrovector publications and in due course I will add such info to the article. In 1999 Walter may have said "just not feasible", but later in 2002 Walter gave the review: "Ungar furnishes the crucial missing element from the panoply of the non-Euclidean style: an elegant nonassociative algebraic formalism". I'm not sure which Sommerfeld paper you are referring to. Could you provide a link ? 89.241.232.203 (talk) 00:28, 10 December 2010 (UTC)
I am glad you agree about the necessity of proof. Yes, Walker expresses different opinions in his two articles. The important point is that Varićak was misquoted. Refer to Fig.2 in the Sommerfeld paper. The link is::[[1]]JFB80 (talk) 19:23, 10 December 2010 (UTC)

## Four models of hyperbolic geometry

Are 4 models of hyperbolic geometry equivalent to each other? How do you show that by maths? Thank you! Milk Coffee (talk) 14:09, 28 October 2010 (UTC)

For each natural number n ≥ 2, there is, up to isomorphism a scale transformation, only one simply connected hyperbolic space of dimension n without a boundary. Thus the four models (the Beltrami–Klein model, the Poincaré disk model, the Poincaré half-plane model, and the hyperboloid model) all describe the same structure for hyperbolic geometry. Where they differ is in how they represent that structure within Eucidean space. In some cases, the articles on those models contain mappings between the model being described in the article and some of the other models. See: Beltrami–Klein model#Relation to the hyperboloid model, Beltrami–Klein model#Relation to the Poincaré disk model, and Poincaré disk model#Relation to the hyperboloid model. Unfortunately, we (at Wikipedia) do not have a formula for conversion between the half-plane model and any of the others (although I am sure that someone knows how to do this). JRSpriggs (talk) 23:39, 28 October 2010 (UTC)
Conversion between disc and upper-half models: Hyperbolic Voronoi diagrams made easy page 6. 89.241.235.146 (talk) 12:05, 1 November 2010 (UTC)
This may be a rather late reply, but may help any new reader trying to get an understanding: The main difference between the 4 models is that they represent different coordinate charts laid down on the same metric space, namely the hyperbolic space. The characteristic feature of the hyperbolic space itself is that it has a constant Ricci scalar curvature of -1. For a given metric space, the matrix representation of the metric tensor, the Ricci curvature tensor, etc. can be different for usage of different coordinate charts (i.e. the different models). However what is characteristic to the metric space and is indifferent to the coordinate chart used, is the Ricci scalar curvature. So, using any of the models, if you compute the Ricci scalar, you'll find that the values match. In short, Ricci scalar is an invariant of Riemannian manifolds. Another similar invariant of metric spaces are the geodesics (i.e. geodesics map to geodesics under coordinate transformation) and hence studying their intersections tell us a lot about the metric space itself. Hyperbolic geometry generally is introduced in terms of this later type of invariant. Once we define a coordinate chart (one of the many possible "models"), we can always embed it in a Euclidean space of same dimension, but the embedding is clearly not isometric (since the Ricci curvature of Euclidean space is 0). There is nothing very special about these 4 models - there are infinitely many more. However the embedding in Euclidean space due to these 4 specific charts show some interesting characteristics.- Subh83 (talk) 07:29, 14 March 2011 (UTC)
Thanks again! Milk Coffee (talk) 15:43, 14 March 2011 (UTC)
If there is popular interest, an abridged version of this explanation may be added in the main article for better understanding. Reference can be found in "Introduction to Hyperbolic Geometry, A. Ramsay" - Subh83 (talk | contribs) 23:04, 17 March 2011 (UTC)
By combining JRSpriggs' discussion and mine above, I am adding a brief subsection under the 4 models section. Feel free to improve. - Subh83 (talk | contribs) 17:37, 18 March 2011 (UTC)

Can Somebody PLEASE just provide an example of this?? Which models? Reference! I would like to read on this.

"Models have been constructed within Euclidean geometry that obey the axioms of hyperbolic geometry, thus proving that the parallel postulate is independent of the other postulates of Euclid." 79.131.121.133 (talk) —Preceding undated comment added 01:57, 2 February 2012 (UTC).

The ones described in the "Models of the hyperbolic plane" section of this article. —David Eppstein (talk) 03:11, 2 February 2012 (UTC)
Especially the Beltrami-Klein model where the relation between Euclidean and hyperbolic planes is quite obvious. JFB80 (talk) 21:37, 2 February 2012 (UTC)

In A Universal Model for Conformal Geometries of Euclidean, Spherical and Double-Hyperbolic Spaces each of the 5 well known models is treated individually up to dimension n and are (conformally) unified together with Spherical AND Euclidean spaces, new theorems are also deduced.Selfstudier (talk) 10:44, 16 February 2012 (UTC)

## Consistency of geometry

thus proving that the parallel postulate is independent of the other postulates of Euclid (assuming that those other postulates are in fact consistent).[2]

I removed the parenthetical qualifier. It was restored and re-removed, so I'll comment here: the qualifier is at best unnecessary, and I think it is actually incorrect or meaningless. If the author was thinking of Gödel's theorem, that fails here, this is not arithmetic. --192.75.48.150 (talk) 17:39, 9 January 2014 (UTC)

It is not necessary that geometry contain arithmetic for this to be an issue. Putting Gödel aside, axiomatic systems have sometimes been found to contain subtle errors which were not discovered until much later. And it should be clear by now that one cannot simply say "look at the physical space of the universe for an example of Euclidean or Hyperbolic geometry" to show its consistency. Aside from general relativity, it is conceivable (but not likely) that the universe could have a finite structure and the apparently flat geometry could simply be an approximation or illusion.
Still I will not fight you further on this since the probability of an inconsistency which cannot be fixed is very low at this point. JRSpriggs (talk) 07:35, 10 January 2014 (UTC)
Well, set theory might be inconsistent, sure, but Tarski's axiomatization of geometry is provably complete and consistent, and there is an elementary recusrive decision procedure. You might think it "possible" (not really) that elementary recursion is a lie, but meta-mathematics without this assumption can get sticky. You should not be certain of the conditional "if there is a model of euclidean geometry, there is a model of hyperbolic geometry" either. So the qualifier is still not right. --192.75.48.150 (talk) 16:42, 13 January 2014 (UTC)
Thanks for explaining that. Do you have a link to an on-line proof? JRSpriggs (talk) 09:14, 14 January 2014 (UTC)
Hmm. The real closed field article cites Ben-Or et al. The complexity of elementary algebra and geometry, 1986.
As for the bit about meta-mathematics without elementary recursion, a quick search didn't turn up any convenient citations. Hand-waving argument: meta-mathematics treats formal systems as mathematical objects in their own right. This involves encoding propositions as integers. The numerical value of a sequence of n symbols is going to be exponential in n. If you don't assume integer exponentiation is a total function, you're going to have a bad time. It can be done, but don't try this at home; metamathematical facts that you'd otherwise take for granted wind up tagged with a lot of caveats. --192.75.48.150 (talk) 17:48, 14 March 2014 (UTC)