# what size is a "unit torus"?

Wikipedia articles on "unit sphere" and "unit circle" say the radius is 1. Articles on the "unit square" and "unit cube" say the length of the side is 1. Would you expect a unit torus to have major radius 1 or major diameter 1?

Admittedly, a torus is a different animal than a sphere, but... It feels most natural to me that the "unit" length should apply to the (major) radius, not the major diameter. Yet I recently came across open source code where someone generated a "unit torus" of major diameter 1.

Is that "wrong enough" that I should change it (in a package of related changes that I'm already preparing to submit)? Can you give me a more solid mathematical basis for advocating that change? Or should I accept the status quo as just a different but legitimate interpretation of "unit torus"?

Edit:

I see from search hits like the following

that the term "unit torus" is used in some fields, like dynamical systems and discrete algorithms. But I'm unable to tell from these papers or abstracts what the authors mean exactly by "unit torus". Dimers and amoebae actually gives this definition:

the unit torus T2 = {(z,w) ∈ C2 : |z| = |w| = 1}

This definition appears to give a definite size. But if it's in the two-dimensional vector space over the complex numbers, I don't know how to apply it to $\mathbb{R}^3$.

If "unit torus" (in $\mathbb{R}^3$) actually means something that does not have any particular size, then that would be important to know.

My question is really not one of programming, but of what this term means in mathematics... including, to what degree is it actually defined (or not) in math? I will base my software decisions on that information.

(Would tag this "torus" if I could create the tag.)

• I would never use the phrase "unit torus." Nov 4, 2010 at 17:23
• It depends on what you want "unit" to mean. I've never seen "unit" used with "torus" before, nor do I see a need for a standardized terminology. I sometimes need to refer to a "square torus", for example. Nov 4, 2010 at 17:24
• @Qiaochu, why not? What would you do in this case: remove the word "unit" and say "of major diameter 1"? @Ryan, one could argue the need for a standard terminology because this code only explained what it was generating by means of the phrase "unit torus". Without a standard, that phrase is ambiguous. Maybe you're saying the programmer should have specified the size independently of this phrase? Nov 4, 2010 at 17:29
• +1 @Ryan. If someone were to tell me "unit torus" I would probably think $\mathbb{R}^d / \mathbb{Z}^d$ with induced flat metric. Unfortunately, the 2d version of this type of torus is not isometrically embedded in $\mathbb{R}^3$. Nov 4, 2010 at 18:13
• What Willie Wong said. Take a unit square, and identify opposite sides with each other. Nov 4, 2010 at 19:16

My best guess is that the unit $d$-torus is the quotient $\mathbb{R}^d/\mathbb{Z}^d$, endowed with some combination of the following additional structures depending on the field of mathematics in consideration:

None of these structures depend on an embedding into $\mathbb{R}^n$. The naming is by analogy with the case $d = 1$, where one gets a description of the usual topological group structure on the circle but, again, without a preferred embedding into $\mathbb{R}^n$. None of these structures suggest a good definition of "unit torus" in $\mathbb{R}^3$. In particular, as Willie notes in the comments, $\mathbb{R}^2/\mathbb{Z}^2$ with the flat metric can't even isometrically embed into $\mathbb{R}^3$

• How does a Riemannian metric not have anything to do with size?
– HJRW
Nov 4, 2010 at 20:21
• Thanks for putting time and thought into this. I guess my conclusion then, as far as the code is concerned, is to say that "unit torus" doesn't have a standard meaning that's clearly applicable to 3D graphics, so we should not use that term in the code as if it unambiguously defined a particular size of torus. We should instead explicitly specify the size, and probably avoid the term. Nov 4, 2010 at 20:22
• @Qiaochu, did you see the definition, which I added to my question, from Dimers and Amoebae - where "the unit torus T<sub>2</sub> = {(z,w) ∈ C<sub>2</sub> : |z| = |w| = 1}"? Does that fit into one of your three bullet points? Nov 4, 2010 at 20:28
• @Henry: ah, right - wasn't thinking straight there. @LarsH: this is a complex torus, and depending on the intentions of the paper it comes with the structure of a topological group as well as that of a complex manifold (en.wikipedia.org/wiki/Complex_manifold). Nov 4, 2010 at 20:34
• But the complex structure on the torus induces a conformal structure, which coincides with the conformal structure that you get on the usual 'square' Euclidean torus.
– HJRW
Nov 4, 2010 at 21:22

The unit torus (in these cases) refers to a torus of major radius $R$ and minor radius $r$ of surface area $4\pi^2 R r=1$. As such it is the "unit square" with periodic boundary conditions.

In random geometric graphs, the boundary of the domain plays an important role in much graph-theoretic behaviour. Sometimes it is useful to analyse graphs not inside the unit square, but rather on the surface of a torus with unit surface area (remembering that the torus is constructed by sewing the parallel edges of a square together). This removes boundary effects on the random geometric graphs.

Below is the unit square with two small obstacle-like regions removed from the domain. We might ask how the obstacles effect the connectivity of the graphs, but this can be difficult when there are "outer" boundary effects that obscure the effects of the obstacles.

We can solve this problem by working on the surface of a torus, since the only impact on the connectivity is (to some extent) the obstacles.

• Interesting answer ... thanks. Jul 18, 2015 at 21:55