Talk:Horosphere

A picture ...
... would be nice... Oleg Alexandrov (talk) 03:36, 30 September 2007 (UTC)

Attribution
The article says:


 * This terminology is due to William Thurston,

But the term appears in H.S.M. Coxeter's 1948 book Regular Polytopes, and Thurston was then only two years old, so I don't think this could be correct.

Google Book Search finds citations for this term back to 1931.

-- Dominus 07:11, 1 October 2007 (UTC)


 * Does Coxeter consider hyperbolic geometry in that book? I don't think so (could be wrong), so it can't be the same usage.  As for the Google Books search, I think it is irrelevant as the dates don't seem to be from the handful of papers it lists.  The hits are using dates that aren't based on the dates of the papers it finds.  For example,  the 1931 date you cite refers to the beginning of Zentralblatt, but the entry you refer to is from v. 839 which dates from 1996.  Note the reviewer appears to be Gabor Toth, who is miraculously still alive and reviewing papers even today.  Also, ironically, the fourth hit (seemingly from 1953) in your search mentions Thurston right above the mention of horoball.  That 1953 seems to be the founding date of the journal in question.  Given this evidence of a hastily done search, I remain skeptical.  Sorry.  I'm going to change it back unless you can come up with something substantive.    --Horoball 08:13, 1 October 2007 (UTC)


 * By the way, why did you remove the other stuff anyway?  Most of the hits you get, if you search Google Scholar, for "horoball" will be for hyperbolic 3-manifolds, and Thurston did popularize the term.  --Horoball 08:23, 1 October 2007 (UTC)


 * I don't think the 1931 reference can be correct since the relevant citation is to a German book---in German the prefix is not "horo" but "grenz", as in Grenzkreis, Grenzkugel, Grenzgebiete. The book's translation using "horoball" is 1970.  On the other hand Coxeter's 1961 Intro to Geometry for sure (I'm looking at a copy) uses "horosphere"---Thurston was only 15 then.  However all that is beside the main point, which is that the term goes at least as far back as Lobachevsky, who wrote about hyperbolic geometry in both German and French.  In German he used the German words of course, but in French he rendered them as horicycle and horisphere.  I don't know whether Lobachevsky invented the French terms though.  The concept itself seems to go as far back as Gauss's student Friedrich Wachter, who, according to Roberto Bonola in 1906, pointed out to Gauss in a letter in 1816 that Euclidean geometry could be recreated in hyperbolic geometry by letting a sphere grow to infinity.  (Wachter died the following year aged only 25 and so never saw his advisor's hyperbolic geometry exposed to the public, and surely did not anticipate the credit going to anyone other than Gauss.)  Incidentally Gauss had like many others come across hyperbolic geometry while trying to prove Euclid's fifth postulate, but like the rest did not "believe in" it for many years; it was not until 1813, judging from his correspondence, that his attitude changed and he came to accept hyperbolic geometry as something real.  Yet he sat on it for another two decades until the younger Bolyai published his proof (which he too had sat on for more than a decade).  --Vaughan Pratt 22:41, 25 October 2007 (UTC)


 * I didn't know Lobachevsky had used "horisphere" before, but the article is about "horoball" strictly speaking. There are probably good reasons "horoball" doesn't appear until more recent literature.  I disagree with your comment about merging, by the way, although at the moment it doesn't much matter.  It makes about as much sense as merging the topic of Euclidean space into Euclidean plane.  While a layman may be more familiar with the 2-dimensional case, a hyperbolic geometer would be surprised to be redirected toward it.  I'm not sure why you list Google search hits anyway.  that would be like arguing that "manifold" should be merged into "surface" because "surface" has many more hits than "manifold".   Horocycle, or more appropriately horodisc, is only for two dimensional hyperbolic geometry and most interesting theorems about horoballs don't apply there.  So it makes little sense from a mathematical viewpoint to include the more interesting case into the less interesting one, don't you think?   Rather than blindly use G-hits, I recommend you take a look at the results that appear when you search for these terms.  Then decide if it is doing a reader a service when he or she searches for "horoball" and is directed toward "horocycle".  As I wrote in the article, there is an expectation of higher dimensional (usually three) geometry in the term.   --Horoball 08:25, 1 November 2007 (UTC)

approximation by hypercycles

 * From the convex side the horocycle is approximated by hypercycles whose distances go towards infinity.

I'm not sure I'm picturing this properly. (The following sentence quoted from Coxeter doesn't illuminate much.) Here's what I'm thinking. Start with a line L which penetrates the intended horocycle, and an equidistant to that line which is tangent to the horocycle at P. Consider the line M which contains P and is perpendicular to L.  Now move L away from P, keeping it perpendicular to M, and increase the distance of the equidistant so that it continues to pass through P.

If this is indeed what's meant, can it be made a bit more explicit? Perhaps with an animation? —Tamfang (talk) 03:30, 8 July 2010 (UTC)

Redundant article
Wikipedia already has a Horocycle article (30,500 google hits), which "horosphere" (19,700 hits) and "horoball" (12,000 hits) should simply be links to and not separate articles, given that none of these articles are more than a couple of paragraphs. Just merge the current material on horospheres and horoballs into the horocycle article, which needs to be boosted up a bit anyway. If the Horocycle article ever gets anywhere near the length of the sphere article it would make sense at that point to separate them out again. --Vaughan Pratt 23:11, 25 October 2007 (UTC)


 * See my comment above. --Horoball 08:26, 1 November 2007 (UTC)Selfstudier (talk) 14:42, 12 February 2012 (UTC)

OK, I merged them, they are same thing, different dimensionSelfstudier (talk) 14:39, 12 February 2012 (UTC)
 * I believe the original suggestion was to create a section of the horocycle article, and move the contents of this article into that section, rather than simply dump the contents of that article here. In any event, the one-dimensional case is studied quite independently of any of the higher-dimensional analogs, similarly to how the hyperbolic plane and hyperbolic space are distinct topics for encyclopedia articles.   Sławomir Biały  (talk) 13:49, 13 February 2012 (UTC)


 * A horocycle is what u get when you slice a horosphere with a plane, alternatively you rotate a horocycle to get a horosphere; both concepts were introduced by Lobachevsky following work by Wachter and are classical predating the planar models of Beltrami and others. I agree that the page now needs work, I am in process of doing that and I believe that when that is done you wouldn't have any complaint.Selfstudier (talk) 15:28, 13 February 2012 (UTC)


 * It seems that people (at least 2) are happy with the pages for horosphere/horocycle so I will leave them in their present form and leave you with a reference Horocycle/sphere)[[User:Selfstudier|Selfstudier] (talk) 15:43, 13 February 2012 (UTC)

confusion sphere/ball
The current version confuses "sphere" and "ball". Tkuvho (talk) 14:35, 13 February 2012 (UTC)

Picture?
The closest example picture I can find is at Apollonian sphere packing. Tom Ruen (talk) 05:27, 6 January 2014 (UTC)

Ridiculous section
The section titled Curvature reads in its entirety as follows:

"A horosphere has a critical amount of (isotropic) curvature: if the curvature were any greater, the surface would be able to close, yielding a sphere, and if the curvature were any less, the surface would be an (N − 1)-dimensional hypercycle."

This makes little or no sense. Especially because the curvature of a horosphere has not been mentioned in the article.

And if it had been, the article would mention that a horosphere is isometric to the Euclidean plane, which is to say that its curvature equals 0 everywhere.

So it could hardly be less clear what is meant by the "amount of curvature" that a horosphere has. 2601:200:C000:1A0:E476:B206:4E5:98D9 (talk) 02:29, 5 September 2021 (UTC)