Image depicting a klein bottle cut into two halves></A> <p> <font size= [Image by Stewart Dickson] Research sponsored by Computer Science and Mathematics Division of Oak Ridge National Laboratory, managed by UT-Battelle, LLC for the U.S. DOE under Contract No. DE-AC05-00OR22725

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.]

Symbolic topology of the Torus vs. the Klein Bottle Half Klein Bottle with Surface Normal Vectors
[Image by: Konrad Polthier, Technische Universitaet Berlin]
Symbolic topology of the Right- vs. Left-Handed Moebius Band Symbolic Assembly of the Moebius bands into the Klein Bottle

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]

[Graphics:Klein_1.gif]

Converted by Mathematica  (January 13, 2003)

Physical klein bottle and two halves

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.

Sofware used to create these objects: