The Klein quartic $\chi$ is given by the set of solutions to the homogeneous equation $$x^3y + y^3z + z^3x=0$$ in $\mathbb{C}P^2$. This has 168 orientation preserving automorphisms, including copies of the 12 element tetrahedral symmetry group.

**Is there a nice way to take the points of $\chi$ in $\mathbb{C}P^2$, map them to $\mathbb{R}^3$ (preserving a tetrahedral symmetry group), and produce an embedded genus three surface?**

There are a number of models of the Klein quartic in $\mathbb{R}^3$ out there, for example by Joe Christy and Greg Egan (see [this webpage](http://www.math.ucr.edu/home/baez/klein.html) by John Baez) and [this 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, not mapping from $\chi\subset \mathbb{C}P^2$ directly in some sensible way.