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The Klein quartic $\mathcal{Q}$ is cut out of $\mathbb{CP}^2$ 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 $\mathcal{Q}$ in $\mathbb{CP}^2$, 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 by John Baez) and also a version by Carlo Sequin. As far as we (Saul Schleimer and I) can tell, these are all "topological" models and not obtained by mapping from $\mathcal{Q} \subset \mathbb{CP}^2$ in some sensible way.

The Klein quartic $\mathcal{Q}$ is cut out of $\mathbb{CP}^2$ 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 $\mathcal{Q}$ in $\mathbb{CP}^2$, 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 by John Baez) and also a version by Carlo Sequin. As far as we (Saul Schleimer and I) can tell, these are all "topological" models and not obtained by mapping from $\mathcal{Q} \subset \mathbb{CP}^2$ in some sensible way.

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 by John Baez) and also a version 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.

The Klein quartic $\mathcal{Q}$ is cut out of $\mathbb{CP}^2$ 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 $\mathcal{Q}$ in $\mathbb{CP}^2$, 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 by John Baez) and also a version by Carlo Sequin. As far as we (Saul Schleimer and I) can tell, these are all "topological" models and not obtained by mapping from $\mathcal{Q} \subset \mathbb{CP}^2$ in some sensible way.

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 by John Baez) and this version 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.

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 by John Baez) and also a version 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.