The Enneper Surface of Genus 4

<<Own`Mesh`

Weierstrass Representation

a=1.81645934660556296`;
b=3.11436011061010598`;
c=3.77509108812262628`;

rho=0.580558059350863508`;

G[z_]:=((rho ((Sqrt[z]) (Sqrt[-(a^2)+(z^2)]) (Sqrt[-(c^2)+(z^2)])))/((Sqrt[-1+(z^2)]) (Sqrt[-(b^2)+(z^2)])));

dh[z_]:=1;

phi1[z_]:=G[z]dh[z];
phi2[z_]:=1/G[z]dh[z];

om1[z_]:=-(1/2)*(phi1[z]-phi2[z]);
om2[z_]:=I/2*(phi1[z]+phi2[z]);
om3[z_]:=dh[z];

f[z_]:=Module[{s},
Re[NIntegrate[{om1[s],om2[s],om3[s]},{s,I,z},PrecisionGoal->5]]]-{-0.855462,0.855462,0.};

n[z_]:=StereographicProjection[G[z]];

Parametrization

eps=10.^-16;

m1=2;
m2=2;
m3=2;
m4=2;
m5=2;
my=3;
rmax=7;

fac=2;

{n1,n2,n3,n4,n5}=fac{m1,m2,m3,m4,m5};
ny=fac my;

dm=PolarDomain[Union[NRange[ϵ,1,n1],NRange[1,a,n2],NRange[a,b,n3],NRange[b,c,n4],NRange[c,rmax,n5],SameTest->AlmostEqual],NRange[ϵ,Pi/2,ny]];

Plotting the Surface

p1=MeshPlot3D[{f[x+I y],n[x+I y]},{x,y},dm];
p2=MeshJoin[p1,MeshRotate[p1,StraightLine[{0,0,0},{1,-1,0}]]];
p3=MeshJoin[p2,MeshReflect[p2,Plane[{0,1,0},0]]];
p4=MeshJoin[p3,MeshReflect[p3,Plane[{1,0,0},0]]];

Show[gr=Mesh3DToGraphics3D[p4],Axes->True,AxesLabel->{x,y,z}]