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

My advisor told me a story about Costa today. It seems that she knew him while they were in grad school. She said while all the other math grad students would have fun on the beautiful beaches of brazil (I believe Rio) after their classes and Costa was always in the library. The story was put in a joking manner but it concluded that it takes hard work/effort to advance maths at that level.

Back in 94, He was my teacher at UFF(University in Niteroi). He was a coll guy kind “hippie” at the time, long hair, jeans, sandals. They say He resolved the equation in he’s dreams!