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WikiProject Mathematics (Rated C-class, High-priority)
WikiProject Mathematics
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 Field:  Geometry


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)


The projection 3-sphere into ordinary 3-dimensional space. Hopf coordinates
The projection 3-sphere into 3d space. Hopf coordinates
Visualization of projection 1
Visualization of projection 2
Visualization of projection 3
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)

I didn't have a time to make a nice pictures of 3-d sphere, I'm sorry. But I have a lot of old ones. These are links to some of them:
You are welcome to use any of these images. They are not perfect and I'll be thankful for suggestions.--TxAlien 18:48, 6 November 2006 (UTC)

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 (I mention this page there too). I discovered also a new, telling video of the 3-sphere itself, here : . Generally I think Wikipedia should take advantage of nice visual material available on the Web.-- (talk) 12:29, 27 June 2016 (UTC)


Hi, I'm not sure if I should upload this directly to the page, so I figured I'd put it here first. I'm fairly sure I've made a successful cross section of a 3-sphere by using the basic idea behind a tesseract as a guide. I believe it's quite straightforward and understandable. Please click on the images for a description. 3sphere1of16.jpg

I have also tesselated the right side image 16 times to create a full 3-sphere:


P.S: This is my first major contribution; I am a new user and not sure how the editing aspect of this site works just yet. If one of those images is acceptable for posting on the page, I would appreciate it if someone would do it. Feel free to edit the descriptions as well.

-- —The preceding unsigned comment was added by TaranVH (talkcontribs) 02:44, 31 October 2006 (UTC)

Well, it's a very nice diagram, but I'm not sure it's a useful addition to the article. All you are really showing here is the intersection of the 3-sphere with 6 coordinate planes, projected in some weird fashion. You're visualizing something here, but it's a stretch to call it the 3-sphere.
In general, I think attempts to add visualizations of the 3-sphere to this article are a lost cause. The only pictures I ever find useful are to just picture the 2-sphere and use analogies, or to picture the one-point compactification of R3. Neither of these makes for a good diagram to add to the article. -- Fropuff 04:54, 31 October 2006 (UTC)
Actually, the "weird fashion" of projection is the exact same fashion used to project a 3D cube to make a 4D tesseract, as you can see here:
Hypercube diagram.svg
I also think the use of planes was nessesary becuase without them, it would just look like a 2-sphere. (likewise a hypercube would just look like a cube)
I also have to disagree when you say a visualization is a lost cause. You could, for example, read thousands of words explaining rainbows, but you'll never know what one truly is without seeing it. As long as the picture is explained so that you know what you're looking at, it should be fine. (Speaking of which, I think my descriptions of my images are somewhat lacking in this respect.)
TaranVH 01:12, 10 November 2006 (UTC)
You are correct that the actual projection of the 3-sphere would look just like a 2-sphere. That is my point — you are not gaining any useful information by such a projection. The projection is useful for the tesseract because it allows you to see the cell structure, edges, vertices, etc.
If there were a nice visualization for the 3-sphere, I would be all in favor of adding it to the article. I just can't think of any. Granted, that may just be a lack of creativity on my part. -- Fropuff 05:04, 10 November 2006 (UTC)
Following your logic, there is no point in showing a picture of a 2-sphere, as seen on the article about spheres:
Can you honestly say you are gaining no useful information about spheres by looking at that?
The picture was made with a wireframe, which is also sort of like a lot of planes intersecting at the top and bottom points of the sphere. You see? There are no edges or verticies on a sphere, but using a wireframe (or intersecting planes) is the best, if not the only, way to actually show it.
My "wireframe" is far too simple, I admit, mainly becuase I did not use 3D software to make the image. I used basic geometry and four axes. It's not even a wireframe. Below is a picture of a 2-sphere using the same method I used to make the 3-sphere.
As you can see, it's quite inferior to the first sphere. (But it still kind of works.)
If someone with 3D software made a 3-sphere using a wireframe, but following my technique of using four axes and using the same projection style, perhaps then we would have a picture suitable for the page.
TaranVH 18:25, 10 November 2006 (UTC)
I think you are missing the point of my argument. My point is that your second picture of the 2-sphere shown above is not a very useful one. So why should the analogous picture of the 3-sphere be useful?
If you (or someone) could do a wireframe projection of the 3-sphere as you suggest, I would very much like to see it. -- Fropuff 19:57, 10 November 2006 (UTC)
I made a picture showing the projection of the 3-sphere in 3d, but I'm not sure it's right. You can find it here: Basically is the same thing as dividing a normal sphere into 2d circles glued together, and if you rotate the sphere you rotate all these crossections. At one point, they will all look flat, but since there are an infinity of them, they will still form the shape of a circle to you. When you look at the 4d sphere sideways, all the 3d spheres that form it will look flat, but they will still project a sphere. When you look it from the front, the 3d spheres that are closer to the observer can be seen in the center of the 3d sphere being projected, and the largest 3d sphere (of the 3-sphere) will be the surface of the projected sphere (assuming it's an opaque 3-sphere). I hope that makes sense. Chronometrier (talk) 04:41, 30 December 2007 (UTC)
I do not completely agree with Fropuff where they write above: "In general, I think attempts to add visualizations of the 3-sphere to this article are a lost cause."
This may be true most of the time, but by no means always. If someone who is both an expert on the 3-sphere and an expert in visualization tries to do this, there may be an excellent chance that the result will be helpful. After all, topologically the 3-sphere is 3-space with only one additional point added.Daqu (talk) 03:46, 17 April 2011 (UTC)

4-sphere images[edit]

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)

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)


I don't understand how the geometric definition differs from the topological one. In fact, the geometric one appears wrong without further explanation. The 3-sphere is normally taken to lie in 4-space, not 3-space, at least, it cannot be embedded in any way in 3-space. Also, how does the geometric structure differ from the topological? ("In the sense of Coxeter" doesn't impart much...are you considering it as a smooth manifold, or Reimannian manifold, or what? What does "geometric structure" mean here?) Revolver 23:21, 11 Jul 2004 (UTC)

I believe he just means what we would call a 2-sphere. It would appear that some people (Coxeter?) use a different notation though I've never come across it personally. I'm thinking about reverting this to how it was. We can add a sentence remarking on different conventions. -- Fropuff 00:14, 2004 Jul 12 (UTC)

I suspect that I am the 'he' referred above?
I disagree with the statement that 'In mathematics, a 3-sphere is...' since this is the topological definition only (with which I am acquainted). Mathematics used to contain both geometry and topology. Geometry uses a different notation - for example, see H. S. M. Coxeter, the 'greatest geometer' of the 20th century - he used 2-sphere for the circle and 3-sphere for the ball. See Eric W. Weisstein. "Hypersphere." From MathWorld--A Wolfram Web Resource. [[1]]. Eric is clear on the distinction between the two terminologies within Mathematics. My edits to this page were intended to make this clear. At the moment, any reader could be forgiven for thinking that Mathematics=Topology and that Geometry doesn't exist. Ian Cairns 00:58, 14 Aug 2004 (UTC)
You haven't given any reference for this supposed use in geometry other than MathWorld, which is not a reliable source. And the MathWorld article you cite doesn't say that Coxeter used the terms 2-sphere and 3-sphere anyway (it says two-dimensional sphere). So you have provided no meaningful evidence for this unusual use of the term 3-sphere. --Zundark 12:33, 14 Aug 2004 (UTC)

I am a little unsure of where you draw the line between topology and geometry. Most consider one to be a branch of the other. They are certainly intimately intertwined. I have never been aware of a "topological" nomenclature for spheres versus a "geometrical" one. Every mathematician I know—geometer, topologist, or otherwise—uses the definition stated here. However, even accepting that there are other systems in common use, I think Wikipedia should stick to one system for consistency. I think it is enough to leave a note mentioning other usages (which I did). -- Fropuff 02:55, 2004 Aug 14 (UTC)

