26
Jun
06

Boy’s Surface

Boy’s Surface (Bryant-Kusner Parametrization) : This one-sided surface was first parametrized correctly by Bernard Morin. The animation looks like it’s turning inside-out, although technically that’s impossible because it only has one side! Robert Bryant told me that the parameters (p,q) = (0,1) give this Willmore immersion of RP2 a trilateral symmetry. The parameters (p,q) = (1,0) should give bilateral symmetry. I’d like to know if it’s possible to make one with pentalateral symmetry. Click here to download some POV-Ray code for this image.

new version: POV-Ray 3.6.1, 6/20/06
old version: Mathematica 4.2, MathGL3d, POV-Ray 3.6.1, 5/24/05

```(* runtime: 1 second *) ParametricPlot3D[Module[{z = r E^(I theta), a, m}, a = z^6 + Sqrt[5]z^3 - 1; m = {Im[z(z^4 - 1)/a], Re[z(z^4 + 1)/a], Im[(2/3) (z^6 + 1)/a] + 0.5}; Append[m/(m.m), SurfaceColor[Hue[r]]]], {r, 0, 1}, {theta, -Pi, Pi}, PlotPoints -> {20, 72}, ViewPoint -> {0, 0, 1}]```

POV-Ray also has an internal function for a different parametrization:
```// runtime: 50 seconds camera{location -1.5*z look_at 0} light_source{-z,1} #declare f=function{internal(8)} isosurface{function{-f(x,y,z,1e-4,1)} pigment{rgb 1}} ```

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