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)
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:
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'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)
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)
I am removing the parenthetical statement about geodesics in Escher's circle limit III. It read:
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)
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.
There we go: I added an intro for you. Let me know what you think. -- Rob 02:00, 4 October 2005 (UTC)
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)
It's also too long. Yanwen 21:13, 22 May 2006 (UTC)
"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)
# 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)
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)
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!
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 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)
I'm surprised that there is an article for hypercycle (aka equidistant) but none for horocycle. — Tamfang 07:17, 21 October 2006 (UTC)
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)
Indeed! :) Tom Ruen 10:40, 10 February 2007 (UTC)
{5,3,4} |
{4,3,5} |
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)
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)
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)
The following sentence is in the article:
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)
An edit I made that was reverted had two rationales.
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)
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)
The difference between hyperbolic vectors and euclidean vectors is that addition of vectors in hyperbolic geometry is nonassociative and noncommutative.
(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)
(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)
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)
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).
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)
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)
I was thinking about this article, and would like to suggest to reorganise it a bit , so that the structure becomes
Intro - History - Main characteristic properties of Hyperbolic Geometry (new heading) - - non intersection lines (existing section) - - Triangles(existing section) - - Circles, disks, spheres and balls (existing section) - Models of the hyperbolic plane(existing section) - - Other models of the hyperbolic plane(existing section) - - Connection between the models(existing section) - Visualizing hyperbolic geometry (Advanced characteristic properties of Hyperbolic Geometry ? ) - Homogeneous structure - See also rest
I would like to add again https://en.wikipedia.org/?title=Hyperbolic_geometry&curid=241291&diff=644565668&oldid=642445150 ( by /info/en/?search=User:Peter_Buch )but they do need referencing.
This will be I think a major structural change so therefore please comment. WillemienH ( talk) 10:47, 30 January 2015 (UTC)
I started with a rewrite , moved history up and rewrote introduction. added links to saddle surface and sectional curvature (was thinking to remove info over ultra parallel and triangles triangles in the introduction but will wait till after rewrite properties)
Was thinking about a section special curves under properties but that is all for later WillemienH ( talk) 02:00, 31 January 2015 (UTC)
I rewrote large parts of the properties section , but kept the old sections, (marked with old at the end of the title) I think the old sections can be removed because they don't contain useful information , but would like to hear the opinion from others on this. (also i think that under the triangle subsection info is that should be moved to hyperbolic triangle where it is lacking. WillemienH ( talk) 17:51, 31 January 2015 (UTC)
Hi, i decided to remove the spheres and balls sections of Hyperbolic geometry#Circles, disks, spheres and balls
I think it is a good idea to limit the scope of this article mainly to the 2 dimensional case, and to add the higher dimension cases to more specific pages like: Hyperbolic space, Hyperbolic manifold or Hyperbolic 3-manifold (do we really need all these pages?)
The text I removed was:
The surface area of a sphere is
The volume of the enclosed ball is
For the measure of an n-1 sphere in n dimensional space the corresponding expression is
The measure of the enclosed n ball is:
PS I am wondering if the formula for the volume of the enclosed ball is correct (it looks like the volume can be negative) can somebody provide an reference? WillemienH ( talk) 14:09, 15 February 2015 (UTC)
The lead currently says
Can someone please clarify this in the lead? How can something "obey the axioms of hyperbolic geometry", such as that there are at least two lines through point P that do not intersect line R, while being "within Euclidean geometry"?