I have removed the note, as it does not appear to be true. --Zundark 12:33, 14 Aug 2004 (UTC)
Revolver writes: "I don't understand how the geometric definition differs from the topological one." Here is a fairly general answer to that kind of question.
A geometrical object is described as an exact shape, either by saying exactly which subset of a larger geometrical object it is, or at least describing what the distance is between any two of its points. Then any geometrical object having an isometry (a bijection that is distance-preserving in both directions) with the originally defined one is considered to be equivalent.
Whereas a topological object may be defined initially the same way, but in this case any topological space that has a homeomorphism (a bijection that is continuous in both directions) is considered to be equivalent. Thus, the surface of a cube is topologically equivalent to a unit-radius 2-sphere.
Note: There are exceptions in more advanced types of geometry, such as conformal geometry. In greater generality there are various types of transformations that define when two objects of a certain type are "equivalent", and from this viewpoint even topology is considered to be one type of geometry. (See Erlangen program for more info on this.)
∗∗ Note2: H.S.M. Coxeter had a very long mathematical career, and lived to age 95. Toward the beginning of his career some mathematicians, including him, were indeed using "2-sphere" to mean a circle. But eventually he changed with the times and his terminology conformed to modern practice. Which, incidentally has become uniform across all fields of math and even physics.Daqu (talk) 03:04, 26 July 2011 (UTC)

any chance of a picture? -- ssam 21:34, 27 jan 2005 (utc)

I am keen to read Baxter's Dante and the 3-Sphere, but can't find any information about it at all. How, where and when was it published? Latch 00:30, 22 May 2006 (UTC)

I'm glad that someone (revision as of 23:59, 22 August 2006 likes my images and gifs files. Actually, I'm working on few others and I'm not sure that all the images should be placed in this article. Links to the images could be enough? TxAlien 05:50, 27 August 2006 (UTC)

n-sphere in n-space?[edit]

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)


3-sphere = sphere, 4-sphere = glome

  1. MathWorld Hypersphere - The -hypersphere (often simply called the -sphere) is a generalization of the circle (called by geometers the 2-sphere) and usual sphere (called by geometers the 3-sphere) to dimensions.
    • MathWorld Glome - A glome is a 4-sphere (in the geometer's sense of the word) The term derives from the Latin 'glomus' meaning 'ball of string.'
  2. Volume of the n-sphere We use the geometers’ nomenclature for n-sphere, n referring to the number of the underlying dimension [3].
  3. Glossary - Sphere - A 3-sphere
  4. Glossary; Glome Glome: A four-dimensional hypersphere

3-sphere = hypersphere in 4-space

  1. Overview of the Sphere The n-dimensional sphere is the set of all points in space at a given radius from a center point.

So perhaps geometers use n-sphere in n-space, and topologists use n-sphere in (n+1)-space? Tom Ruen 03:02, 6 October 2006 (UTC)

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)

There is alternate, and far less common,...

I'm content, but funny how a FAR LESS COMMON scheme is the primary one referenced on the web. Seems like what's considered common is whose talking. How about some references! Tom Ruen 05:15, 6 October 2006 (UTC)

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)
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)
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)
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)


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)

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)
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)
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)


In the article there are the volume and the hypervolume , but, ¿a n-sphere hasn't got an area? --Daniel bg 16:09, 26 January 2007 (UTC)

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)
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., . Disclaimer: I haven't explicitly researched this, this should only serve to address the initial query. — Preceding unsigned comment added by Tjips (talkcontribs) 12:07, 10 October 2012 (UTC)

Stephen Baxter reference.[edit]

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 [2], in which the story is named Dante Dreams, published in 1998 (later published in Phase Space (book), according to [3]) and he was aware of Peterson's article, as opposed to the other way around or something. Akriasas 22:04, 20 March 2007 (UTC)

I just added a reference to Baxter's short story Shell. The story basically concerns a group of people who are trapped in (i think) a hypersphere-type world. From their perspective, they're on a normal world that has another world on the sky (i.e. it looks like you're standing on a sphere and there's shell enclosing the sky, with another apparent world on the inside of it, upside down). However, in the course of the story, one character manages to get to shell only to realize that it's part of the same world: The sphere that they left appears to curve into a shell, and the shell they were traveling to turns into a sphere.

