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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.)

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    I would never use the phrase "unit torus."2010-11-04
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    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.2010-11-04
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    @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?2010-11-04
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    If an open source programmer uses a phrase, that does not mean mathematicians need to accept it as a standard, nor even to define the term. To me this is an issue between you and your software, not for mathematics.2010-11-04
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    @LarsH: in pure mathematics, most people who study tori don't care what size they are.2010-11-04
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    +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 $\mathbb{R}^3$.2010-11-04
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    If this is a poll: when you say unit torus I think of a torus with area equal to one.2010-11-04
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    What Willie Wong said. Take a unit square, and identify opposite sides with each other.2010-11-04
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    @Sam: why? a unit circle doesn't have area one, nor does a unit sphere.2010-11-04
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    @Qiaochu: when I google "unit torus" I find it in papers like http://en.scientificcommons.org/29544141 and http://portal.acm.org/citation.cfm?id=1496770.1496776. I'm not sure I buy the idea that "most people who study tori don't care what size they are", given that terms like unit interval, unit circle, etc. are well-used even we limit ourselves to pure math. Think Cantor Set or Fatou sets.2010-11-04
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    @LarsH: in both of those cases the torus is not being regarded as embedded in R^3. The unit interval and unit circle are special (the former because its natural measure is a probability measure and the latter because it is naturally the image of the exponential function) and I have trouble buying a corresponding definition of "unit torus" which isn't what Willie suggested.2010-11-04
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    @Ryan, why should mathematics not inform the "issue between me and (someone's) software"? In my view, software that involves math should strive to be as consistent with math terms and conventions as is possible. Hence the question about what math terms mean. Clearly the phrase is used in the math world, but maybe not widely enough for people on this site to be familiar with what it means.2010-11-04
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    @TonyK, what size of torus does that give me?2010-11-04
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    @Qiaochu, it sounds like they're embedded in 'n' or 'd' dimensions, which could be 3. Does that mean they don't have a defined radius? If not, what does "unit torus" mean here as opposed to just "torus"? As for @Willie's answer, it sounds like you think I was responding to it, but in fact it went well over my head.2010-11-04
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    @All, I edited the question to clarify what I'm asking.2010-11-04
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    @Willie, can you explain to me what "$\mathbb{R}^d / \mathbb{Z}^d$ with induced flat metric" is? I understand $\mathbb{R}^d$ and $\mathbb{Z}^d$, but not the `/` operator nor a flat metric.2010-11-04
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    "unit torus" is a non-standard term in mathematics, so there's no common definition to appeal to. Your example torus in $S^3$ I would call a "square torus". IMO "unit torus" is far too ambiguous to be a term worthy of standardization.2010-11-04
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    The reason why "unit torus" sucks as terminology is that it implies that only one parameter is required to specify its shape/size, when in fact you have two radii to contend with. There's also now the question of whether you intended a "ring torus", a "horn torus", or a "spindle torus", to use *the* standard terminology.2010-11-04
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    @J.M. - Yeah, I felt that way too (2 parameters, but "unit" only seems to specify one). Regarding the latter, that is a valid point, but according to Wikipedia "In most contexts it is assumed that the axis does not touch the circle (in this case the surface has a ring shape and is called a ring torus or simply torus if the ring shape is implicit)."2010-11-05

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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 $\mathbb{R}^3$

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    How does a Riemannian metric not have anything to do with size?2010-11-04
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    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.2010-11-04
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    @Qiaochu, did you see the definition, which I added to my question, from *Dimers and Amoebae* - where "the unit torus T2 = {(z,w) ∈ C2 : |z| = |w| = 1}"? Does that fit into one of your three bullet points?2010-11-04
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    @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 (http://en.wikipedia.org/wiki/Complex_manifold).2010-11-04
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    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.2010-11-04
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    @Henry, I'm not sure what all that means... doesn't sound like it means we can infer a size for a corresponding unit torus in $\mathbb{R}^3$?2010-11-05
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    @LarsH: unfortunately, no. A conformal structure lets you measure angles, not lengths.2010-11-05
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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.

rgg inside a corridor, courtesy A.P. Giles

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.

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    why the downvote?2015-07-18
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    Interesting answer ... thanks.2015-07-18