TITLE: Tori in three-space
QUESTION [11 upvotes]: 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 $R^3$.

REPLY [8 votes]: 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 (and answers thereto).
EDIT Pinkall, in the paper cited by j.c. in his comment, explicitly solves the question for the case of genus 1 (as asked by the OP).