# [GIF] The Chase (real part of Klein quartic)

Posted 4 years ago
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| The ChaseThe Klein quartic is the zero locus of the homogeneous polynomial $x^3 y + y^3 z + z^3 x$ inside $\mathbb{CP}^2$. With real eyes, it is a genus-3 surface with the largest possible automorphism group. Echoing Henry Segerman, I would love to find a geometric immersion in $\mathbb{R}^3$ by applying some nice map $\mathbb{CP}^2 \to \mathbb{R}^3$, but in the absence of that, I took the real part, thought of as living in the real projective plane $\mathbb{RP}^2$.Of course, I'm actually taking the curve as sitting in the sphere $S^2$, which double-covers $\mathbb{RP}^2$, so you see two curves (green and pink) which are antipodal images of each other. Then I let the sphere rotate around the $y$-axis and take the stereographic projection to the plane.Here's the code, which required a slightly fancy bit of business with RegionFunction to make the two curves different colors: Stereo[{x_, y_, z_}] := {x/(1 - z), y/(1 - z)}; InverseStereo[{x_, y_}] := {2 x/(1 + x^2 + y^2), 2 y/(1 + x^2 + y^2), (x^2 + y^2 - 1)/(1 + x^2 + y^2)}; Module[{cols}, cols = RGBColor /@ {"#FF358B", "#AEEE00", "#333333"}; ParallelTable[ Show[ContourPlot[ Evaluate[(Quiet[ x[]^3 x[] + x[]^3 x[] + x[]^3 x[]] /. {x -> (RotationMatrix[t, {0, 1, 0}].InverseStereo[{u, v}])}) == 0], {u, -#, #}, {v, -#, #}, PlotPoints -> 20, PlotRange -> #, ContourStyle -> Directive[cols[], Thickness[.015]], Frame -> False, ImageSize -> 500, RegionFunction -> Function[{u, v}, (-2 #[] - 2 #[] + 1 - #[]^2 - #[]^2 &[Stereo[RotationMatrix[t, {0, 1, 0}].InverseStereo[{u, v}]]]) > 0], Background -> cols[]] &, ContourPlot[ Evaluate[(Quiet[ x[]^3 x[] + x[]^3 x[] + x[]^3 x[]] /. {x -> (RotationMatrix[t, {0, 1, 0}].InverseStereo[{u, v}])}) == 0], {u, -#, #}, {v, -#, #}, PlotPoints -> 20, PlotRange -> #, ContourStyle -> Directive[cols[], Thickness[.015]], RegionFunction -> Function[{u, v}, (-2 #[] - 2 #[] + 1 - #[]^2 - #[]^2 &[Stereo[RotationMatrix[t, {0, 1, 0}].InverseStereo[{u, v}]]]) < 0]] &], {t, 0, 2 ?}] ] Answer
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Posted 4 years ago - another post of yours has been selected for the Staff Picks group, congratulations !We are happy to see you at the tops of the "Featured Contributor" board. Thank you for your wonderful contributions, and please keep them coming! Answer
Posted 4 years ago
 @Moderation Team Awesome, thanks so much! Answer