Talk:3-sphere

message: = a*x^2+b*y^2+z^2=16
?  — Preceding unsigned comment added by 79.72.81.99 (talk) 02:05, 14 November 2017 (UTC)
thread_title:  Comments 

message: I cut the following pictures out of the article.  If you understand them, feel free to put them back into the page, but with a good explanation, please! Sam nead 21:59, 17 September 2006 (UTC)
replies: text: I think that exact formulas for the projections give just a little for explanation of these images. I'll make a few new images of 2 and 3-spheres which, I hope, will be useful for the article and understanding the pictures.--TxAlien 22:13, 19 September 2006 (UTC)
replies: text: These links do not work anymore. As for the projections shown, they concern ((4D-to-3D)-in-2D) renderings. 
Projections 1 and 2 are apparently Clifford toruses rotating in their 3-sphere, and covered with Villarceau circles. 
Projection 3 represents a family of toruses of which the Clifford torus is a special case, and which together fill the embedding 3-sphere. The toruses would obey equations [in a (x,y,u,v) coordinate system] of the form x^2 + y^2 = r^2, and u^2 + v^2 = (2 - r^2). The overall sum = 2, so the toruses reside all in the 3-sphere "r = sqrt(2)". The x,y circles vary from radius 0 to sqrt(2), when the u,v circles do so from sqrt(2) to 0. They meet "half way" with r = 1, the Clifford torus. The animation "runs through" the family with r varying from 0 through 1 to sqrt(2) and back again, the successive toruses visit the entire 3-sphere in doing so. The toruses are covered not with Villarceau circles here, but with ordinary, poloidal and toroidal (?) circles that represent the radius variation.
I obtained the same pictures, see Youtube videos mentioned on my website  http://home.scarlet.be/wugi/qbComplex.html (I mention this page there too).
I discovered also a new, telling video of the 3-sphere itself, here : https://www.youtube.com/watch?v=dy_MUfBuq2I .
Generally I think Wikipedia should take advantage of nice visual material available on the Web.-- 83.134.156.161 (talk) 12:29, 27 June 2016 (UTC)
thread_title: Visualizations

message: ::I think that exact formulas for the projections give just a little for explanation of these images. I'll make a few new images of 2 and 3-spheres which, I hope, will be useful for the article and understanding the pictures.--TxAlien 22:13, 19 September 2006 (UTC)
replies: text: These links do not work anymore. As for the projections shown, they concern ((4D-to-3D)-in-2D) renderings. 
Projections 1 and 2 are apparently Clifford toruses rotating in their 3-sphere, and covered with Villarceau circles. 
Projection 3 represents a family of toruses of which the Clifford torus is a special case, and which together fill the embedding 3-sphere. The toruses would obey equations [in a (x,y,u,v) coordinate system] of the form x^2 + y^2 = r^2, and u^2 + v^2 = (2 - r^2). The overall sum = 2, so the toruses reside all in the 3-sphere "r = sqrt(2)". The x,y circles vary from radius 0 to sqrt(2), when the u,v circles do so from sqrt(2) to 0. They meet "half way" with r = 1, the Clifford torus. The animation "runs through" the family with r varying from 0 through 1 to sqrt(2) and back again, the successive toruses visit the entire 3-sphere in doing so. The toruses are covered not with Villarceau circles here, but with ordinary, poloidal and toroidal (?) circles that represent the radius variation.
I obtained the same pictures, see Youtube videos mentioned on my website  http://home.scarlet.be/wugi/qbComplex.html (I mention this page there too).
I discovered also a new, telling video of the 3-sphere itself, here : https://www.youtube.com/watch?v=dy_MUfBuq2I .
Generally I think Wikipedia should take advantage of nice visual material available on the Web.-- 83.134.156.161 (talk) 12:29, 27 June 2016 (UTC)
thread_title:  Visualizations 

message: I am no mathematician, and this is a straight-forward question having recently come across this article: The images futher down the article say they are of a 4-sphere. Is it right to have 4-sphere images in the 3-sphere article? - CharlesC 22:16, 31 August 2006 (UTC)
replies: text: I cut those out of the article, because I didn't understand them, and because they were not explained. Sam nead 22:01, 17 September 2006 (UTC)
thread_title: 4-sphere images

