Klein bottle explicit parametrization
Bruce Bowen
> have anyone tried to draw KLEIN BOTTLE on his/her display? I'll be greatly
> pleased if you send me its parametrization. Please, send me mail directly:
> NIKI...@mx.ihep.su Igor Nikitin.
One parametrization is a "figure 8 torus with a half twist".
Although it does not "look" like the traditional rendering, it is
topologically the same. In 4 dimensions you can twiddle with the
parameters so that the surfaces miss each other at the "intersection"
region of the "8".
To get a parametrization, take any standard 2-D figure 8 curve, such
as a lemniscate. Starting in the x-z plane, rotate it around the
z-axis, getting a torus, but rotate your lemniscate theta/2, where
theta is the angle of rotation about the z-axis. The figure 8's will
match up when they come back together, but one will be rotated 180
degrees with respect to the other.
Another way to view it is as a mobius strip with the top and bottom
halves joined.
If you're really into warping your brain, try viewing the real
projective plane. This is a hemisphere with diametrically opposed
points on the equator identified. You can view this as a sort of
figure 8 bowl (the rim of the bowl looks like a figure 8), with the
two lobes of the 8 folded up to touch each other. In 4 dimensions you
can again twiddle with the parameters to avoid self intersection.
-Bruce bbo...@megatest.com