Talk:Torus/Archive 1

Archive 1

n-torus

We could also write the n-torus as something like this right: $\prod _{i=1}^{n}S^{1}$ -- Ævar Arnfjörð Bjarmason 00:32, 21 November 2004 (UTC)

Sure, if you wanted to. -- Fropuff 02:17, 21 November 2004 (UTC)

@Fropuff g-i dono the latest test results were in my Beezer images and the only time I have to look at the difference is when they confirm the timer switch on the living room table Geeze the first infin faller was a reversing Shoot from the bowe of binomial polynomial innuenda5ion. 174.250.210.57 (talk) 12:03, 30 October 2022 (UTC)

Topological group

Hi User:Fropuff I notice you removed the brief sentence about "topological group" (and the formulas to go with it). Not a big deal, but I think there is some utility in keeping the presentation simple, for the less educated reader. Thus, of course Lie groups are toplogical groups, but if the reader doesn't know what a Lie group is (which is quite possible), I think its very nice to have a simple, concrete formula showing how the multiplication is defined. Assume the reader is a (smart) teen-ager. Also, I tend to prefer formulas over words ... again, think of the slightly dyslexic, slightly autistic reader, who might have trouble converting sentences into formulas: Group multiplication on the torus is then defined by coordinate-wise multiplication: you and I know how to convert that sentence into a formula, but I am afraid that first-time readers/younger students may struggle or fail to make the effort. Can we have the formula back? Specifically:

The n-torus is also an example of a topological group. This follows from the fact that the unit circle is a topological group. Group multiplication on the torus is then defined by coordinate-wise multiplication. That is, for $x=(x_{1},x_{2},...,x_{n})\in \mathbb {T} ^{n}$ and $y=(y_{1},y_{2},...,y_{n})\in \mathbb {T} ^{n}$, the point $z=xy$ given by $z=(x_{1}y_{1},x_{2},y_{2},...,x_{n}y_{n})$ is again a point on the torus.

I concur with this comment. That overly terse sentence "Group multiplication (...)" is pretty opaque. I have a BS in applied math, but I didn't study topology or abstract/modern algebra as part of my undergrad work. This needs at least one of two improvements: give a more readable description, or create a new page that defines what "coordinate-wise multiplication" means, or better yet a definition of what it means to perform an operation in a "coordinate-wise" manner.

linas 01:21, 2 April 2005 (UTC)

I'm not opposed, but if that's the level of readership you want to shoot for then it might be better to be more explicit still. Such a reader may not know what $x_{1}$ or $y_{1}$ are or how to multiply them. Perhaps

For $x=(e^{i\theta _{1}},e^{i\theta _{2}},...,e^{i\theta _{n}})\in \mathbb {T} ^{n}$ and $y=(e^{i\phi _{1}},e^{i\phi _{2}},...,e^{i\phi _{n}})\in \mathbb {T} ^{n}$, the point $z=xy$ given by $z=(e^{i(\theta _{1}+\phi _{1})},e^{i(\theta _{2}+\phi _{2})},...,e^{i(\theta _{n}+\phi _{n})})$ is again a point on the torus.

That said, we may lose said readership in the next paragraph (if we haven't already lost them in the previous section). -- Fropuff 05:44, 2 April 2005 (UTC)

Ohh, I like the exp i θ bit. I believe that horrible articles should target all levels, high and low; its also OK to lose readers at some point, it gives them a challenge, something to work up to. Most math articles are never going to be accessible to beginers, they're compilcated cause the topic is complicated. There are a few that are simple, and stay simple (and are thus pretty darn boring). This is one of the few topics that can go all the way from grade-school simple to complicated, and so it should ramp up in this gentle way. (I was 9 years old when the teacher wrote actual, honest formulas for sphere, cylinder and torus on the blackboard. I think it would be great if future 9-year olds could get some traction in an uproar like this... or teens for that matter). I'm thinking we should have a special category, "Educational trampoline" or something like that, for this article and e.g.Pi that can go from dirt-simple to current-research level, providing a portal for smart teens (or older) to get into math. linas 02:29, 3 April 2005 (UTC)

I like Linas's vision. Except that you need to "lose" readers, rather than "loosing" them. :) Oleg Alexandrov 02:44, 3 April 2005 (UTC)

Hypertoroid

I ran across the Hypertoroid article when randomly browsing wikipedia, and decided the article finally should be merged (someone put the tag on ages ago). But when I read this article and the Toroid-article, I got a little confused, Shouldn't Hypertoroid be merged with Toroid? Also I always thought hyper- was for n dimensions instead of just 4. - Dammit 23:06, 23 April 2006 (UTC)

optimal packing of toruses

Can anybody suggest how I would go about figuring the optimum packing arrangement for fitting several toruses (each with an inner radius of 1 and an outer radius of 2) into a cube that measures 16 on a side? Is there a particular place in Wiki that is best suited for describing this? What if I try packing those same toruses into a sphere of radius 16?

