Talk:3-sphere - Knowledge

1288:

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.,

601:. 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. 1515:

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 Knowledge 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.--

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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

95: 85: 64: 31: 198:& output answers, because honestly, rationalising spheres is almost a punishment because its impossible, so, take pi & work from its latest revision, and in maths do not study spheres nor circles, unless you want to lose hope, please only invest time in absolution if you wish to study this field, squares are math and this is art, so please contribute efforts in changing the equations from images to symbols, to save our contributed time. 441: 22: 1068:

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.)

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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

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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

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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

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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.

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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,

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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

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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

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please may I request the images to be made into symbols? If these maths are applied in actual format, logically the computer will resolve the calculation? if the symbol contains the information ;] - the laborious task may be worth it - if the computer will understand the equations like a calculator

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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

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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

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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

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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,

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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

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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

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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

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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

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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 of the

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___ SECONDED! you wrote that in 2010 & it stands the same in 2017, it is entirely art or way above my understanding but maybe, this is just wrong & i believe that is so, this implies a hollow sphere in 3d spacetime from a 4d perspective, cryptic would be an understatement about most the

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JUST PLEASE SOMEONE CHANGE THE EQUATIONS FROM IMAGES TO SYMBOLS - ill do some when I get time, but in what format? unicode? utf-8? hmmm, just think though, when we can not copy & paste the equations as text we can not find out more of each individual symbol & therefore i believe these

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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.

1537:. 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. 513:

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

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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

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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,

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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

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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.

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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.

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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:

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topologist, or otherwise—uses the definition stated here. However, even accepting that there are other systems in common use, I think Knowledge should stick to one system for consistency. I think it is enough to leave a note mentioning other usages (which I did). --

709:]. 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. 702:

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

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I'm glad that someone (revision as of 23:59, 22 August 2006 84.145.108.141) 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?

1014:, 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. -- 1009:

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

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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?

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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

2142:. 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 414:

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

1496:... 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?) — 1028:

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. --

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It's because of the second image. But it only happens at some window sizes and scalings. The image could be moved to the next section, but then it might mess up something else - it's hard to get everything to look good at all sizes.

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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? -

691:. 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. -- 1264:

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

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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 Knowledge as a way of inflating their egos. The word does not occur

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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

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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

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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.

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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?

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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

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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. --

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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

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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.--

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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.

2243: 707:, 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. 201:

and precision in this is to get pi to the precision you need it for, because infinite precisions of pi is way ahead of applied engineering so why study or invest time in this field?

1389:. 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 1203: 1319: 2071:

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.

2233: 2111:. 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. 2248: 978:

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.

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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.

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I vote to remove "glome" from Knowledge. It seems to exist nowhere but in Mathworld (which is unreliable) and this Knowledge article. Am I correct, or not?

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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.

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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. —

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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.

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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!

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I think this article needs major revision from the get go, because as it stands it is amongst 1,2,3 & 4 sphere & makes no sense at all.

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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.

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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

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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

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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:

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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

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The second paragraph mentions the phrase "length, width, and height". What phrase is the analog for talking about the "3-sphere"??

904:- 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.' 2223: 1997:{\displaystyle \sum _{i=0}^{3}(x_{i}-C_{i})^{2}=(x_{0}-C_{0})^{2}+(x_{1}-C_{1})^{2}+(x_{2}-C_{2})^{2}+(x_{3}-C_{3})^{2}=r^{2}.} 623:

where they write above: "In general, I think attempts to add visualizations of the 3-sphere to this article are a lost cause."

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I'm content, but funny how a FAR LESS COMMON scheme is the primary one referenced on the web. Seems like what's considered

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The third paragraph mentions the phrase "it has no volume". What phrase is the analog for talking about the "3-sphere"??

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If you (or someone) could do a wireframe projection of the 3-sphere as you suggest, I would very much like to see it. --

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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.

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I suspect there's different definitions from different sources. Anyone want to help straighten this inconsistency out?