Thanks. Loraof ( talk) 20:37, 11 April 2015 (UTC)
I was thinking about removing the whole passage, it is to complicated at the point where it is . Allready under history it is mentioned that "In 1868, Eugenio Beltrami provided models of hyperbolic geometry, and used this to prove that hyperbolic geometry was consistent if and only if Euclidean geometry was." and further on there is a section on "Models have been constructed within Euclidean geometry " so the whole passage doesn't really add any thing. WillemienH ( talk) 10:21, 12 April 2015 (UTC)
But that also made me think about the next paragraph maybe "Because each of Euclidean geometry and hyperbolic geometry is consistent, and both have similar, small sectional curvatures, an observer will have a hard time determining whether his environment is Euclidean or hyperbolic. Thus we cannot decide whether our world is Euclidean or hyperbolic."(old version 12/04/2015) Is better replaced by something like
"Because both Euclidean geometry and hyperbolic geometry are consistent, and in the physical world we cannot physically construct lines long enough to decide if they will eventually meet or not we cannot decide if space is curved or not and with that if the geometry of our world is Euclidean or hyperbolic. we can only decide that the curvature of space is higher than (latest measure of space curvature with ref) "
There are some problems with this passage , what are those lines really? elastic string, to short; light rays , bend under gravity; gravity lines don't seem straight either and so on. (the last calculation of curvature I know off was made when it was still believed that light rays were straight and there are people who disagree with the used method) WillemienH ( talk) 10:21, 12 April 2015 (UTC)
Hyperbolic geometry § Geometry of the universe does not give me the sense of really "belonging" in this article. The relevance of hyperbolic geometry in special relativity is that all subluminal velocities form a hyperbolic space, which is unrelated to the geometry of the universe. Describing our universe as three-dimensional (by thinking of it as Euclidean, elliptic or hyperbolic) is a throwback to Newtonian/Galilean thinking that time and space are in some sense separable. Since as Minkowski geometry displaces Galilean geometry (= 3-d Euclidean + time of Galilean relativity) in special relativity, if one wishes to discuss deviations from flat space (Minkowski geometry), rather than the Euclidean/hyperbolic/elliptic geometry discussion, the appropriate geometries to consider are Minkowski space/ anti-de Sitter space/ de Sitter space. — Quondum 16:45, 3 May 2015 (UTC)
References
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help)
I was wondering what is the right name of this model?
I do prefer "hemisphere model" but that is just personal.
I did find literature that refers to this model as the "hemisphere model":
while i couln not find any for "hemispherical model"
but let us discuss before changing. WillemienH ( talk) 08:00, 7 May 2015 (UTC)
In the second paragraph of Hyperbolic geometry#Geometry of the universe (pre dating relativity), it refers to "the absolute length is at least one million times the diameter of the earth's orbit (20 000 000 AU, 10 parsec)". This implies that these three distances are (at least approximately) equal, but they are not. An astronomical unit is the radius of the Earth's orbit, so one million times the diameter of the Earth's orbit would be 2 000 000 AU, not 20 000 000 AU. A parallax second is about 206 265 AU. So ten parsecs is slightly more than 2 000 000 AU. JRSpriggs ( talk) 19:13, 7 May 2015 (UTC)
The Hyperbolic geometry#Geometry of the universe (including relativity) section is written on the mistaken assumption that Euclidean, hyperbolic and elliptic geometries are inherently 3-dimensional, which of course is rubbish. Sorry I don't have the expertise to correct it. Please could someone rewrite it to make sense? -- Stfg ( talk) 13:06, 26 May 2015 (UTC)
Hi
I nominated this page for good article ( /info/en/?search=Wikipedia:Good_article_nominations ) As part of this i moved the complete talk page as now to /Archive 1 Later I will remove sections I find "out of date" "irrelevant"or "solved problems" and shorten discussions still old but still relevant. WillemienH ( talk) 07:42, 26 June 2015 (UTC)
Removed some sections of this page as described above, they can still be found at /Archive 1 WillemienH ( talk) 08:28, 26 June 2015 (UTC)
I am not sure about where the GA discussion should be hold, I guess we could keep it here. Or when you want to staer a reviwe of it on a dedicated review page (I am not sure how this works yet) About the archiving, I did remove the bits of this page that were (according to me) not any longer relevant. Bits of which i was not sure of the (present) relevancy I kept here. (but there are copies of them in the archive) WillemienH ( talk) 20:11, 17 July 2015 (UTC)
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)
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:
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'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)
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)
I am removing the parenthetical statement about geodesics in Escher's circle limit III. It read:
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)
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.