The bigger the cube, the more ways there must be to arrange toruses inside of it. At what point does the arrangement of toruses in a cube become non-trivial, and multiple solutions become possible? Perhaps the computational time necessary to generate arrangements and rearrangements might be useful for cryptography, just as the computation of prime numbers is?

It's not enough to describe a torus by its inner radius and outer radius, you also have to describe the cross-sectional shape of the tube itself, which I will assume to be perfectly circular. I guess you could call it the "box full of doughnuts" problem. If you do nothing more than generate lots of toruses at various angles and tilts, and relying on a random function throw them into the 16x16x16 cube, discounting improper intersections (but admitting pairs of toruses linked through each other's middle) you're sure to get a good practical idea how many can be made to fit. — Preceding unsigned comment added by 198.177.27.29 (talk) 21:30, 11 May 2006 (UTC)

Toroidal polyhedra

What about adding a few words about toroidal polyhedrons, smth like this (please, edit this with better english):

There exists polyhedra, which are topologically equivalent to a torus. Their Euler characteristic is 0, so the number of edges is equal to the sum of number of verticies and number of faces. Toroidal polyhedra gives us examples of polyhedra with very unusual properties:

Szilassi toroidal polyhedron has 7 faces with every pair of faces has an edge in common (so it gives a nice proof that 7 colours is minimum for colouring a torus).

Császár toroidal polyhedron (the dual of Szilassi one) has 7 vertices, and every pair of vertices is connected by one of its edges, so it has no diagonals (like tetrahedron). It is not known, whether there exist polyhedra other then tetrahedron and Császár polyhedron with such property.

Sources: Császár, Á. "A Polyhedron without Diagonals." Acta Sci. Math. 13, 140-142, 1949-1950. http://www.minortriad.com/szilassi.html

I can make pictures for them. Markelov

This would be nice, especially with pictures. But I would think that it would be better for this to be a new article, toroidal polyhedron, and one one could add a blurb or link to torus mentioning the new article. What do you think? Oleg Alexandrov (talk) 16:37, 17 February 2007 (UTC)

Yes, a new article would be pretty good! — Kieff | Talk 18:33, 18 February 2007 (UTC)

Corresponding adjectiv?

What is the corresponding adjective for "torus" with the meaning "shaped as a torus" (as "cylindric" corresponds to "cylinder")? --84.172.164.89 18:21, 12 May 2007 (UTC)

That would be toroidal. — Kieff | Talk 23:50, 12 May 2007 (UTC)

Hm. Doesn't the word "toroidal" mean "shaped as a toroid"? Is it ambiguous? --84.172.164.92 23:54, 13 May 2007 (UTC)

Well, the word "toroid" means "torus-like" (similarly to humanoid, human-like), so I think toroidal is the correct word. — Kieff | Talk 00:30, 14 May 2007 (UTC)

RPG Map

Should there be a mention of the fact that the world map of a typical RPG video game implies that the world is a torus because it wraps both North-South and East-West?

Not in particular. A lot of games have this sort of thing going on. Pac-Man and Asteroids, for example. — Kieff | Talk 01:22, 20 May 2007 (UTC)

Verifiability

I'll quote WP:V that is cited: "Editors should provide a reliable source for quotations and for any material that is challenged or likely to be challenged". I don't think anything here is likely to be chanllenged. Please stop putting citation tags. This is not an Israeli-Arab wars article or something. Saaska (talk) 15:48, 24 January 2009 (UTC)

Apparently you didn't make it past the "nutshell" description of WP:V. I agree that not every statement needs to have an individual citation, but for an article to totally lack any references is not appropriate, and is a clear violation of policy. Either a {{citations missing}} template or a {{unreferenced}} template is totally appropriate for this article, and removing one without addressing the concern is not appropriate. 15:55, 24 January 2009 (UTC)

There you go. Saaska (talk) 02:31, 25 January 2009 (UTC)

