Talk:Torus

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

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

Contents

[edit] 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 = (x1y1,x2,y2,...,xnyn) is again a point on the torus.

linas 01:21, 2 Apr 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 x1 or y1 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, 2005 Apr 2 (UTC)

Ohh, I like the exp i θ bit. I believe that ideally 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 article 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 Apr 2005 (UTC)
I like Linas's vision. Except that you need to "lose" readers, rather than "loosing" them. :) Oleg Alexandrov 02:44, 3 Apr 2005 (UTC)

[edit] 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)

[edit] 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
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)

[edit] 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)

[edit] 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.

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)