Klein Bottle

In mathematics,
the Klein bottle is a certain non-orientable
surface with
no distinction between the “inside” and “outside” surfaces.
The Klein bottle was first described in 1882
by the German
mathematician Felix
Klein. It is closely related to the Möbius
strip. The initial name given was “Klein Fla-e-che” (Fläche =
Surface). However, this was wrongfully interpreted as Fla-s-che, which
ultimately, due to the dominance of the English language in science, led to the
adoption of this term in the German language, too.

Picture a bottle with a hole in the bottom. Now extend the neck. Curve the
neck back on itself, insert it through the side of the bottle without touching
the surface (an act which is impossible in three-dimensional space), and extend
the neck down inside the bottle until it joins the hole in the bottom. A true
Klein bottle in four dimensions does not intersect itself where it crosses the
side.

Unlike a drinking glass, this object has no “rim” where the surface
stops abruptly. Unlike a balloon, a fly can go from the outside to the inside
without passing through the surface (so there isn’t really an “outside”
and “inside”).

Mathematics

The Figure-8 surface shown on the right has an even simpler parameterization
than the standard Klein bottle.

In this immersion, the self-intersection circle is a geometric circle
in the XY plane. The positive constant r is
the radius of this circle. The parameter u
gives the angle in the XY plane, and v
specifies the position around the 8-shaped cross section.

Topologically, the Klein bottle can be defined as the square
[0,1] x [0,1] with sides identified by the relations (0,y) ~ (1,y)
for 0 ≤ y ≤ 1 and (x,0) ~ (1-x,1)
for 0 ≤ x ≤ 1, as in the diagram on the
left.

Properties

If we dissecting a Klein bottle into halves along its plane of symmetry we
get two mirror image Möbius
strips, the one with a left-handed half-twist and the other with a
right-handed half-twist. In fact, it is also possible to
cut the Klein bottle into a single Möbius strip.

Otherwise, a Klein bottle can be represented as two Möbius bands connected by an ordinary two-sided
band whose back and front sides are colored white and blue respectively as you
can see at the image below.

Klein bottle can be constructed from single cylinder. We loop one
end back through the cylinder and glue it to the other end, with the two
boundary circles given opposite orientations. To accomplish this with a pleasing
shape we adjust the thickness of the cylinder. This allows us to glue inside to
outside, and obtain a one-sided surface. In the diagram below we have used white
and green to distinguish the two sides of the original cylinder. When the Klein
bottle is finished, the colors still show where the cylinder was glued together,
but gluing at any other parallel circle would have been just as good.

The article was created by matherials of Wikipedia
and “Imaging
maths – Inside the Klein bottle” by Konrad Polthier

See also Collection of various kinds of Klein bottles.