Thank you for the reference. Regrettably, its discussion of tori is rather peripheral. It does not appear to discuss in detail the topology of tori, nor tori of dimension greater than two. Since about two-thirds of the article are dedicated to such matters, the reference you gave is clearly inadequate, and the removal of the template was unjustified. Nevertheless, I have changed it to a {{refimprove}} template instead. siℓℓy rabbit (talk) 05:21, 25 January 2009 (UTC)

See also section

I believe that links to Doughnut, Torus mandibularis, Torus palatinus should better fit in a disambiguation page than in "See also" section. TommasoT (talk) 16:19, 10 March 2009 (UTC)

I thought so too, it's done.--RDBury (talk) 18:32, 2 October 2009 (UTC)

Helix

Would it be appropriate to add a section about a "torical helix" on the page? Jeanlovecomputers (talk) 02:24, 9 January 2010 (UTC)

Ring Torus

In the introduction it says that a ring torus is a 2-manifold of genus 2. Is this different from a torus? (it doesn’t seem like it from the description), because a torus should be genus 1. If it is actually different, I think it is a little unclear. —Preceding unsigned comment added by GromXXVII (talk • contribs) 18:30, 24 January 2010 (UTC)

Merge with Toroid

The article Toroid (geometry) is virtually empty, lists no sources and much of the material that might be covered there is already covered here. In general non-mathematical usage, the words torus and toroidal can refer to either the surface or the solid so having separate articles is somewhat confusing. Also, the subjects are so closely related that having separate articles creates an artificial division. Therefore I'm proposing that the other article be merged with this one.--RDBury (talk) 16:56, 22 October 2009 (UTC)

Merge or not, the two disagree about what the distinction is between the two terms. That apparently needs some work. Ccrrccrr (talk) 19:51, 22 October 2009 (UTC)

Toroid (geometry) includes objects generated by revolving a non-circular object, while Torus does not, so the two deal with clearly separate – though closely related – subjects, and therefore I do not think a merge is a good idea. brtkrbzhnv 01:54, 1 November 2009 (UTC)

I'm not sure what material you could have for the non-circular toroid. Any such material should be included in Solid of revolution to avoid a content fork.--RDBury (talk) 04:06, 1 November 2009 (UTC)

If anything, toroid should merge here, since torus is by far the more common term. A sentence could be added to this article saying that a generalization of the torus, obtained by rotating an non-circular curve about a center, is sometimes called a toroid, provided there is no self-intersection. Rick Norwood (talk) 12:26, 1 November 2009 (UTC)

I think that "toroid" should remain separate, it introduces additional issues and definitions and concepts. Toroidal forms aren't necessarily solids of revolution, and the word "toroidal" is often handy when referring to the topology of an object, rather than to a strict shape or set of dimensions (we can also have irregular, scalene toroids). With toroidal transformers, the topology of the coil and core is definitionally important, the exact cross-section isn't. For faceted, toroidal polyhedra, the cross-section usually changes abruptly at places, the cross-section of a conventional toroidally-wound coil or a Villarceau coil actually rotates around the minor axis as we sweep the cross-section around the major axis, and we even have exotica like the the Szilassi polyhedron. There are also the shapes that you get by setting up vibrational standing waves on a toroidal membrane and taking snapshots - the resulting shapes have cross-sectional proportions that vary in a regular way around the major axis, the shape is toroidal, but it's not a solid of revolution. Toroidal configurations of hardware (and toroidal structures in biochemistry) don't always suit definitions based on "Let's take a perfect geometrical shape and sweep it simply and continuously around an axis" idea.

I'd also point people at the existence of separate pages for sphereoid and ellipsoid - we don't attempt to define all the additional parameters and properties of sphereoids and ellipsoids on the "sphere" and "ellipse" pages. ErkDemon (talk) 13:22, 1 November 2009 (UTC)

It seems there's no consensus to merge the two pages, and they have only a few common properties. I'm going to remove the merge notice. —Pengo 22:11, 7 October 2010 (UTC)

Merge with Standard torus

The article Standard torus has considerable overlap with this one. It appears that what happened is the other article is based on a MathWorld article, but MathWorld tends to split subjects into different articles that would be covered in a single article in Wikipedia. There might be a case for having separate articles for the topological object and the algebraic surface but that isn't the split that was done. I am therefore proposing that the articles be merged with a possible split to a new article "Torus (topological space)" if the merged article becomes unwieldy.--RDBury (talk) 19:12, 2 October 2009 (UTC)