There we go: I added an intro for you. Let me know what you think. -- Rob 02:00, 4 October 2005 (UTC)
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)
It's also too long. Yanwen 21:13, 22 May 2006 (UTC)
"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)
# 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)
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)
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!
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 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)
I'm surprised that there is an article for hypercycle (aka equidistant) but none for horocycle. — Tamfang 07:17, 21 October 2006 (UTC)
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)
Indeed! :) Tom Ruen 10:40, 10 February 2007 (UTC)
{5,3,4} |
{4,3,5} |
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)
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)
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)
The following sentence is in the article:
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)
An edit I made that was reverted had two rationales.
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)
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)
The difference between hyperbolic vectors and euclidean vectors is that addition of vectors in hyperbolic geometry is nonassociative and noncommutative.
(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)
(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)
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)
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).
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)
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)
I was thinking about this article, and would like to suggest to reorganise it a bit , so that the structure becomes
Intro - History - Main characteristic properties of Hyperbolic Geometry (new heading) - - non intersection lines (existing section) - - Triangles(existing section) - - Circles, disks, spheres and balls (existing section) - Models of the hyperbolic plane(existing section) - - Other models of the hyperbolic plane(existing section) - - Connection between the models(existing section) - Visualizing hyperbolic geometry (Advanced characteristic properties of Hyperbolic Geometry ? ) - Homogeneous structure - See also rest
I would like to add again https://en.wikipedia.org/?title=Hyperbolic_geometry&curid=241291&diff=644565668&oldid=642445150 ( by /info/en/?search=User:Peter_Buch )but they do need referencing.
This will be I think a major structural change so therefore please comment. WillemienH ( talk) 10:47, 30 January 2015 (UTC)
I started with a rewrite , moved history up and rewrote introduction. added links to saddle surface and sectional curvature (was thinking to remove info over ultra parallel and triangles triangles in the introduction but will wait till after rewrite properties)
Was thinking about a section special curves under properties but that is all for later WillemienH ( talk) 02:00, 31 January 2015 (UTC)
I rewrote large parts of the properties section , but kept the old sections, (marked with old at the end of the title) I think the old sections can be removed because they don't contain useful information , but would like to hear the opinion from others on this. (also i think that under the triangle subsection info is that should be moved to hyperbolic triangle where it is lacking. WillemienH ( talk) 17:51, 31 January 2015 (UTC)
Hi, i decided to remove the spheres and balls sections of Hyperbolic geometry#Circles, disks, spheres and balls
I think it is a good idea to limit the scope of this article mainly to the 2 dimensional case, and to add the higher dimension cases to more specific pages like: Hyperbolic space, Hyperbolic manifold or Hyperbolic 3-manifold (do we really need all these pages?)
The text I removed was:
The surface area of a sphere is
The volume of the enclosed ball is
For the measure of an n-1 sphere in n dimensional space the corresponding expression is
The measure of the enclosed n ball is:
PS I am wondering if the formula for the volume of the enclosed ball is correct (it looks like the volume can be negative) can somebody provide an reference? WillemienH ( talk) 14:09, 15 February 2015 (UTC)
The lead currently says
Can someone please clarify this in the lead? How can something "obey the axioms of hyperbolic geometry", such as that there are at least two lines through point P that do not intersect line R, while being "within Euclidean geometry"?