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The problem (in this one's humble opinion) is not so much that it's distorted as that there's nothing familiar in the

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I don't understand how the geometric definition differs from the topological one. In fact, the geometric one appears

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rounded away, ie. a single curved cell. It doesn't have an area any more than a normal sphere has an edge length… --

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I also think the use of planes was nessesary becuase without them, it would just look like a 2-sphere. (likewise a

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please put these images back in, I agree that they highlight more understanding than the ones currently there - PDG

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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:

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has four equations, three centered at the origin as your suggestion is. Preferable is the first definition given:

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Following your logic, there is no point in showing a picture of a 2-sphere, as seen on the article about spheres:

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Actually, the "weird fashion" of projection is the exact same fashion used to project a 3D cube to make a 4D

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These links do not work anymore. As for the projections shown, they concern ((4D-to-3D)-in-2D) renderings.

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distorted version. You can look at a series of map projections and say aha, Australia is stretched like

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We use the geometers’ nomenclature for n-sphere, n referring to the number of the underlying dimension .

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You are welcome to use any of these images. They are not perfect and I'll be thankful for suggestions.--

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I cut those out of the article, because I didn't understand them, and because they were not explained.

338:(I mention this page there too). I discovered also a new, telling video of the 3-sphere itself, here : 94: 1101: 2135: 2042: 1635: 1325: 1166: 540: 179: 1419: 1399: 342:. Generally I think Knowledge should take advantage of nice visual material available on the Web.-- 21: 1674: 1423: 1403: 1274: 304: 300: 567: 463: 392: 378:

hmm, not sure what is happening here? is this image on the talk page because it is incorrect? PDG

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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

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left appears to curve into a shell, and the shell they were traveling to turns into a sphere.

1291: 1050:-dimensional manifold in at least 99% of all math writing. And that Mathworld is often wrong. 460:(Speaking of which, I think my descriptions of my images are somewhat lacking in this respect.) 2204: 2175: 2151: 2083: 2020: 1682: 1542: 1501: 1471: 1444: 396: 1466:. There ought to be a visual comparison like this to help explain why it looks so distorted. 1435:

equations are un-moderated & should be removed or replaced with symbols of the same math

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Can you honestly say you are gaining no useful information about spheres by looking at that?

1373:) and he was aware of Peterson's article, as opposed to the other way around or something. 2139: 1385: 1333: 946: 835:, but can't find any information about it at all. How, where and when was it published? 598: 288: 308: 1583: 1352: 1270: 1255: 1144: 1073: 815: 680:

manifold, or Reimannian manifold, or what? What does "geometric structure" mean here?)

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infinity. This helps understand the 3-sphere's topology, but not so much its geometry.

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As you can see, it's quite inferior to the first sphere. (But it still kind of works.)

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When it says such that, there is a big gap. Can someone fix this? If so, thank you!

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are like longitude, and two orthogonal latitude angles (in ±90°) off an equator.

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juxtaposes the sines and cosines, indicating multiplication as one would expect.

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

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pictures, but no they stay because we have no idea what to replace it with.

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incidentally has become uniform across all fields of math and even physics.

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directions) with the originally defined one is considered to be equivalent.

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The projection 3-sphere into ordinary 3-dimensional space. Hopf coordinates

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I'm wondering (for my own mental clarity) if it would be accurate to call

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maps to the minor radius of the torus, call it "r". The next coordinate

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will also look strange when projected onto a plane, like earth maps

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http://img133.imageshack.us/img133/1369/graphicsnsph4p2sslx2.gif

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http://img160.imageshack.us/img160/9052/graphicsnsph4p2sgyl8.gif

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http://img145.imageshack.us/img145/2584/graphicsnsph4p5gwl6.gif

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I have removed the note, as it does not appear to be true. --

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Many years ago the way to view the three-sphere was through

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

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Yep, very confusing because its a perspective projection (

1664:, round spheres are sent to round spheres or to planes.) 1677:

is conformal, but it doesn't take circles to circles. —

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The projection 3-sphere into 3d space. Hopf coordinates

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http://img261.imageshack.us/img261/3038/3spherefw2.jpg

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http://img148.imageshack.us/img148/8538/sph04smac7.gif