• Yes good idea. I simply don't see how the "standard torus" article could grow, and its awfully short to sit on its own. The name "standard torus" even sounds like a bit of a neologism to me. linas (talk) 01:01, 9 January 2011 (UTC)

• Yes as well. I've added some of its material to torus (which is very little). The only additional, possibly relevant material seems to be Johannes Kepler's notation, but that doesn't seem very useful. Chris857 (talk) 23:37, 3 May 2011 (UTC)

• In fact, given the slow pace of response, I will be bold and merge Standard torus in. Chris857 (talk) 23:53, 3 May 2011 (UTC)

File:Sphere-like degenerate torus.gif

As the distance to the axis of revolution decreases, the ring torus becomes a spindle torus and then degenerates into a sphere.

I fear this animation could be misleading resp. contain an error. The height of the torus considerably grows while the torus contracts. In the caption, there is only said that the axis of revolution decreases. For a normal reader, there does not seem to be a hint that, together with this, the volume of the body should stay the same (which could eventually explain the increasing height). --Hans Dunkelberg (talk) 09:48, 20 June 2011 (UTC)

Proof of a minimum of 7

Proving 4 is the minimum number of colors needed on a plane can be done as follows:

To prove 1 color doesn't suffice, all we need is the fact that the map has more than one region.

To prove 2 colors don't suffice, all we need is a vertex inside the map where an odd number of regions meet. They have to alternate red-green-red-green-red-green, which cannot be done if the number of regions meeting at the vertex is odd.

To prove 3 colors don't suffice, we need a region not touching the edge of a map surrounded by an odd number of regions. Color the central region red, and the surrounding regions have to alternate green-blue-green-blue-green-blue, which is impossible if their total is odd.

But a torus needs 7 colors. Can we prove it's impossible with 4, 5, or 6 colors?? Please write the proofs here:

Georgia guy (talk) 17:49, 14 July 2011 (UTC)

Here is a proof. Cover a torus with this rectangle. You'll find that each of the seven parallelograms touches all six of the others. —Tamfang (talk) 05:27, 23 July 2011 (UTC)

Flat torus

Anyone willing to add info about recently-constructed flat torus in 3D? See "Flat tori in three-dimensional space and convex integration", http://www.pnas.org/content/early/2012/04/18/1118478109. OverQuantum (talk) 20:22, 9 May 2012 (UTC)

There are some pictures here near the bottom of the page. This is totally cool! You basically have an infinite sequence of corrugations. The pictures are probably not public domain or I would attach them to the article. Cloudswrest (talk) 05:15, 10 May 2012 (UTC)

One quibble I have, while it is isometric to a flat torus, in the three dimensional embedding the local curvature is not defined, and is in fact non-definable. So is it really valid to call it a flat torus where the said "flatness" does not exist? Cloudswrest (talk) 23:38, 10 May 2012 (UTC)

This section says both that the flat torus cannot be isometrically embedded in 3-space, and that an embedding has been found. Either something is incorrect or the wording is unclear. (76.77.34.250 (talk) 18:18, 28 October 2012 (UTC))

LaTeX

I tried to type theta and phi with backslashes inside wedge math and got a lexing error. Then I entered Theta and Phi, and that worked just fine. Further, after getting Theta and Phi to work, I could change them back to theta and phi. WTF? Rick Norwood (talk) 14:48, 18 November 2012 (UTC)

Please explain

Please explain what the line u, v ∈ [0, 2π), means in the Geometry section. Tractor 19:09, 5 March 2007 (UTC)

That means u and v are in the interval from 0 to 2pi. I clarified that in the article, although this is rather standard math notation. Oleg Alexandrov (talk) 02:26, 6 March 2007 (UTC)

I'm yet another reader who has trouble understanding u and v. They are angles alright, but this is only understood from context, i.e. their variation from 0 to 2pi. There is no indication inside the figures. What do they refer to exactly? The explanation states clearly what R and r refer to, but when it comes to the two angles I believe too much is taken for granted. Have pity on us poor non-mathematicians. I've always come across angles marked with Greek letters such as theta and phi. The Italian version of the article has their two roman counterparts, t and p (and it's clearer than the English but provides no clear explanation either). Had I seen two Greek letters I would have understood it sooner. All in all this is very confusing and - sorry but I have to point it out - it is partly due to a lack of consistency in mathematical notation. But the most important player is the figure -- the more detailed, the better. The 3D animations look impressive, but do not really help in every respect. No offense intended. Viktor Laszlo (talk)