Like I said, I could be wrong, but I'm pretty sure in the story someone mentions that the world is a "hypersphere", which, if a "hypercube" is a four dimensional cube, that would make a hypersphere a four dimensional sphere, a 3sphere, right? Anyone smarter than me want to verify/correct me? Would this count as a 3sphere reference? Shnakepup (talk) 18:32, 10 January 2008 (UTC)

At the penalty of sounding stupid[edit]

What does all this mean? I don't understand this at all how does a set of points equidistant from a fixed central point in space-time end up looking like that weird inside out twisted, and warped 3-sphere? It looks all convoluted and stuff? AVKent882 (talk) 04:28, 22 June 2010 (UTC)

The picture is rather cryptic. Perhaps some of the Jenn3d projections of polychora (treated as tilings of the 3sphere) would make the point more clearly. —Tamfang (talk) 19:26, 22 June 2010 (UTC)
Yep, very confusing because its a perspective projection (Stereographic projection), so a sphere will also look strange when projected onto a plane, like earth maps [4]. There ought to be a visual comparison like this to help explain why it looks so distorted. Tom Ruen (talk) 23:18, 22 June 2010 (UTC)
The problem (in this one's humble opinion) is not so much that it's distorted as that there's nothing familiar in the undistorted version. You can look at a series of map projections and say aha, Australia is stretched like this and South America is blown up like that ... but a mess of randomly-selected circles looks pretty much like a mess in any projection! It helps that the circles belong to orthogonal families; it would help more if they belonged to patterns within those families. (Or do they?) —Tamfang (talk) 04:56, 23 June 2010 (UTC)
Well, I think people just need to get used to the fact that higher-dimensional structures are hard to visualize. The best anyone can do is a projection. If you still find it difficult, guess what: doesn't matter. It is an abstract concept. Imagining certain vector spaces is difficult as well. And from what I understand, in the context of the article it is a subset of Euclidean-space, spacetime is a different beast. But other than that, it seems you have grasped the most important feature: the definition. In my opinion, you already understand it as much as you need to. Unless you're a mathematician or a physicist (and physicists probably don't need to either), you probably don't need to know anything more about it than that. And if you do, then Wikipedia is definitely not the best place to learn more about math. My honest advice is to buy a book and read it; if you are not willing to do that, then you are probably just looking for pretty pictures.-- (talk) 13:59, 8 January 2011 (UTC)
Many years ago the way to view the three-sphere was through versors. This approach is not a projective one. Rather, the elements of the three-sphere correspond to directed great circle arcs. The model is richer than mere geometry as quaternion multiplication is represented by laying these arcs head to tail to form a third arc. See references in that article.Rgdboer (talk) 19:28, 8 January 2011 (UTC)
I don't mess around with quaternions much TBH, but wouldn't that multiplication only work for unit quaternions? Maybe that is what you meant anyway, since we are talking about 3 spheres.-- (talk) 14:08, 7 May 2011 (UTC)
One of the coolest thing about the 3-sphere is that there are so many different ways to visualize it that are all valid. Eventually, the more of these ways you become familiar with, the better you will understand this shape.
I'd say that without a doubt the simplest way of all is just to imagine 3-space with one point added at "infinity". This makes the most sense when you've understood what it means to say the circle is just the line with a point at infinity, and the ordinary sphere is just the plane with a point at infinity. This helps understand the 3-sphere's topology, but not so much its geometry.Daqu (talk) 03:51, 20 June 2011 (UTC)

Hyperspherical coordinates: accidental commas?[edit]

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)

Bogus commas removed. Cloudswrest (talk) 16:50, 20 September 2010 (UTC)


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 (talk) 21:13, 8 December 2011 (UTC)

spheres to spheres[edit]

(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)


Checking out this URL:

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)

Would this equation be easier?[edit]

I found a simpler equation to make a 3-sphere: 4√(x^2)+(y^2)+(z^2)+(w^2) = 2√r where w is the 4th dimention and r is the 3-sphere's radius. (talk) 08:47, 8 April 2014 (UTC)

Thank you Melbourne for your idea. But the square roots are unnecessary and you have an extra factor of 2 on the left. The section 3-sphere#Definition has four equations, three centered at the origin as your suggestion is. Preferable is the first definition given:

which has an arbitrary center C and points x on the 3-sphere.Rgdboer (talk) 21:08, 8 April 2014 (UTC)

notational clean-up[edit]

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) — Preceding unsigned comment added by 'Ff'lo (talkcontribs)