# Plotting the Schwarz D minimal surface

With the aid of the following paper I'm trying to plot the Schwarz D minimal surface: paper. So far I have followed all the instructions to the best of my abilities and have come up with the following code:

zeta[u_, v_] := u Exp[I v] - Exp[I Pi/4]

kappa = 2/EllipticE[1/4];
xi = 1/Sqrt[2];

x[u_, v_] :=
kappa Re[1/(2 Sqrt[2]) EllipticF[
ArcSin[2 Sqrt[2] zeta[u, v]/
Sqrt[zeta[u, v]^4 + 4 zeta[u, v]^2 + 1]], 1/4]]

y[u_, v_] :=
kappa Im[1/(2 Sqrt[2]) EllipticF[
ArcSin[-2 Sqrt[2] zeta[u, v]/
Sqrt[zeta[u, v]^4 + 4 zeta[u, v]^2 + 1]], 3/4]]

z[u_, v_] :=
kappa Re[1/4 EllipticF[ArcSin[4 zeta[u, v]^2/(zeta[u, v]^4 + 1)],
97 - 56 Sqrt[3]]]

f1 = ParametricPlot3D[{xi x[u, v] - xi y[u, v],
xi x[u, v] + xi y[u, v], z[u, v]}, {u, 0, Sqrt[2]}, {v, 0, Pi/4},
RegionFunction ->
Function[{x, y, z, u, v},
Re[zeta[u, v]] >= 0 && Im[zeta[u, v]] >= 0],
PerformanceGoal -> "Quality", MaxRecursion -> 10,
AxesLabel -> {x, y, z}, PlotRange -> {{0, 1}, {0, 1}, {0, 1}}];

f2 = ParametricPlot3D[{-xi x[u, v] + xi y[u, v] + 1, z[u, v],
xi x[u, v] + xi y[u, v]}, {u, 0, Sqrt[2]}, {v, 0, Pi/4},
RegionFunction ->
Function[{x, y, z, u, v},
Re[zeta[u, v]] >= 0 && Im[zeta[u, v]] >= 0],
PerformanceGoal -> "Quality", MaxRecursion -> 10,
AxesLabel -> {x, y, z}, PlotRange -> {{0, 1}, {0, 1}, {0, 1}}];


The fundamental patch f1 seems to have the correct shape, so neither the parameterization nor the choice of parameter domain is likely to be the source of my problem. The problem being that the patches f1 and f2, which according to the authors should be identical modulo rotation around an axis of symmetry, don't match up (I've tried the other four as well, and they don't fit together either). Either the authors have made a mistake, or I'm missing something - quite possibly something obvious.

Could someone please give me some pointers to help me sort this out? Many thanks in advance!

• Why did you use zeta[u_, v_] := u Exp[I v] - Exp[I Pi/4]? I can't see this anywhere in the paper. – Enigma123 Jun 18 '18 at 16:36