I've tried to clear this up. Let me know what you think. Rick Norwood (talk) 14:45, 18 November 2012 (UTC)

I apologize, but I have only just now returned to the page. If you look at the Italian counterpart to this article http://it.wikipedia.org/wiki/Toro_%28geometria%29 you'll find that the figure there shows the angles t and p. It's definitely me; but can you please explain? In that figure, is p the angle between the red and the blue line? (and if that's the case, how can the angle vary without turning the cross-section from a circle into some sort of an ellipsis? does that still make a torus?). And is t enclosed between the two reddish lines? If I'm wrong, I suppose I can't be blamed for not looking. I originally came to the English article because I hadn't found the Italian one very helpful, and I suppose international users who are disappointed in their own language article might well do the same. However, there is a need for an explanatory figure where all the members of the parametric equation are shown clearly. The Italian figure I was referring to is not exceedingly good either. I hope this doesn't sound rude, but the fact that it's pleasing to the eye does not mean that it can do the job. Now speaking in general terms, for all science articles in the Wikipedia, I feel I should argue that a good figure PLUS a good side explanation will do the job. In this case, one saying something like "the angle theta is between points A, O and B"; or "AOB" (with ^ on O). (I'm sorry if it doesn't sound right but it's just to provide a suggestion). Thank you for the time you've dedicated to this article. Viktor Laszlo (talk) —Preceding undated comment added 14:56, 29 November 2012 (UTC)

Trivia

I was very facinated when I realized, that the map of the game I was just playing was actually an abstract torus. The intrinsic view of a manifold is not an easy concept and illustrating it with computer games might help people to understand it. Furthermore it would connect the abstract world of differential geometry with something more people are familiar with. Markus Schmaus 14:32, 26 July 2005 (UTC)

Yes, but the place to explain this is in the article on computer games, and not on the article on torii. Similarly, mirrored rooms have (more or less) the same topology. The topology of multiply-connected polytopes in general, (known as knot theory), and not just torii, shows up in various sci-fi books as as models of space and wormholes and killed-my-grandfather-etc (they cut the math out in the made-for-tv versions). And truth being stranger than fiction, the Wilkinson microwave anisotropy probe has measured a CMB that is compatible with the topology of a soccer ball (i.e. with opposite faces of the soccer ball connected, the way a torus is.) linas 17:05, 26 July 2005 (UTC)

Nanotechnology contains a section Nanotechnology in fiction. We could make a similar section Torii in popular culture and list games and fiction. If I elaborated the idea in the article on The Settlers and made a backlink

• The Settlers is using a torus as map.

would this be ok with you?

It might also be a nice idea to describe how being in a three dimensional torus would look like.

P.S.: A room with ordanary mirrors on its walls, floor and ceiling? What's its topology? Be formal if you like. Markus Schmaus 23:33, 26 July 2005 (UTC)

I do think it would be a good idea to somewhere state that Torus are often used in Computer Games, with a link to the computer games article. But the Torus article shouldn't try to directly maintaign that list. Joncnunn 18:45, 28 April 2006 (UTC)

Funny thing, I just texture-mapped Asteroids in a torus the other day. It's rather fun to watch. ☢ Ҡi∊ff⌇↯ 06:50, 1 October 2006 (UTC)

I disagree with the above responses: I think it would be great to mention the use of the torus in computer games here. It's used in many references I know, for instance in Jeff Weeks' book The Shape of Space, and makes a great way to visualize the flat torus. I don't think it makes sense to mention any particular current game, though. --Dylan Thurston (talk) 19:26, 11 February 2013 (UTC)

Lead

There seems to be a bit of an edit war going on about doughnuts, so I thought it best to discuss it here.

The first sentence used to include:

a torus (pl. tori) is, generally speaking, a doughnut-shaped object. More precisely, it is...

that got removed with the comment:

Removed the circular reference to doughnuts, a torus cannot be described as doughnut-shaped when a doughnut is torus-shaped. It is an unnecessary attempt at description and would be better placed as an example of a torus.

The argument about circular references is a red herring. Firstly, the doughnut article does not refer back to this article until the second paragraph. And secondly, an examination of any dictionary or encyclopedia necessarily turns up many circular loops of references. (For more on this or mostly as a fun aside, see User:Jesseoffy/Wiki_Loop_Theorem and [1]).