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http://img522.imageshack.us/img522/4195/sph06svg6.gif

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http://img143.imageshack.us/img143/610/sph03sfw5.gif

112:, a collaborative effort to improve the coverage of 1700:

http://mathforum.org/library/drmath/view/54696.html

1996: 1313: 1246: 1197: 2038:Lisa "Buffalo" Nichols 20:14, 17 May 2016 (UTC) 1383:I just added a reference to Baxter's short story 1660:(Notice that, since stereographic projection is 1104:. I see no reason it should be in Knowledge. -- 2244:Knowledge level-5 vital articles in Mathematics 687:I believe he just means what we would call a 2- 1594:Hyperspherical coordinates: accidental commas? 1063:As a mathematician I totally concur with what 1673:preservation of shape) isn't sufficient; the 1247:{\displaystyle {\frac {1}{2}}\pi ^{2}r^{4}\,} 997:is whose talking. How about some references! 8: 699:I suspect that I am the 'he' referred above? 1708:The first paragraph need not be re-worded. 989:There is alternate, and far less common,... 340:https://www.youtube.com/watch?v=dy_MUfBuq2I 538: 336:http://home.scarlet.be/wugi/qbComplex.html 177: 58: 1985: 1972: 1962: 1949: 1933: 1923: 1910: 1894: 1884: 1871: 1855: 1845: 1832: 1816: 1806: 1793: 1780: 1769: 1763: 1305: 1293: 1254:, but, ¿a n-sphere hasn't got an area? -- 1236: 1226: 1212: 1210: 1187: 1177: 1168: 1366:, published in 1998 (later published in 2234:Knowledge vital articles in Mathematics 1242: 1193: 60: 19: 2192:2A02:C7C:9413:5F00:9181:193B:37EB:6F04 2249:C-Class vital articles in Mathematics 1100:I believe the term "glome" is due to 930:Glome: A four-dimensional hypersphere 7: 1163:In the article there are the volume 106:This article is within the scope of 2076:3-sphere#Hyperspherical_coordinates 1492:and South America is blown up like 1398:this count as a 3sphere reference? 49:It is of interest to the following 14: 2259:Mid-priority mathematics articles 1413:At the penalty of sounding stupid 126:Knowledge:WikiProject Mathematics 2229:Knowledge level-5 vital articles 2162: 2134:The "See also" section links to 1198:{\displaystyle 2\pi ^{2}r^{3}\,} 527: 497: 439: 373: 363: 129:Template:WikiProject Mathematics 93: 83: 62: 29: 20: 619:I do not completely agree with 403:) 02:44, 31 October 2006 (UTC) 146:This article has been rated as 2239:C-Class level-5 vital articles 2156:16:06, 14 September 2022 (UTC) 2121:20:02, 18 September 2018 (UTC) 2088:20:38, 17 September 2018 (UTC) 2007:which has an arbitrary center 1969: 1942: 1930: 1903: 1891: 1864: 1852: 1825: 1813: 1786: 1731:Would this equation be easier? 