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
- Spectral Analysis of Virus Spreading in Random Geometric Graphs
- Unconditional Proof of the Boltzmann-Sinai Ergodic Hypothesis
- The cover time of random geometric graphs
- Dimers and amoebae
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.)
Best Answer
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 $\mathbb{R}^3$