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Sunday, April 12, 2015

Math Gems: the Breather

Previous Math Gem: the Klein Bottle
Next Math Gem: the Snail Shell

Gem

This image was rendered with Blender 2.74


For the gold shader used in this scene I want to thank E. M. Malo and his Gold Shader

Math

The Breather is a surface parameterized in 3 dismensions by (see e.g. Wikipedia):

 The following Mathematica script produces output in the PLY 3d-format which can be readily imported into Blender for rendering/postprocessing:

r := 1 - b^2;
w := Sqrt[r];
denom := b ((w Cosh[b u])^2 + (b Sin[v w])^2);
x = (2 r Sinh[b u] Cosh[b u])/denom - u
y = (2 w Cosh[b u] (-(Sin[v] Sin[v w])
     - w Cos[v] Cos[v w]))/denom
z = (2 w Cosh[b u] (Cos[v] Sin[v w]
     - w Sin[v] Cos[v w]))/denom
breather = {x, y, z}
thePlot = ParametricPlot3D[
  Evaluate[breather /. b -> 0.4],
  {u, -13.2, 13.2}, {v, -37.4, 37.4},
  PlotRange -> All, PlotPoints -> {60, 150},
  Axes -> None,   Boxed -> False, PlotPoints -> 40,
  Mesh -> None, NormalsFunction -> None]

Export["Breather.ply", thePlot, "VertexNormals" -> Automatic]