message: The definitions in hypersphere imply an n-sphere is in n-space. So a 3-sphere ought to be a "normal" sphere, and this article then defines a 4-sphere.
I suspect there's different definitions from different sources. Anyone want to help straighten this inconsistency out? Tom Ruen 02:46, 6 October 2006 (UTC)
replies: text: In rereading hypersphere also considered an n-sphere in (n+1)-space. I rewrote the intro there for clarity. I'd still like some more clarity here on ambiguity, but I'll leave for now. Tom Ruen 04:00, 6 October 2006 (UTC)
replies: text: I challenge you to find a mathematics textbook that uses the altenate indexing. I'm aware of only one (at it was written in the 1940's; subsequent works by the same author use the standard indexing system). As stated elsewhere on this page and at Talk:Hypersphere, MathWorld is an unreliable reference. It uses funny (and sometimes just plain wrong) definitions all over the place. Unfortunatly, many so-called web references derive from it. I'm tempted to delete the external link to MathWorld altogether. -- Fropuff 05:25, 6 October 2006 (UTC)

text: The MathWorld assertion about the difference in terminology between geometers and topologists is just wrong.  It mixes up a now archaic usage by some classical geometers with modern terminology.  Those kinds of links isn't really going to convince an actual geometer or topologist (of which I interact with a good number) that the MathWorld terminology is indeed common.  You can look in any serious mathematical book (say a Springer GTM, for example) and see that regardless of field of specialty, an n-sphere really is an n-manifold.  --C S (Talk) 19:02, 18 November 2006 (UTC)

text: There's no doubt that an n-sphere is an n-dimensional manifold in at least 99% of all math writing.  And that Mathworld is often wrong. Zaslav 04:09, 20 November 2006 (UTC)
replies: text: As a mathematician I totally concur with what C S wrote: that among professional mathematicians, there is complete consistency -- at least nowadays -- in the way that the dimension of a manifold (and a sphere in particular) is referred to.  Geometers and topologists use exactly the same convention: If the manifold is locally homeomorphic to an open set of n-dimensional Euclidean space, then it is an n-manifold.  Same for the sphere: A 1-sphere is topologically a circle; a 2-sphere is the usual sphere; a 3-sphere is the "hypersphere" that this article is about, etc.  (I.e., the n-sphere is homeomorphic to the set of points at a distance of exactly 1 from the origin in (n+1)-dimensional Euclidean space.)Daqu (talk) 03:58, 17 April 2011 (UTC)
thread_title:  n-sphere in n-space? 

message: I vote to remove "glome" from Wikipedia.  It seems to exist nowhere but in Mathworld (which is unreliable) and this Wikipedia article.  Am I correct, or not? Zaslav 04:14, 20 November 2006 (UTC)
replies: text: I believe the term "glome" is due to George Olshevsky. I see no reason it should be in Wikipedia. -- Fropuff 05:56, 20 November 2006 (UTC)
replies: text: I removed the three uses of "glome" (actually, "hemiglome") from this article. The obscure word made that paragraph harder to understand for no good reason. MrRedact 05:10, 20 April 2007 (UTC)
replies: text: I have just removed a sentence claiming that the 3-sphere is also called a "glome".  I am so weary of people inserting their own private coinages into Wikipedia as a way of inflating their egos.  The word does not occur even once in the MathSciNet database.Daqu (talk) 03:34, 17 April 2011 (UTC)
thread_title: Glome

message: In the article there are the volume 2 \pi^2 r^3 \,and the hypervolume \frac {1}{2} \pi^2 r^4 \,, but, ¿a n-sphere hasn't got an area? --Daniel bg 16:09, 26 January 2007 (UTC)
replies: text: Well no, that's one reason it's so hard to visualize. Think of a normal sphere as a polyhedron with the edges rounded away, ie. a single curved, edgeless surface: a 3-sphere, analogously, would be a polychoron with all the faces rounded away, ie. a single curved cell. It doesn't have an area any more than a normal sphere has an edge length… --Tropylium (talk) 16:45, 20 February 2008 (UTC)
replies: text: Another way of explaining why this is the case is observing that something similar happens when considering normal spheres and "circumference". A circle has both a circumference and surface area, but a sphere does not. Of the two it only has a surface area. Thus, in going to the higher dimensional case (the sphere in this case) one of the quantities was lost. This isn't the end of the story though. On a sphere we can define a great circle, which does have a circumference. Maybe it would be useful to discuss the analogue of the great circle hear, or the "great sphere". Just as a sphere can be intersected by a plane containing the central point (thus finding a great circle), so too a 3-sphere can be intersected by a 3 dimensional space containing the central point. In the latter case the sphere obtained would be the one with the maximal possible surface area. We can then of course find the great circle of that sphere and have the circle with maximal circumference obtainable by intersection from our original 3-sphere. So I guess a 3-sphere does have something akin to a surface area, in the same way as a sphere has something akin to a circumference. The expression for it, of course is just the expression for the surface area of a sphere with the same radius as our 3-sphere, i.e., 4 \pi r^2. Disclaimer: I haven't explicitly researched this, this should only serve to address the initial query.  — Preceding unsigned comment added by Tjips (talk • contribs) 12:07, 10 October 2012 (UTC)
thread_title:  Area 

