Family Resemblance

This is an isometric projection of one fundamental domain of the Scherk surface and its associate family. This is obviously not a projection that reveals much of the geometry of the surface, but I like how it visually recalls string art parabolas.

Here's the code for the corresponding Manipulate object:

scherk[u_, v_, \[Theta]_] := Re[E^(I \[Theta]) {ArcTanh[z], I ArcTan[z], I ArcTanh[z^2]} /. z -> u + I v] Manipulate[Module[{a, projector}, a = .99; projector = {{1, 0, 0}, {0, 1, 0}}; Show[ParametricPlot[ Table[projector.scherk[u, v, \[Theta]], {u, -a, a, a/10}], {v, -a, a}, PlotRange -> 3.5, Axes -> False, PlotStyle -> Directive[Thickness[.0045], RGBColor["#fbfbfb"]]], ParametricPlot[ Table[projector.scherk[u, v, \[Theta]], {v, -a, a, a/10}], {u, -a, a}, PlotStyle -> Directive[Thickness[.0045], RGBColor["#fbfbfb"]]], ImageSize -> 540, Background -> RGBColor["#08327d"]]], {\[Theta], 0, \[Pi]}]

I originally wrote this as a single ParametricPlot using projector.# & /@ {scherk[u, v, \[Theta]], scherk[v, u, \[Theta]]} , but breaking the mesh drawing code into two separate ParametricPlot s ended up being considerably faster. (Also, note that I've projected down to 2D, which generally seems to lead to higher image quality and faster processing.)

Here's a more conventional projection of the same surface:

And here's the code for this view:

Manipulate[Module[{a, projector}, a = .99; projector = {{1, 0, 0}, {0, 1, 0}}; Show[ParametricPlot3D[ Table[scherk[u, v, \[Theta]], {u, -a, a, a/10}], {v, -a, a}, PlotRange -> 3.5, Axes -> None, Boxed -> False, ViewAngle -> \[Pi]/12, PlotStyle -> Directive[Thickness[.0045], RGBColor["#fbfbfb"]]], ParametricPlot3D[ Table[scherk[u, v, \[Theta]], {v, -a, a, a/10}], {u, -a, a}, PlotStyle -> Directive[Thickness[.0045], RGBColor["#fbfbfb"]]], ImageSize -> 540, Background -> RGBColor["#08327d"]]], {\[Theta], 0, 2 \[Pi]}]