1623:16:50, 20 September 2010 (UTC) 1609:21:27, 19 September 2010 (UTC) 1362:, in which the story is named 665:22:01, 17 September 2006 (UTC) 280:22:13, 19 September 2006 (UTC) 217:21:59, 17 September 2006 (UTC) 1: 1687:05:13, 16 February 2013 (UTC) 1279:16:45, 20 February 2008 (UTC) 1109:05:56, 20 November 2006 (UTC) 1095:04:14, 20 November 2006 (UTC) 1055:04:09, 20 November 2006 (UTC) 1036:19:02, 18 November 2006 (UTC) 964:use n-sphere in (n+1)-space? 960:use n-sphere in n-space, and 611:04:41, 30 December 2007 (UTC) 589:19:57, 10 November 2006 (UTC) 571:18:25, 10 November 2006 (UTC) 553:02:10, 14 November 2017 (UTC) 486:05:04, 10 November 2006 (UTC) 467:01:12, 10 November 2006 (UTC) 267:Visualization of projection 3 259:Visualization of projection 2 251:Visualization of projection 1 192:02:05, 14 November 2017 (UTC) 120:and see a list of open tasks. 2254:C-Class mathematics articles 1648:21:13, 8 December 2011 (UTC) 1408:18:32, 10 January 2008 (UTC) 1342:12:07, 10 October 2012 (UTC) 1259:16:09, 26 January 2007 (UTC) 456:would just look like a cube) 424:04:54, 31 October 2006 (UTC) 317:18:48, 6 November 2006 (UTC) 2209:19:12, 1 October 2024 (UTC) 2074:But I suppose more clearly 1547:19:28, 8 January 2011 (UTC) 1525:13:59, 8 January 2011 (UTC) 1139:in the MathSciNet database. 1019:05:25, 6 October 2006 (UTC) 1002:05:15, 6 October 2006 (UTC) 983:04:00, 6 October 2006 (UTC) 969:03:02, 6 October 2006 (UTC) 871:02:46, 6 October 2006 (UTC) 850:05:50, 27 August 2006 (UTC) 831:I am keen to read Baxter's 655:22:16, 31 August 2006 (UTC) 2275: 1726:14:45, 18 March 2013 (UTC) 1378:22:04, 20 March 2007 (UTC) 1314:{\displaystyle 4\pi r^{2}} 1149:03:34, 17 April 2011 (UTC) 1122:05:10, 20 April 2007 (UTC) 1078:03:58, 17 April 2011 (UTC) 639:03:46, 17 April 2011 (UTC) 362:images for a description. 2180:07:29, 1 March 2023 (UTC) 2069:3-sphere#Hopf_coordinates 2025:21:08, 8 April 2014 (UTC) 1747:08:47, 8 April 2014 (UTC) 1588:03:51, 20 June 2011 (UTC) 1506:04:56, 23 June 2010 (UTC) 1476:23:18, 22 June 2010 (UTC) 1449:19:26, 22 June 2010 (UTC) 1428:04:28, 22 June 2010 (UTC) 1042:There's no doubt that an 820:03:04, 26 July 2011 (UTC) 751:02:55, 2004 Aug 14 (UTC) 695:00:14, 2004 Jul 12 (UTC) 684:23:21, 11 Jul 2004 (UTC) 352:12:29, 27 June 2016 (UTC) 145: 78: 57: 1457:Stereographic projection 840:00:30, 22 May 2006 (UTC) 759:12:33, 14 Aug 2004 (UTC) 740:12:33, 14 Aug 2004 (UTC) 713:00:58, 14 Aug 2004 (UTC) 537:- request removal ASAP 152:project's priority scale 2194:16:07, 1 October 2024‎ 1696:Checking out this URL: 1561:14:08, 7 May 2011 (UTC) 800:for more info on this.) 109:WikiProject Mathematics 2224:C-Class vital articles 1998: 1785: 1613:Bogus commas removed. 1315: 1248: 1199: 947:Overview of the Sphere 910:Volume of the n-sphere 833:Dante and the 3-Sphere 730:two-dimensional sphere 436:, as you can see here: 268: 260: 252: 244: 236: 1999: 1765: 1316: 1249: 1200: 893:MathWorld Hypersphere 391:comment was added by 266: 258: 250: 242: 234: 173:= a*x^2+b*y^2+z^2=16 36:level-5 vital article 2136:rotation group SO(3) 1762: 1292: 1209: 1205:and the hypervolume 1167: 855:n-sphere in n-space? 