Embedding of Klein Bottle in ${\Bbb E}^3$

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 $\{a,b \vert a^2=b^2\}$ and its first homology group is ${\Bbb Z}+ {\Bbb Z}/2.$

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.