I believe the real discussion point is this: should the very start of the lead introduce the concept by way of a familiar example, or should it introduce it with a precise mathematical definition. I favor the familiar example, because I believe that general readers (without much mathematical/geometrical background) who have seen a reference to a torus and they're trying to figure out what in the world that is, will find that more helpful and could be bewildered by the precise mathematical definition. Readers who know the general concept, and wish to obtain the precise mathematical definition, will not be bewildered by the familiar example. So I think that starting with the familiar example makes the article more universally helpful.

I would agree that the phrasing should make it clear that the doughnut is there as an example, not as a definition. I thought that "generally speaking" and "more precisely" worked fine for that, but perhaps other ways would satisfy others objections.

Ccrrccrr (talk) 05:16, 21 October 2011 (UTC)

Part of the problem is that this should really be split into two articles. Part of the article is about a specific family of surfaces, namely surfaces of revolution of circles, the study of which dates back to antiquity, and the other part is about a homeomorphism class of 2-manifolds studied in topology. There are issues with using the word doughnut-shaped in either context, but I agree that a certain amount of imprecision should be allowed if it helps people identify the subject.--RDBury (talk) 01:34, 22 October 2011 (UTC)

I agree, the two different topics in one article is a problem, and I think that splitting into Torus (geometry) and Torus (topology) would help. Would you agree with those names? Ccrrccrr (talk) 15:54, 25 October 2011 (UTC)

I agree, with corresponding updates to the dab page. 'Torus' (with no parenthetical specification) should probably either redirect to or replace the dab page, but I'm no sure how that would go without losing page histories. Doing a copy and paste is frowned upon.--RDBury (talk) 09:21, 26 October 2011 (UTC)

OK, so if that complicated solution didn't happen in the last year, it probably won't happen in the next year...so I'm back to my original question: should be lead explain in lay language or obscure technical language. This guideline is pretty clear: Wikipedia:Make_technical_articles_accessible#Lead_section. So I pretty much think that settles it. Any further discussion before I put doughnuts back in the lead? Ccrrccrr (talk) 04:16, 20 January 2013 (UTC)

I think the lede reads better and more clearly, as at the moment, without references to doughnuts. Firstly, there's a good description of constructing it in three-dimensional space, and an illustration. Secondly, not all doughnuts are toroidal. Indeed, in the UK, including most from chain bakers, most are spheroidal, and the only toroidal things they are likely to sell are beigels. --Cedders^tk 12:31, 24 February 2013 (UTC)

Geometry: θ vs φ

In the Geometry section, there appears to be inconsistency in the usage of θ and φ. The parametric definition shows θ as the angle of big-R, and φ as the angle of the little-r tube. However, in the paragraph on the modified spherical coordinate system and its associated image, θ is the "poloidal" angle (with the little-r), and φ is the "toroidal" angle (associated with big-R). Have I misunderstood, and there isn't a discrepancy? (If so, please explain.) Or should there at least be a note that coordinate system has changed in that paragraph? (In the Spherical coordinate system article, there is the discrepancy between the mathematics usage and physics usage. I assume it's probably the same case here.) -- Pryrt (talk) 15:24, 13 May 2014 (UTC)

Given the intense silence, I made the change. -- Pryrt (talk) 23:41, 22 August 2014 (UTC)

Area and volume formulas appear dubious

The article correctly states that reducing R to 0 turns the torus into a sphere, but the area and volume formulas don't agree. Setting R=0 in those formulas makes the area and volume both disappear. Could someone help debug this? -- 128.46.119.2 (talk) 18:52, 6 February 2015 (UTC)

This is not surprising. When surfaces move through each other, sometimes signs change. The formulae do apply unchanged to horn tori, and I strongly suspect to spindle tori if the interior volume is omitted (and hence the volume measured tends to zero in the spherical limit) and the inner surface area regarded as negative. This would make it a matter of interpretation of what is being measured. —Quondum 19:10, 6 February 2015 (UTC)

(edit conflict)The thing is setting R = 0 makes it into a sphere but it does so by overlapping and self intersecting itself, so it is no longer a proper torus. As such the formula breaks down. It's not the formula for the volume or area of a sphere obviously. In that case I think what's happening is it is the torus in effect loops round twice, wholly overlapping itself. If you think of it doing so once normally and once 'inside out' then the volume of the sphere gets counted twice, once as a positive volume once negative, so giving zero. Something similar happens for the surface. So basically it's wrong to try and calculate the volume of the sphere from the formula as by that point you have a degenerate not a proper torus.--JohnBlackburne^words_deeds 19:13, 6 February 2015 (UTC)

