Klein Bottles

There exist mirror symmetric Klein Bottles and also left- or right-twisting ones.
They cannot be deformed into each other.
The symmetric Klein Bottle is made by starting from a loop with parallel touching ends. Around this loop there is a tube of varying radius, as in the animation. One family of parameter lines are circles.
The symmetry plane cuts the symmetric Klein Bottle into a pair of mirror symmetric Moebius Strips.
moebius to kleinbottle 012
Increasing the width of Moebius Strip to become a twisting Klein Bottle.
kleinbottle left right 001
Twisting a left-handed klein bottle into a figure-8 torus, then into a right-handed klein bottle.
kleinbottle 8 move 001
Cross-section of the figure-8 Klein Bottle.
klein bottle patch ana
twisting klein bottle (anaglyph)
klein bottle wire ana
twisting klein bottle (anaglyph, wireframe)
                                   Klein Bottle

                               Hermann Karcher

    Parametric Equations for the Twisting Klein Bottle

 x = (aa + cos(v / 2) * sin(u) - sin(v / 2) * sin(2 * u)) * cos(v)
 y = (aa + cos(v / 2) * sin(u) - sin(v / 2) * sin(2 * u)) * sin(v)
 z = sin(v / 2) * sin(u) + cos(v / 2) * sin(2 * u)

See the Mobius Strip first.
The non-orientable surfaces are "one-sided",
and this concept can best be understood if one starts from a Mobius Strip.

Imagine that we modify a Torus by rotating a figure-eight instead of a
circle. If we color the two sides of this surface differently then one
loop of the figure-eight has one color, the other loop the other color.
(One can make such a surface in the Surface Category, by taking the
Lemniscate as the meridian curve after choosing "User Defined (Rotation)"
from the Surface menu.  (This is in fact is the default meridian. )

The default Klein Bottle is obtained by one further modification:
Rotate the meridian figure-eight in its plane by 180 degrees as the
curve is being rotated 360 degrees about the axis of rotation. One can
see the figure-eight better if in "Set u,v ranges" one sets  vmin = 0.5.
Use ``Distinguish Sides by Color'' in this cut open view. One can see this
in a morph with

 aa= 3, umin=0,  umax=2 * π, vmin=0,  0.5 < umax < 2 * π.

The default morph starts from the Mobius Strip

  -0.4 < u < 0.4,  0 < v < 2 * π,  aa = 3

and increases the width of the Mobius Strip until it closes
to the Klein Bottle at   -  π <  u  <  π.

There are in fact three different kinds of Klein Bottles which
cannot be deformed into each other:
   (i) The present one, where the figure-eight rotates to the left in its plane;
   (ii) The mirror image of the present one, where the figure-eight rotates
   (iii) a Klein Bottle with mirror symmetry, glass models are frequently made.

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Moebius_Strip and Klein_Bottle.pdf