Inspired by a [discussion][1] with [Neil Strickland][2] I am very interested to know examples of the following type. 

A compact Riemann surface can be presented in many different ways.  For example:  

 1. A smoothly embedded embedded surface in the three-sphere $S^3$. 
 2. A smooth projective curve (say cut out of $\mathbb{CP}^2$ by a single equation).
 3. A quotient of the hyperbolic plane by some fuchsian group. 

In each of these we can accept some minor modifications. In (1) we accept embeddings into three-space $\mathbb{R}^3$ or the three-torus $\mathbb{T}^3$. In (3) we accept quotients of $\mathbb{C}$ by a lattice $\mathbb{Z} + \mathbb{Z}\omega$.  We also accept tilings of the upper half-plane by a tiling without "moduli" (from which the fuchsian group can be deduced, with sufficient amount of hyperbolic trig).  (We could also replace (3) in various other ways - for example by square-tiled surfaces or more generally by gluing *explicitly* described polygons in $S^2$, $\mathbb{E}^2$, or $\mathbb{H}^2$.)
I am not algebraic enough to deform the condition given in (2) - I hope some reader will suggest the correct modifications.

> Question: Give *explicit* examples of pairs (or triples) of
> isomorphic Riemann surfaces of the above types.

As a motivating example, we have tori. For (3), we specify $\omega$ and thus the lattice $\mathbb{Z} + \mathbb{Z}\omega$.  For (2), we have the Eisenstein series giving the modular invariants.  For (1) we have the *Hopf tori* embedded in $S^3$ (and thus, after stereographic projection, embedded in three-space).  Pinkall [[Inventiones, 1985][3]] defines these, and has explicit constructions (see Figure 4b (reproduced below) and Figure 6b).  See also Figure 5 of Sullivan [[Bridges, 2011][4]].  I give further examples in the comments [here][1].

[![Hopf torus taken from Figure 4b of Pinkall's 1985 paper.][5]][5]

  [1]: https://mathoverflow.net/a/421359/1650
  [2]: https://mathoverflow.net/users/10366/neil-strickland
  [3]: https://link.springer.com/article/10.1007/BF01389060
  [4]: https://archive.bridgesmathart.org/2011/bridges2011-593.html
  [5]: https://i.sstatic.net/kLcLl.png