The Klein quartic $\mathcal{X}$ is cut out of $\mathbb{C}P^2$ by the homogeneous equation $$x^3 y + y^3 z + z^3 x = 0.$$ It has 168 orientation preserving automorphisms and includes several copies of the tetrahedral group (with twelve elements).

>Is there a nice way to take the points of $\mathcal{X}$ in $\mathbb{C}P^2$, map them to $\mathbb{R}^3$ (preserving one of the tetrahedral symmetry groups) and so produce an embedded, compact, genus three surface?

There are already a number of models of the Klein quartic in $\mathbb{R}^3$.  So far we've found the two by Joe Christy and Greg Egan (see [this webpage](http://www.math.ucr.edu/home/baez/klein.html) by John Baez) and also [a version](http://www.cs.berkeley.edu/~sequin/GEOM/TILES/LizardTetrus1.JPG) by Carlo Sequin. As far as we (Saul Schleimer and I) can tell, these are all "topological" models and not obtained by mapping from $\mathcal{X} \subset \mathbb{C}P^2$ in some sensible way.