I see what you both are saying. Thank you for the explanation. Is it then correct to say that the formulas as they are written would apply only to ring and horn tori but not to spindle tori or spheres (taking only the "intuitive" interpretation of area and volume, not the sign-dependent one)? Some note along these lines would be appropriate to the article. -- 128.46.119.2 (talk) 21:02, 6 February 2015 (UTC)

It does in the paragraph immediately above, starting at "When R ≥ r, ...". I agree it's not obvious though; your eyes can latch onto the formulae and not notice the associated restriction. So I've merged the two paragraphs to make it clearer it's all about the same R ≥ r case. The resulting paragraph is not I think too long and if anything the prose flows better.--JohnBlackburne^words_deeds 21:27, 6 February 2015 (UTC)

Dimensions

So is this 3-torus 4-dimensional? --anon

Well, the 2-torus is the torus whose surface has 2 dimensions, that is, the interior of the torus is actually three dimensional. This is the usual torus, the doughnut if you wish.

Then, the 3-torus is a torus whose surface (boundary) is in 3 dimensions. Then, the body enclosing this three dimensional surface must be four dimensional. So you are right. This torus is an imaginary torus of course. One could have a hard time imaginging a body whose boundary has three dimensions. Oleg Alexandrov 14:36, 15 May 2005 (UTC)

The n-torus Tⁿmeans the topological space that is the cartesian product of n copies of the circle S¹. This is an n-dimensional manifold. It is not "in" anyplace. (Unless you decide to put it there.) It has no boundary,.Daqu (talk) 05:32, 28 October 2015 (UTC)

I have removed irrelevant section

I have removed the irrelevant section Toratopes in 4 dimensions since there is no justification for it. Please do not revert this deletion.Daqu (talk) 15:08, 28 October 2015 (UTC)

What do you mean "There is no justification for it"? It's included in the higher dimensional extensions section, with cited reference. Who decides what's "relevant". Cloudswrest (talk) 16:49, 28 October 2015 (UTC)

The reference is to another wiki, which is not a reliable source. I have not heard of these and if no proper source can be found I agree with the deletion.--JohnBlackburne^words_deeds 17:06, 28 October 2015 (UTC)

I have edited section to remove reference to colloquial name "Tiger". Cloudswrest (talk) 16:13, 29 October 2015 (UTC)

The problem was not (just) the colloquial name "Tiger". The problem is that this is not mainstream mathematics. In fact, it looks like someone trying to promote some kind of fringe mathematics that they have some particular reason for wanting to promote. Wikipedia is not the place to do that.Daqu (talk) 01:01, 30 October 2015 (UTC)

Hi, I'm Keiji, the maintainer of Higher Space, the wiki sourced by Cloudswrest. I've re-removed the section in question as I agree it's not relevant to Wikipedia (nor is most of what we do tbh), and I've explained why over here. Just wanted to give an official stance on the matter, as such. ~ Keiji (iNVERTED) (Talk) 22:55, 30 October 2015 (UTC)

Torus inversion description

I'm having a problem with the torus inversion description, specifically the "garden hose and sock" analogy. As I see it, they are not inverts. Once you connect two opposite edges of a rectangle to form a piece of a cylinder, there is no way to get a pair of inverts by connecting the circles in different ways. If you imagine lines running parallel to the two edges connected (i.e., not circles around the cylinder), then the garden hose and the sock both yield circles whose planes pass thru the central axis (i.e. not the equator). PAR (talk) 00:59, 1 November 2015 (UTC)

I imagined a cylinder with lines running parallel to the two edges connected: i.e. straight lines along the surface from one end to the other. Say eight lines equidistant around the circumference of the ends. Then, by imagining the cylinder is a garden hose (the lines now run along from one end of the hose to the other) and connecting the ends together you end up with each line becoming a circle that runs all the way along the hose back to where it started. The lines do not pass through the "hole" that is formed in the middle of the loop. If you use the sock-rolling construction, i.e. you take one end and pass it through the inside of the cylinder to join the other, now the lines are all perpendicular to where they were with the hose construction: the lines each travel through the "hole". Does this help explain it? Feel free to edit my edit to the article if there's a better way to word it. ~ Keiji (iNVERTED) (Talk) 22:48, 1 November 2015 (UTC)

Yes, I see now what you mean. If the circles are connected in the interior of the cylinder (piece) the torus is an invert of the case where you connect them outside. A sock is closed at one end, and that was confusing to me. I will try to edit it, see what you think. PAR (talk) 14:21, 14 November 2015 (UTC)