132:mathematics articles 2030:notational clean-up 1754:3-sphere#Definition 1675:Mercator projection 922:Sphere - A 3-sphere 859:The definitions in 1994: 1654:spheres to spheres 1601:Dependent Variable 1368:Phase Space (book) 1311: 1244: 1243: 1195: 1194: 269: 261: 253: 245: 237: 101:Mathematics portal 45:content assessment 2059: 2045:comment added by 1638:comment added by 1345: 1328:comment added by 1220: 705:H. S. M. Coxeter 555: 543:comment added by 404: 194: 182:comment added by 166: 165: 162: 161: 158: 157: 2266: 2170: 2166: 2165: 2063:Hopf coordinates 2058: 2039: 2015:on the 3-sphere. 2003: 2001: 2000: 1995: 1990: 1989: 1977: 1976: 1967: 1966: 1954: 1953: 1938: 1937: 1928: 1927: 1915: 1914: 1899: 1898: 1889: 1888: 1876: 1875: 1860: 1859: 1850: 1849: 1837: 1836: 1821: 1820: 1811: 1810: 1798: 1797: 1784: 1779: 1650: 1344: 1322: 1320: 1318: 1317: 1312: 1310: 1309: 1253: 1251: 1250: 1245: 1241: 1240: 1231: 1230: 1221: 1213: 1204: 1202: 1201: 1196: 1192: 1191: 1182: 1181: 1102:George Olshevsky 1012:Talk:Hypersphere 798:Erlangen program 728:anyway (it says 531: 501: 443: 386: 377: 367: 134: 133: 130: 127: 124: 103: 98: 97: 87: 80: 79: 74: 66: 59: 42: 33: 32: 25: 24: 16: 2274: 2273: 2269: 2268: 2267: 2265: 2264: 2263: 2214: 2213: 2188: 2163: 2161: 2140:charts on SO(3) 2132: 2110: 2103: 2065: 2040: 2032: 1981: 1968: 1958: 1945: 1929: 1919: 1906: 1890: 1880: 1867: 1851: 1841: 1828: 1812: 1802: 1789: 1760: 1759: 1739:220.237.115.109 1733: 1694: 1656: 1633: 1630: 1596: 1415: 1370:, according to 1357: 1323: 1301: 1290: 1289: 1232: 1222: 1207: 1206: 1183: 1173: 1165: 1164: 1161: 1088: 927:Glossary; Glome 902:MathWorld Glome 857: 673: 647: 645:4-sphere images 387:—The preceding 359: 229: 210: 171: 131: 128: 125: 122: 121: 99: 92: 72: 43:on Knowledge's 40: 30: 12: 11: 5: 2272: 2270: 2262: 2261: 2256: 2251: 2246: 2241: 2236: 2231: 2226: 2216: 2215: 2212: 2211: 2187: 2184: 2183: 2182: 2131: 2125: 2124: 2123: 2108: 2101: 2064: 2061: 2031: 2028: 2005: 2004: 1993: 1988: 1984: 1980: 1975: 1971: 1965: 1961: 1957: 1952: 1948: 1944: 1941: 1936: 1932: 1926: 1922: 1918: 1913: 1909: 1905: 1902: 1897: 1893: 1887: 1883: 1879: 1874: 1870: 1866: 1863: 1858: 1854: 1848: 1844: 1840: 1835: 1831: 1827: 1824: 1819: 1815: 1809: 1805: 1801: 1796: 1792: 1788: 1783: 1778: 1775: 1772: 1768: 1757: 1732: 1729: 1693: 1690: 1669:Conformality ( 1667: 1666: 1655: 1652: 1629: 1626: 1595: 1592: 1591: 1590: 1574: 1573: 1568: 1567: 1566: 1565: 1564: 1563: 1528: 1527: 1511: 1510: 1509: 1508: 1479: 1478: 1452: 1451: 1414: 1411: 1382: 1356: 1353:Stephen Baxter 1350: 1349: 1348: 1347: 1346: 1308: 1304: 1300: 1297: 1282: 1281: 1239: 1235: 1229: 1225: 1219: 1216: 1190: 1186: 1180: 1176: 1172: 1160: 1157: 1156: 1155: 1154: 1153: 1152: 1151: 1127: 1126: 1125: 1124: 1112: 1111: 1087: 1084: 1083: 1082: 