Talk:Torus

Mistake in Volume Formula

I think the formula for Volume of a toroid using P & Q is wrong, it differs by a factor of 2 from the correct result obtained using the formula based on R & r. I don't know how to edit equations on wikipedia so please can someone else fix this. The final formulation on the right hand side of the line with the equation for volume as a functions of P & Q should start with 1/4 instead of 1/2. — Preceding unsigned comment added by 2A00:23C5:7E07:7501:14B1:E13A:8B04:968D (talk) 09:56, 24 September 2020 (UTC)

And also there's a simpler proof to just cut the Torus into a cylinder. Base area of the cylinder is (pi)r² and height is (2)(pi)(R). Multiply them to get the volume of the torus and we get 2(pi²)(r²)(R). Same with surface area by finding the lateral surface area of the cylinder which is (2)(pi)(R) • (2)(pi)(r)= (4)(pi²)(R)(r). 223.184.77.204 (talk) 14:22, 26 July 2021 (UTC)

That formula is exactly what Pappus's theorem says the volume of the solid torus must be, just as for any solid of revolution: The area of a cross-section times the distance that its centroid travels as the cross-section revolves to sweep out the solid. — Preceding unsigned comment added by 2601:200:C082:2EA0:D90D:F15A:D939:72E8 (talk) 04:14, 11 November 2023 (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?

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)

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.

Quotients of T^3?

There are eleven(??) quotient spaces of $T^{3}$, six of which are orientable. These are the dicosm $T^{3}/\mathbb {Z} _{2}$, the tricosm $T^{3}/\mathbb {Z} _{3}$, the tetracosm $T^{3}/\mathbb {Z} _{4}$, the hexacosm $T^{3}/\mathbb {Z} _{6}$, and the didicosm aka Hantzsche-Wendt space $T^{3}/\mathbb {Z} _{2}\times \mathbb {Z} _{2}$. Notice all the red links! What can be said about these?? The orientable ones, listed above, have interiors. What's the shape of the interiors? What can be said about the non-orientable ones? (Hint for constructing these: take a cube, glue opposing faces together, but first twist the faces by 90 degrees before gluing them. The non-orientable ones go the same way, but you glue like a Klein bottle).

What about quotient spaces for higher $T^{n}$? What's the pattern for the ability to form a quotient? How can I count them? Enquiring minds want to know ... 67.198.37.16 (talk) 07:07, 20 December 2018 (UTC)

On the classification of finite subgroups of special orthogonal group

A cplete 197.239.5.234 (talk) 07:15, 17 January 2023 (UTC)

toric lenses

I agree with the removal of this unclear mention of toric lenses, "Eyeglass lenses that combine spherical and cylindrical correction are toric lenses.". I agree with that not so much because it's unsourced, but because it doesn't belong with the list of objects whose shapes approximate a torus. In that context it would be important to explain that it's only a cap of a solid torus, not the whole thing.

The list of objects with that shape is to help people understand the concept, whereas the example of lenses is an application of the specific geometry. it would be cool to have a section titled applications that would include that with maybe just a little bit more explanation, but it would be odd to have a whole section titled applications with only one application in it. Maybe we should collect additional applications. There are applications of toroids in inductors and transformers, but there's a separate article on toroids. Tokamaks are also toroids. Inner tubes for typical motor vehicles don't count because once they're in their real application, they are deformed into a different shape, but some bicycle tires are actually toruses. O-rings are the only other example I can think of, and although they do get deformed in application, I think the initial torus shape is specifically intended whereas in an inner tube it's more of a general approximation of the shape needed.

Would it make sense to have an application section with o-rings, bicycle tires and toric lenses?

The toric lenses article is missing sources, but that's a different issue. Ccrrccrr (talk) 11:35, 30 April 2023 (UTC)

I believe that it would be mainly confusing to mention toric lenses to anyone without a strong background in optics. — Preceding unsigned comment added by 2601:200:C082:2EA0:8912:BC6C:2C06:BADF (talk) 23:33, 9 November 2023 (UTC)

Conformal classification

I have added a subsection to the section Flat torus describing the conformal classification of tori.

Any feedback on that section — which is less elementary than the rest of the article — will be welcomed.

Poincaré

I respectfully question the relevance here of the Poincaré conjecture. It is no surprise that a 2-manifold does not meet the criteria of a conjecture about 3-manifolds. I get that it's making a point about simply connected manifolds vs others, but it would be more natural to contrast with a 2-sphere. —Tamfang (talk) 05:51, 27 December 2023 (UTC)

I agree; this section should be removed. The torus is not a helpful non-example of the Poincaré conjecture, and so discussing the Poincaré conjecture does not give more understanding of the torus, which is of course the purpose of this page. 207.237.230.170 (talk) 21:42, 25 February 2024 (UTC)

Hypertoruses (4D geometrical toruses)

I'd be interested in seeing the Hypertorus article expanded to describe the several 4D generalizations of the usual 3D donut shape:

• The spheritorus [1] occurs when taking a circle and 'inflating' it to give it some thickness in 4D. I.e. taking a spherinder and bending it around a donut.
• The torisphere [2] occurs when taking an ordinary sphere surface and inflating it. I.e. taking an uncapped spherinder and bending it inwards.
• The ditorus [3] and tiger [4] which are weirder yet.

I am leaning towards a single article, but perhaps they could be separate articles like duocylinder and spherinder are.

Note these are distinct from the topological hypertorus that is currently described in the article. --Nanite (talk) 16:46, 28 March 2024 (UTC)