message: Anyone who can actually read Spanish is welcome to correct me on this, but I've changed the section on this page refering to Baxter per , in which the story is named Dante Dreams, published in 1998 (later published in Phase Space (book), according to ) and he was aware of Peterson's article, as opposed to the other way around or something. Akriasas 22:04, 20 March 2007 (UTC)
thread_title:  [[Stephen Baxter]] reference. 

message: In the first set of equations of this section, those defining hyperspherical coordinates in terms of a transformation to Cartesian coordinates, commas are placed between each pair of trigonometric functions. Is this a mistake? The section of hypersphere linked to from here has no such commas. It just juxtaposes the sines and cosines, indicating multiplication as one would expect. Dependent Variable (talk) 21:27, 19 September 2010 (UTC)
thread_title:  Hyperspherical coordinates: accidental commas? 

message: In the section for gluing, what is a 3-ball?  The term is not mentioned anywhere else in the article, and I can't find it referenced anywhere on the web.  — Preceding unsigned comment added by 68.2.132.51 (talk) 21:13, 8 December 2011 (UTC)
thread_title:  3-ball 

message: : (Notice that, since stereographic projection is conformal, round spheres are sent to round spheres or to planes.) 
Conformality (local preservation of shape) isn't sufficient; the Mercator projection is conformal, but it doesn't take circles to circles. —Tamfang (talk) 05:13, 16 February 2013 (UTC)
thread_title:  spheres to spheres 

message: Checking out this URL:
http://mathforum.org/library/drmath/view/54696.html
Go to the section written by someone named "Dr. James". Looking at the first 3 paragraphs of that section, I think similar statements can be made about the "3-sphere", but the only problem is I don't know what terminology to use for higher dimensions. For clarification:
The first paragraph need not be re-worded.
The second paragraph mentions the phrase "length, width, and height". What phrase is the analog for talking about the "3-sphere"??
The third paragraph mentions the phrase "it has no volume". What phrase is the analog for talking about the "3-sphere"??
Georgia guy (talk) 14:45, 18 March 2013 (UTC)
thread_title:  Terminology 

message: Looks as if we could use some font consistency help with some "S"es and "H"es and at least one "R" in the notation here, in "Definition" and (in particular) "Topological properties", at least.  I'm not sufficiently confident of my understanding to take care of it myself.
Lisa "Buffalo" Nichols 20:14, 17 May 2016 (UTC)
thread_title:  notational clean-up 

message: I'm wondering (for my own mental clarity) if it would be accurate to call 3-sphere#Hopf_coordinates 3 angles as latitude, longitude, and colongitude, since the last two are 0 to 360° range, although the first angle is only 0 to 90° rather than ±90° of real latitude. 
But I suppose more clearly 3-sphere#Hyperspherical_coordinates are like longitude, and two orthogonal latitude angles (in ±90°) off an equator. Tom Ruen (talk) 20:38, 17 September 2018 (UTC)
replies: text: The way I like to think of the coordinates.   Start with a great circle that is the center of a torus.  The first "0 to 90°" coordinate η maps to the minor radius of the torus, call it "r".  The next coordinate ξ1 maps to the angle of this radius vector pivoting around the original great circle. The tip of the minor radius vector now points to a new circle and the angle coordinate around this circle is ξ2. I guess those names could work, especially the last two as they are dual, although they are allusions to spherical coordinates rather than toroidal coordinates. Cloudswrest (talk) 20:02, 18 September 2018 (UTC)
thread_title:  Hopf coordinates 

message: The "See also" section links to rotation group SO(3) and to charts on SO(3). This is confusing. The basic source of the confusion is that SO(3) is the rotational symmetry group of the 2-sphere, SO(4) is the rotational symmetry group of the 3-sphere, etc. Having those links here, when they'd be more appropriate in sphere, exacerbates the confusion. I propose removing those links. Maproom (talk) 16:06, 14 September 2022 (UTC)
replies: text: Maproom (talk) 07:29, 1 March 2023 (UTC)
thread_title:  [[3-sphere#See_also|See also]] section