Recently I was talking to an alien who does not know complex function theory. I was trying to convince her that the set of conformal equivalence classes of smooth embedded tori in $R^3$ is two parametric. Is there a nice two-parametric family of tori in $R^3$ which are pairwise conformally non-equivalent? 


I do not have an exact definition of ``nice''. Maximally symmetric, canonical in some sense. Or such that it is intuitively clear that they are all in different conformal classes. The standard model (a circle rotated around a line) gives only
one-parametric family up to conformal equivalence.


EDIT. I understand that all conformal tori EXIST in $R^3$. But is it possible to
show (describe, draw, make a 3-d model) a torus in $R^3$ which is not conformally equivalent to a ``rectangular'' torus ? And this can be proved just by looking at its shape in $R^3$.



Tori in three-space
It is a 1961 result by Adriano Garsia that *every* conformal class can be represented by an embedded surface. His proof seems to be reasonably constructive, see [this question](https://mathoverflow.net/questions/53999/conformal-embedding-of-riemann-surfaces-into-3-space) (and answers thereto).


**EDIT** Pinkall, in [the paper cited by j.c. in his comme](http://www3.math.tu-berlin.de/geometrie/Lehre/SS12/MathVisII/Pinkall%20-%201985%20-%20Hopf%20tori%20in%20S3.pdf)nt, explicitly solves the question for the case of genus 1 (as asked by the OP).