1081: 1080: 1058: 1057: 1046:-sphere is an 1039: 1038: 1024: 1023: 1022: 1021: 986: 985: 954: 953: 952: 934: 933: 932: 924: 915: 907: 906: 905: 877: 856: 853: 827: 825: 824: 823: 822: 804: 803: 802: 801: 790: 789: 788: 787: 780: 779: 778: 777: 770: 769: 768: 767: 761: 760: 744: 743: 742: 741: 715: 714: 700: 672: 669: 668: 667: 646: 643: 642: 641: 625: 624: 616: 615: 614: 613: 592: 591: 581: 580: 576: 575: 574: 573: 565: 562: 535: 534: 533: 532: 525: 518: 517: 516: 515: 511: 505: 504: 503: 502: 495: 489: 488: 478: 477: 472: 471: 470: 469: 461: 457: 447: 446: 445: 444: 437: 427: 426: 411: 410: 358: 357:Visualizations 355: 344:83.134.156.161 331: 330: 329: 328: 321: 311: 307: 303: 299: 295: 291: 285: 284: 283: 282: 228: 227:Visualizations 225: 224: 223: 209: 208:Visualizations 206: 170: 167: 164: 163: 160: 159: 156: 155: 144: 138: 137: 135: 118:the discussion 105: 104: 88: 76: 75: 67: 55: 54: 48: 26: 13: 10: 9: 6: 4: 3: 2: 2271: 2260: 2257: 2255: 2252: 2250: 2247: 2245: 2242: 2240: 2237: 2235: 2232: 2230: 2227: 2225: 2222: 2221: 2219: 2210: 2206: 2202: 2197: 2196: 2195: 2193: 2185: 2181: 2177: 2173: 2169: 2160: 2159: 2158: 2157: 2153: 2149: 2145: 2141: 2137: 2129: 2126: 2122: 2118: 2114: 2107: 2100: 2096: 2092: 2091: 2090: 2089: 2085: 2081: 2077: 2072: 2070: 2062: 2060: 2056: 2052: 2048: 2044: 2036: 2029: 2027: 2026: 2022: 2018: 2014: 2010: 1991: 1986: 1982: 1978: 1973: 1963: 1959: 1955: 1950: 1946: 1939: 1934: 1924: 1920: 1916: 1911: 1907: 1900: 1895: 1885: 1881: 1877: 1872: 1868: 1861: 1856: 1846: 1842: 1838: 1833: 1829: 1822: 1817: 1807: 1803: 1799: 1794: 1790: 1781: 1776: 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1123: 1120: 1116: 1115: 1114: 1113: 1110: 1107: 1103: 1099: 1098: 1097: 1096: 1093: 1085: 1079: 1075: 1071: 1066: 1062: 1061: 1060: 1059: 1056: 1053: 1049: 1045: 1041: 1040: 1037: 1034: 1031: 1026: 1025: 1020: 1017: 1013: 1008: 1007: 1006: 1005: 1004: 1003: 1000: 996: 991: 990: 984: 981: 977: 974:In rereading 973: 972: 971: 970: 967: 963: 959: 951: 948: 945: 944: 943: 942: 940: 931: 928: 925: 923: 919: 916: 914: 911: 908: 903: 900: 899: 898: 894: 891: 890: 889: 888: 887: 884:, 4-sphere = 883: 876: 873: 872: 869: 864: 862: 854: 852: 851: 848: 842: 841: 838: 834: 829: 821: 817: 813: 808: 807: 806: 805: 799: 794: 793: 792: 791: 784: 783: 782: 781: 774: 773: 772: 771: 765: 764: 763: 762: 758: 754: 753: 752: 750: 739: 735: 731: 727: 723: 719: 718: 717: 716: 712: 708: 706: 701: 698: 697: 696: 694: 690: 685: 683: 678: 670: 666: 663: 659: 658: 657: 656: 653: 644: 640: 636: 632: 627: 626: 622: 618: 617: 612: 608: 604: 603:Chronometrier 600: 596: 595: 594: 593: 590: 587: 583: 582: 578: 577: 572: 569: 566: 563: 560: 559: 558: 557: 556: 554: 550: 546: 542: 530: 526: 522: 521: 520: 519: 512: 509: 508: 507: 506: 500: 496: 493: 492: 491: 490: 487: 484: 480: 479: 474: 473: 468: 465: 462: 