Cartesian equation

The Cartesian equation of the torus can also be written $(x^{2}+y^{2}+z^{2}-[R^{2}+r^{2}])^{2}=4R^{2}(r^{2}-z^{2}).$. SophieAthena (talk) 16:29, 19 April 2016 (UTC)

non-standard teminology

This article introduces the non-standard terminology of "n-torus" for higher genus surfaces. It is usual to call a connect sum of n tori a "surface of genus n" rather than an "n-torus" - which is usually reserved for the cartesian product of n copies of the 1-sphere. — Preceding unsigned comment added by 160.39.158.68 (talk) 02:46, 4 January 2013 (UTC)

This may be a problem for inexpert readers like myself. I also read

The torus has a generalization to higher dimensions, the n-dimensional torus, often called the n-torus or hypertorus[5] for short. (This is one of two different meanings of the term "n-torus".)

What's the second meaning? Is it the "n-fold torus" that appears later (a 2-fold torus being a double torus, a two-dimensional surface with Euler characteristic -2)? Since a 3-sphere is a 3-dimensional surface (embedded in 4 or more dimensions), I'd imagine a 3-torus would be too. Maybe the parenthetical comment could be removed if it is misleading?

Another problem I notice is that the n-dimensional torus is illustrated with a Clifford torus, which is surely a 2-torus in R⁴, and so not the "generalisation to higher dimensions" described in the section?

In general, there's a confusion between a torus in geometry and one in topology. A caption reads "As the distance to the axis of revolution decreases, the ring torus becomes a horn torus, then a spindle torus, and finally degenerates into a sphere." Topologically, it's still not a sphere, is it? Roughly, there are two rotations of the circle, so two surfaces or skins in the same place. I wonder because I was looking for information on the non-heterogeneous Gaussian curvature of a 2-torus in R³ compared to R⁴. Just some ideas to improve the article further. --Cedders^tk 12:53, 24 February 2013 (UTC)

This is the first time I've ever seen "n-fold" torus. I'm going to change this to the more standard "genus g" surface. Sam nead (talk) 17:45, 22 February 2017 (UTC)

Cutting the torus section impenetrable

The last section of this article is so terse and confusing that I can't even parse the grammar of the first sentence (first out of two). It's also unclear (to someone who doesn't already know, like yours truly) what the section is even about -- i.e. what type of mathematical result is this, what does it mean even remotely concretely. Could someone either fix or remove? --Rainspeaker (talk) 14:45, 21 August 2017 (UTC)

I added a line that might be helpful. But currently the problem is this: If you are allowed to make 3 planar cuts in a solid torus of revolution sitting in 3-space (that is, you remove all points of the solid torus that lie in one or more of the 3 planes) ... then what is the maximum possible number of pieces in the remaining portion of the solid torus, obtained by a judicious arrangement of the 3 planes?

Interior of T^n ?

What's the interior of $T^{n}$ like? Is it simply connected, or does it have some more complex homotopy? 67.198.37.16 (talk) 06:53, 20 December 2018 (UTC)

All points of Tⁿ look the same, so a priori it doesn't have any "interior" defined.

However, it is possible to embed Tⁿ into (n+1)-dimensional euclidean space. Once it is in (n+1)-space, it will be the boundary of an (n+1)-dimensional manifold.

A standard way to do this can be defined inductively, so that the "interior" of Tⁿ in (n+1)-space is topologically the cartesian product of the circle S¹ and the "interior" of the torus of dimension n-1.

1D

How to explain :"1-dimensional torus T= R/Z " ? from : https://www.researchgate.net/publication/266020405_The_Mandelbrot_set_and_kneading_sequences

Is it related with "the quotient space obtained from ℝ by the relation x~y IFF x−y∈Z " ?

TIA

--Adam majewski (talk) 15:48, 5 September 2017 (UTC)

Hi Adam! Your mandelbrotset-buddy, here. The 1D torus is the circle! Yes, the 1D torus is "the quotient space obtained from ℝ by the relation x~y IFF x−y∈Z". That is, it is just x mod 1: given some real x, just keep the fraction, always throw away the integer parts. To get a circle, just write $z=e^{i2\pi x}$ for real x. So the "flat" 1D torus can be mapped to the "round" circle this way. 67.198.37.16 (talk) 06:58, 20 December 2018 (UTC)