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Embedding of Klein Bottle in

C.T.J. Dodson

The equations below give an embedding of the Klein bottle (originally called a Kleinsche Fläche [surface] in German but mistranslated into English as Kleinsche Flasche [flask or bottle]). This Klein bottle could be made thus: glue two tori (tyre tubes) together one above the other; cut through both, perpendicularly to the circle of contact, twist one side of the cut through 180 degrees and rejoin.

The Klein bottle is an example of a non-orientable closed surface. It has other embeddings in 3-space but all involve self-intersections and in fact it requires 4 dimensions to allow an injective embedding.

The graphic shown here is a LiveGraphic3D applet object which you can rotate using the left mouse button--provided that you have enabled Java in your browser preferences. Try dragging and releasing to spin it.

The fundamental group of the Klein bottle is and its first homology group is

Parametric equations:

```     k1=(2+Cos[u/2]Sin[t]-Sin[u/2]Sin[2t])Cos[u];
k2=(2+Cos[u/2]Sin[t]-Sin[u/2]Sin[2t])Sin[u];
k3=Sin[u/2]Sin[t]+Cos[u/2]Sin[2t];
```
If they are pasted into a Mathematica notebook together with this definition
```kkk[z_]:=ParametricPlot3D[{k1,k2,k3},{t,0,2Pi},{u,0,z},
Boxed->False, Axes->None, PlotPoints->{40,70},
LightSources\[Rule]{{{1.3,-2.4,2.},RGBColor[0.9,1,0]},{{2.7,0.,2.},
RGBColor[0,1,.1]},{{-5.3,-1.4,2.},RGBColor[0,1,1]}}]
```
Then the command kkk[2 Pi]; will produce a graphic like that above.

C.T.J. Dodson
12/9/1999