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.

- Mathematica
- ThreeScript [Mathematica Standard Package]
- fromThreeScript [Program written by Stewart Dickson]
- thicken [Program written by Stewart Dickson]
- Maya [by Alias|Wavefront]
- tostl, towaveobj [Program written by Stewart Dickson]
- fromobj [Program written by Stewart Dickson ]