## Background

In my previous post I discussed rather technically how Blender can be used as render backend for Graphics3D output created with Mathematica. Dealing with that matter, I thought it could be nice to create a couple of image of mathematically described 3D surface objects, which I call "Math Gems", because they look like jewelry.## The Klein Bottle

Klein Bottle faceted

Klein Bottle smooth with mesh from edges of surface

## Description in Mathematica

I my previous post I had some problems creating usable Mathematica output for the the parametrized immersion in 3 dimensions, so called "Klein Bottle" based on the equations given in Wikipedia. Finally I found the proper way to create the shape in Mathematica.The Klein bottle can be parametrized in the following way (see e.g. Paul Bourkes article):

In Mathematica formulation:

r = 4 (1 - cos(u)/2)The resulting PLY file can directly imported to Blender. Doing some Blender post processing yields in the images above.

x = Piecewise[({

{r cos(u) cos(v) + 6 (sin(u) + 1) cos(u), 0 <= u < \[Pi]},

{r cos(v + \[Pi]) + 6 (sin(u) + 1) cos(u), \[Pi] <= u <= 2 \[Pi]}

})]

y = Piecewise[({

{r sin(u) cos(v) + 16 sin(u), 0 <= u < \[Pi]},

{16 sin(u), \[Pi] <= u <= 2 \[Pi]}

})]

z = r sin (v)

thePlot =

ParametricPlot3D[{x, y, z}, {u, 0, 2 \[Pi]}, {v, 0, 2 \[Pi]},

Axes -> None, Boxed -> False, PlotPoints -> 50, MaxRecursion -> 10,

Mesh -> None, NormalsFunction -> None]

Export["KleinBottle1.ply", thePlot, "VertexNormals" -> Automatic]