# 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." –  Qiaochu Yuan Nov 4 '10 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. –  Ryan Budney Nov 4 '10 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? –  LarsH Nov 4 '10 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 $\mathbb{R}^3$. –  Willie Wong Nov 4 '10 at 18:13
What Willie Wong said. Take a unit square, and identify opposite sides with each other. –  TonyK Nov 4 '10 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 $\mathbb{R}^3$