The Klein bottle can be formed from two Moebius bands twisted in opposite directions and joined at their edge.
[Note that the edge of the Klein bottle halves (curve B below)
can be traced in a single, closed loop.]
[Please see the
physical models of the Klein bottle and its two halves, at the
bottom of this page.]
[Image by:
Konrad Polthier, Technische Universitaet Berlin]
Here is the Mathematica code which generated this Klein bottle:
In[1]:=
(* File: klein_PC.m Klein bottle of Paul Chang, paul@math.ucla.edu *)
(* << Graphics/ParametricPlot3D.m *)
bot={(2.5+1.5 Cos[v]) Cos[u], (2.5+1.5 Cos[v]) Sin[u], -2.5 Sin[v]};
mid={(2.5+1.5 Cos[v]) Cos[u], (2.5+1.5 Cos[v]) Sin[u], 3v};
han={2-2 Cos[v]+Sin[u], Cos[u], 3v};
top={2+(2+Cos[u]) Cos[v], Sin[u], 3Pi + (2+Cos[u]) Sin[v]};
bottom=ParametricPlot3D[bot,{u,0,2Pi},{v,0,Pi},PlotPoints->{32, 16}];
middle=ParametricPlot3D[mid,{u,0,2Pi},{v,0,Pi},PlotPoints->{32, 16}];
topper=ParametricPlot3D[top,{u,0,2Pi},{v,0,Pi},PlotPoints->{32, 16}];
handle=ParametricPlot3D[han,{u,0,2Pi},{v,0,Pi},PlotPoints->{32, 16}];
all=Show[handle,topper,middle,bottom]
Display["klein_pc.ps",all]
Converted by Mathematica (January 13, 2003)
The Klein bottle and two halves [congruent to Moebius bands twisted in opposite directions] manufactured via Stereolithography, material: DSM SOMOS 8120 photopolymer. [Image by Stewart Dickson, Rapid Prototyping was done on a 3D Systems SLA-3500 Stereolithography Apparatus by the Rapid Prototyping and Manufacturing Institute Georgia Institute of Technology, Andrew Layton, Program Manager.