Gridded
The associate family of Scherk's surfaces.
The code below implicitly uses the Weierstrass–Enneper parametrization $f(z) = \frac{4}{1-z^2}$, $g(z) = iz$, since
$\int_0^\zeta \frac{f(z)(1-g(z)^2)}{2} dz = 2 \operatorname{arctanh} \zeta$, $\int_0^\zeta i \frac{f(z)(1+g(z)^2)}{2} dz = 2 i \arctan \zeta$, and $\int_0^\zeta f(z) g(z) dz = 2 i \operatorname{arctanh}(\zeta^2)$.
This is too slow to give a useful Manipulate, so I just give the GIF-generating code:
scherk2[u_, v_, ?_] :=
Re[E^(I ?) {ArcTanh[z], I ArcTan[z], I ArcTanh[z^2]} /. z -> u + I v]
scherkfamily = Module[{a = .9, cols},
cols = RGBColor /@ {"#07588A", "#E4FCF9"};
ParallelTable[
Show[
ParametricPlot[
Table[scherk2[u, v, ?][[2 ;;]], {v, -a, a, a/7}],
{u, -a, a}, PlotRange -> 2, Axes -> None,
Background -> Last[cols], ImageSize -> 540,
PlotStyle -> cols[[1]]],
ParametricPlot[
Table[scherk2[u, v, ?][[2 ;;]], {u, -a, a, a/7}],
{v, -a, a}, PlotStyle -> cols[[1]]]], {?, 0., 2 ? - #, #}] &[2 ?/250]
];
Export[NotebookDirectory[] <> "scherkfamily.gif", scherkfamily,
"DisplayDurations" -> {1/50}]
