Paul Nylander

Costa’s Minimal Surface is a classic example of a minimal surface with holes in it, also called “handles”. The number of holes is called the genus of the surface. This surface was discovered by a graduate student. I think it would be interesting to see someone create an actual soap film with this shape.

Here is some Mathematica code:
(* runtime: 5 seconds *)
c = 189.07272; e1 = 6.87519;
Costa[u_, v_] := Module[{z =u + I v}, zeta = WeierstrassZeta[z, {c, 0}]; zeta1 = WeierstrassZeta[z - 1/2, {c, 0}]; zeta2 = WeierstrassZeta[z - I/2, {c, 0}]; p = WeierstrassP[z, {c, 0}]; x = Re[Pi (u + Pi/(4 e1) ) - zeta + Pi(zeta1 - zeta2)/(2 e1)]/2; y = Re[Pi (v + Pi/(4 e1)) - I(zeta + Pi(zeta1 - zeta2)/(2 e1))]/2; z = (Sqrt[2 Pi]/4)Log[Abs[(p - e1)/(p + e1)]]; {x, y, z, EdgeForm[]}];
ParametricPlot3D[Costa[u, v], {u, 0.0001, 1}, {v, 0.0001, 1}, PlotPoints -> 40, PlotRange -> {{-3.5, 3.5}, {-3.5, 3.5}, {-2, 2}},Compiled -> False]
Here is another parametrization:
(* runtime: 5 seconds *)
Costa[z_] := Module[{phi1 = -2 Sqrt[z] Sqrt[1 - z^2] Hypergeometric2F1[1/4, 3/2, 5/4, z^2]/Sqrt[z^2 - 1], phi2 = -(2/3) z^(3/2) Sqrt[z^2 - 1] Hypergeometric2F1[3/4, 1/2, 7/4, z^2]/Sqrt[1 - z^2]}, Re[{phi2 - phi1, I(phi1 +phi2), Log[z - 1] - Log[z + 1]}]/2];
surface = ParametricPlot3D[Costa[Sqrt[Exp[r - I theta] + 1]], {r, -3.5, 6}, {theta, -Pi, Pi}, PlotPoints -> {20, 18}, Compiled -> False][[1]];
<< Graphics`Shapes`; surface = {surface, RotateShape[surface, Pi, 0, 0]}; Show[Graphics3D[{surface, RotateShape[surface, Pi/2, Pi, 0]}]]

Links

• Mathematica code and minimal surface art – by Matthias Weber
• Alfred Gray – differential geometry gallery
• soap bubble light interference – by Kei Iwasaki
• Eva Hild – beautiful ceramic sculptures
• minimal surfaces with metal frames
• Helaman Ferguson – math sculptor, see his Costa snow sculpture