Thanks. Loraof ( talk) 20:37, 11 April 2015 (UTC)
I was thinking about removing the whole passage, it is to complicated at the point where it is . Allready under history it is mentioned that "In 1868, Eugenio Beltrami provided models of hyperbolic geometry, and used this to prove that hyperbolic geometry was consistent if and only if Euclidean geometry was." and further on there is a section on "Models have been constructed within Euclidean geometry " so the whole passage doesn't really add any thing. WillemienH ( talk) 10:21, 12 April 2015 (UTC)
But that also made me think about the next paragraph maybe "Because each of Euclidean geometry and hyperbolic geometry is consistent, and both have similar, small sectional curvatures, an observer will have a hard time determining whether his environment is Euclidean or hyperbolic. Thus we cannot decide whether our world is Euclidean or hyperbolic."(old version 12/04/2015) Is better replaced by something like
"Because both Euclidean geometry and hyperbolic geometry are consistent, and in the physical world we cannot physically construct lines long enough to decide if they will eventually meet or not we cannot decide if space is curved or not and with that if the geometry of our world is Euclidean or hyperbolic. we can only decide that the curvature of space is higher than (latest measure of space curvature with ref) "
There are some problems with this passage , what are those lines really? elastic string, to short; light rays , bend under gravity; gravity lines don't seem straight either and so on. (the last calculation of curvature I know off was made when it was still believed that light rays were straight and there are people who disagree with the used method) WillemienH ( talk) 10:21, 12 April 2015 (UTC)
Hyperbolic geometry § Geometry of the universe does not give me the sense of really "belonging" in this article. The relevance of hyperbolic geometry in special relativity is that all subluminal velocities form a hyperbolic space, which is unrelated to the geometry of the universe. Describing our universe as three-dimensional (by thinking of it as Euclidean, elliptic or hyperbolic) is a throwback to Newtonian/Galilean thinking that time and space are in some sense separable. Since as Minkowski geometry displaces Galilean geometry (= 3-d Euclidean + time of Galilean relativity) in special relativity, if one wishes to discuss deviations from flat space (Minkowski geometry), rather than the Euclidean/hyperbolic/elliptic geometry discussion, the appropriate geometries to consider are Minkowski space/ anti-de Sitter space/ de Sitter space. — Quondum 16:45, 3 May 2015 (UTC)
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I was wondering what is the right name of this model?
I do prefer "hemisphere model" but that is just personal.
I did find literature that refers to this model as the "hemisphere model":
while i couln not find any for "hemispherical model"
but let us discuss before changing. WillemienH ( talk) 08:00, 7 May 2015 (UTC)
In the second paragraph of Hyperbolic geometry#Geometry of the universe (pre dating relativity), it refers to "the absolute length is at least one million times the diameter of the earth's orbit (20 000 000 AU, 10 parsec)". This implies that these three distances are (at least approximately) equal, but they are not. An astronomical unit is the radius of the Earth's orbit, so one million times the diameter of the Earth's orbit would be 2 000 000 AU, not 20 000 000 AU. A parallax second is about 206 265 AU. So ten parsecs is slightly more than 2 000 000 AU. JRSpriggs ( talk) 19:13, 7 May 2015 (UTC)
The Hyperbolic geometry#Geometry of the universe (including relativity) section is written on the mistaken assumption that Euclidean, hyperbolic and elliptic geometries are inherently 3-dimensional, which of course is rubbish. Sorry I don't have the expertise to correct it. Please could someone rewrite it to make sense? -- Stfg ( talk) 13:06, 26 May 2015 (UTC)
Hi
I nominated this page for good article ( /info/en/?search=Wikipedia:Good_article_nominations ) As part of this i moved the complete talk page as now to /Archive 1 Later I will remove sections I find "out of date" "irrelevant"or "solved problems" and shorten discussions still old but still relevant. WillemienH ( talk) 07:42, 26 June 2015 (UTC)
Removed some sections of this page as described above, they can still be found at /Archive 1 WillemienH ( talk) 08:28, 26 June 2015 (UTC)
I am not sure about where the GA discussion should be hold, I guess we could keep it here. Or when you want to staer a reviwe of it on a dedicated review page (I am not sure how this works yet) About the archiving, I did remove the bits of this page that were (according to me) not any longer relevant. Bits of which i was not sure of the (present) relevancy I kept here. (but there are copies of them in the archive) WillemienH ( talk) 20:11, 17 July 2015 (UTC)