458: 455: 451: 450: 449: 448: 442: 438: 435: 431: 430: 429: 428: 425: 422: 418: 413: 412: 407: 406: 405: 402: 398: 394: 390: 383: 379: 376: 371: 368: 366: 356: 354: 353: 349: 345: 341: 337: 326: 325: 324: 323: 322: 319: 318: 315: 310: 306: 302: 298: 294: 290: 281: 278: 273: 272: 271: 270: 265: 257: 249: 241: 233: 226: 221: 220: 219: 218: 215: 207: 205: 202: 199: 195: 193: 189: 185: 181: 174: 168: 153: 149: 143: 140: 139: 136: 119: 115: 111: 110: 102: 96: 91: 89: 86: 82: 81: 77: 71: 68: 65: 61: 56: 52: 46: 38: 37: 27: 23: 18: 17: 2189: 2167: 2133: 2105: 2098: 2094: 2073: 2066: 2041:— Preceding 2037: 2033: 2012: 2008: 2006: 1737: 1734: 1716: 1713: 1710: 1707: 1703: 1698: 1695: 1670: 1668: 1659: 1634:— Preceding 1631: 1612: 1597: 1493: 1489: 1485: 1433: 1416: 1396: 1390: 1384: 1381: 1364:Dante Dreams 1363: 1358: 1324:— Preceding 1266: 1162: 1136: 1089: 1047: 1043: 994: 992: 988: 987: 961: 957: 955: 949: 936: 935: 929: 921: 912: 896: 879: 878: 875:References? 874: 865: 858: 843: 832: 830: 826: 745: 733: 729: 725: 721: 686: 676: 674: 648: 629:point added. 539:— Preceding 536: 416: 384: 380: 372: 369: 360: 332: 320: 286: 211: 203: 200: 196: 178:— Preceding 175: 172: 148:Mid-priority 147: 107: 73:Mid‑priority 51:WikiProjects 34: 2113:Cloudswrest 2011:and points 1718:Georgia guy 1692:Terminology 1640:68.2.132.51 1615:Cloudswrest 1553:75.80.43.80 1517:75.80.43.80 976:hypersphere 962:topologists 956:So perhaps 939:hypersphere 937:3-sphere = 880:3-sphere = 861:hypersphere 545:79.72.81.99 275:pictures.-- 184:79.72.81.99 123:Mathematics 114:mathematics 70:Mathematics 2218:Categories 1355:reference. 941:in 4-space 711:Ian Cairns 1662:conformal 1420:AVKent882 1400:Shnakepup 1271:Tropylium 1256:Daniel bg 1137:even once 958:geometers 786:2-sphere. 524:3-sphere. 454:hypercube 434:tesseract 409:3-sphere. 39:is rated 2128:See also 2080:Tom Ruen 2055:contribs 2043:unsigned 1636:unsigned 1468:Tom Ruen 1459:), so a 1375:Akriasas 1338:contribs 1326:unsigned 1119:MrRedact 999:Tom Ruen 980:Tom Ruen 966:Tom Ruen 918:Glossary 868:Tom Ruen 734:3-sphere 726:3-sphere 722:2-sphere 682:Revolver 662:Sam nead 652:CharlesC 541:unsigned 401:contribs 389:unsigned 214:Sam nead 180:unsigned 169:Comments 2201:Zundark 2186:w h y ? 2172:Maproom 2148:Maproom 2138:and to 2130:section 2017:Rgdboer 1679:Tamfang 1539:Rgdboer 1535:versors 1498:Tamfang 1441:Tamfang 1106:Fropuff 1016:Fropuff 847:TxAlien 757:Zundark 749:Fropuff 738:Zundark 693:Fropuff 621:Fropuff 586:Fropuff 568:TaranVH 483:Fropuff 464:TaranVH 421:Fropuff 393:TaranVH 314:TxAlien 277:TxAlien 150:on the 41:C-class 2144:sphere 2047:'Ff'lo 1628:3-ball 1572:shape. 1461:sphere 1092:Zaslav 1052:Zaslav 1033:(Talk) 995:common 882:sphere 689:sphere 47:scale. 1671:local 1386:Shell 1330:Tjips 1267:faces 1086:Glome 886:glome 837:Latch 796:(See 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