23
Mar
05

Costa’s Minimal Surface

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]}]]

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2 Responses to “Costa’s Minimal Surface”


  1. April 29, 2009 at 1:08 am

    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.

  2. 2 Alex
    September 29, 2009 at 2:13 pm

    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!


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